Social Perspective of Mobility Sharing · From the sharing economy perspective, being a passenger...

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Social Perspective of Mobility Sharing: Understanding, Utilizing, and Shaping Preference by Hongmou Zhang Master of City Planning, University of Pennsylvania (2014) Bachelor of Science in Applied Mathematics, Peking University (2012) Bachelor of Engineering in Urban Planning, Peking University (2012) Submitted to the Department of Urban Studies and Planning in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Urban Science and Planning at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019 © Hongmou Zhang. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Author ................................................................................... Department of Urban Studies and Planning May 21, 2019 Certified by ............................................................................... Jinhua Zhao Edward H. and Joyce Linde Associate Professor Department of Urban Studies and Planning Dissertation Supervisor Accepted by .............................................................................. Lawrence Vale Ford Professor of Urban Design and Planning Chair, Ph.D. Committee Department of Urban Studies and Planning

Transcript of Social Perspective of Mobility Sharing · From the sharing economy perspective, being a passenger...

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Social Perspective of Mobility Sharing:Understanding, Utilizing, and Shaping Preference

by

Hongmou Zhang

Master of City Planning, University of Pennsylvania (2014)Bachelor of Science in Applied Mathematics, Peking University (2012)Bachelor of Engineering in Urban Planning, Peking University (2012)

Submitted to the Department of Urban Studies and Planningin partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Urban Science and Planning

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2019

© Hongmou Zhang. All rights reserved.

The author hereby grants to MIT permission to reproduce and to distribute publicly paperand electronic copies of this thesis document in whole or in part in any medium now known

or hereafter created.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of Urban Studies and Planning

May 21, 2019

Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Jinhua Zhao

Edward H. and Joyce Linde Associate ProfessorDepartment of Urban Studies and Planning

Dissertation Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lawrence Vale

Ford Professor of Urban Design and PlanningChair, Ph.D.Committee

Department of Urban Studies and Planning

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Social Perspective of Mobility Sharing:Understanding, Utilizing, and Shaping Preference

byHongmou Zhang

Submitted to the Department of Urban Studies and Planningon May 21, 2019, in partial fulfillment of the

requirements for the degree ofDoctor of Philosophy in Urban Science and Planning

Abstract

Advances in information and communications technologies are enabling the growth of real-timeride sharing—whereby drivers and passengers or fellow passengers are paired up on car trips withsimilar origin-destinations and proximate time windows—to improve system efficiency by movingmore people in fewer cars. Lesser known, however, are the opportunities of shared mobility as atool to foster and strengthen human interactions.

In this dissertation, I used preference as a lens to investigate the social interaction in mobilitysharing, including how the interpersonal preference in mobility sharing can be understood, utilizedand reshaped. More specifically, I answered the questions of how preference could be used tomatch fellow passengers and to improve trip experiences; how gender, one of the key factors maycontribute to this preference; and in the reverse direction, if there are factors in the preference whichare unrespectable and need to be changed, whether mobility sharing can be used as a tool to changeit, and improve the integration of cities. Besides, I also studied how time flexibility of trips can beincorporated into mobility sharing models to reduce congestion.

For policy makers and planners, this dissertation could partially answer or provide a frameworkof analysis to the following questions. 1) How could preference in mobility sharing services be usedor misused? What is the efficiency trade-off, and how to regulate the use of it? 2) What factors mayimpact the preference for fellow passengers? Are the preference factors respectable, and what factorsshould be included/excluded in the mobility sharing services from a regulation perspective? 3) Howcan mobility sharing be actively used as a tool to encourage more social interaction, especially acrossdifferent social groups? What is the short-term cost, and the long-term benefit?

For the system designers of mobility sharing services, this dissertation can be used as a referencefor the development of a preference-based mobility sharing platform. The following questions havebeen traced, and the methods can be improved when more data are available to the system designers.1) If preference is to be used, what input data are needed, and how they need to be processed for thepreference-matching model? 2) What preference factors should be included in the system design,what factors should be used with caution, and what factors should be eliminated? 3) If timeflexibility of trips can be included in the system design, how much congestion can be reduced?What system design is needed in order to achieve this congestion reduction?

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Dissertation Supervisor:Jinhua ZhaoEdward H. and Joyce Linde Associate Professor of City and Transportation PlanningDepartment of Urban Studies and Planning

Dissertation Committee Members:Paolo SantiResearch ScientistSenseable City Laboratory, Department of Urban Studies and PlanningSenior ResearcherInstitute of Informatics and Telematics, National Research Council, Italy

Justin SteilAssistant Professor of Law and Urban PlanningDepartment of Urban Studies and Planning

Benjamin G. EdelmanEconomistMicrosoft

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Acknowledgments

Throughout my years at MIT, I am indebted to many people for their companion, support, encour-agement, and help.

First, I would like to express my deepest gratitude to my advisor and committee chair, Prof.Jinhua Zhao. Jinhua always greatly inspired me during our discussion of research. He showed mehow research ideas can grow from small seeds into luxuriant and fruitful trees. Moreover, from himI learned what is a great mentor. I feel very honored to have him as my advisor during my PhDstudies at MIT.

I would also like to express my gratitude to my committee members: Dr. Paolo Santi, Dr. BenEdelman, and Prof. Justin Steil. I would like to thank Paolo for bringing me to the Senseable CityLab before I started my PhD at DUSP, instructing me on research in the lab, and showing me howto make models rigorous and neat. I thank Ben for offering insights from a different angle fromplanners and modelers, and always encouraging me to reach out to the real world. I would like tothank Justin for joining the committee and offering advice with great expertise on a very importantpiece of this dissertation.

I would like to extend my gratitude to Prof. Joseph Ferreira, Prof. Ingrid Gould Ellen, Prof. EzraGlenn, for all of whom I worked as teaching assistants at MIT. From them, I did not only learn howto be a good teacher, but also how to communicate with students and help them.

I am thankful to my friends at JTL Urban Mobility Lab. I would like to especially thankZhan Zhao, with whom I discussed and discovered a lot of good research ideas during afternooncoffee time. Thanks to M. Elena Renda for collaborative works and very helpful comments. I alsothank Nate Bailey, Joanna Moody, Peyman Noursalehi, Saeid Saidi for sharing good comments,suggestions, and encouragement during my last stage of dissertation writing.

I would also like to thank my fellow PhD students at DUSP. Thanks to Liyan Xu, AnthonyVanky, and Lyndsey Rolheiser for sharing your experiences as PhDs at DUSP when I was stillnew to here. Thanks to my fellow PhD students: Shenhao Wang, Colleen Chiu-Shee, Andrea Beck,Nicholas Kelly, Elise Harrington, and Shekhar Chandra for always sharing challenges and progressesduring this long journey. Thanks to my colleagues at the Singapore–MIT Alliance for Research andTechnology center, especially Yu Shen, Hui Kong, and Xiaohu Zhang for the help in Singapore.Thanks to my friends outside of DUSP: Andong Liu and Zichen Ma for the friendship, especiallyduring the long winters of Massachusetts.

I am grateful to MIT Presidential Fellowship, the Singapore–MIT Alliance for Research andTechnology Future Mobility program, and MIT Institute of Data, System and Society Seed Fundfor providing financial support for this dissertation.

I am extremely grateful to my parents, Xuelan Niu and Jincheng Zhang. Thank you for fosteringme and providing me with opportunities to pursue my dreams. I am also grateful to my fiancée,Xin Tan’s parents, Zhihong Cao and Dongfeng Tan, for their support and understanding over theyears.

Finally, to you, Xin: thanks for your companion and understanding throughout the years thoughthousands of miles apart. During my hard times, you guided me out of difficulties and helped meovercome challenges. You bring me love, care, and encouragement. This dissertation is dedicatedto you.

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Contents

1 Introduction 151.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 Conceptual Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.1 Procedure-wise framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.2 Substantive framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Data and Approach Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Mobility Sharing as a Preference Matching Problem 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Data and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Synthetic preference orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Irving–Tan algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.1 Efficiency vs. preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Result stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.3 Sensitivity analysis for ∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.4 Two-group scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Gender, Social Interaction, and Mobility Sharing 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Gender and mobility sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Gender difference in social interaction . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Data and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Survey summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.1 Factor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.2 Structural equation modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Matching with Gender Preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5.1 Combining trip data and gender preference . . . . . . . . . . . . . . . . . . . 483.5.2 Matching results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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4 Congestion–Sensitive Mobility Sharing with Time Flexibility 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.1 Time flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.2 Time flexibility and mobility sharing . . . . . . . . . . . . . . . . . . . . . . . 564.2.3 Mobility sharing and traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.4 Time flexibility and traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3.1 Traffic assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3.2 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.3 Collecting travel demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.4 Travel demand and traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.5 Natural constraints about sharing and pairing . . . . . . . . . . . . . . . . . . 614.3.6 Generic time flexibility constraints . . . . . . . . . . . . . . . . . . . . . . . . 624.3.7 Time flexibility constraints for shared trips . . . . . . . . . . . . . . . . . . . 624.3.8 Formulation summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.9 Dependency diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.1 Sioux Falls network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.2 Trip generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.3 Time flexibility generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5.1 TR-decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5.2 Traffic Assignment (T-step) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.3 Linearization of R-step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.4 Pruning infeasible trip pairs and discretizing departure time in the R-steps . . 714.5.5 Making TR-decomposition converge . . . . . . . . . . . . . . . . . . . . . . . 72

4.6 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.7 Result Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7.1 Impact of proportion of shareable trips . . . . . . . . . . . . . . . . . . . . . . 774.7.2 Impact of time flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.7.3 User equilibrium vs. system optimum traffic assignment . . . . . . . . . . . . 79

4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Mobility Sharing as a Tool for Social Integration: A Reverse Schelling Model 835.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.1 The Schelling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2.2 Social contact theory: positive and negative intergroup contact . . . . . . . . 865.2.3 Mobility sharing for social integration . . . . . . . . . . . . . . . . . . . . . . 865.2.4 Mathematical formulation of dynamic systems . . . . . . . . . . . . . . . . . . 87

5.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3.1 Overall structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3.2 Relationship between contact time and preference threshold . . . . . . . . . . 905.3.3 Manhattan distance and maximum detour . . . . . . . . . . . . . . . . . . . . 905.3.4 Shareability network and matching algorithms . . . . . . . . . . . . . . . . . . 915.3.5 Index of dissimilarity and Gini coefficient . . . . . . . . . . . . . . . . . . . . 925.3.6 Unit system and equivalence of parameters . . . . . . . . . . . . . . . . . . . 93

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5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.4.1 Matching algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.4.2 Maximum detour allowed, actual detour, and trip distance . . . . . . . . . . . 955.4.3 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4.4 Group shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4.5 Speed of convergence and the “tipping point” . . . . . . . . . . . . . . . . . . 99

5.5 Counteracting Forces and Adversarial Agents . . . . . . . . . . . . . . . . . . . . . . 1005.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Epilogue and Prospects 1076.1 Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Areas for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2.1 Boundary of respectable preference . . . . . . . . . . . . . . . . . . . . . . . . 1086.2.2 Duality of efficiency and preference . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3 Theoretical and Practical Implications . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A Universe of Preference 113

B Results of Structural Equation Models 115

Bibliography 117

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List of Figures

1-1 Structure of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2-1 Distribution of the origins and destinations of taxi trips in Manhattan and distribu-tion of trip duration without sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2-2 Frequency of the number of shareable trips with a trip in shareability networks with∆t = 100–600 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2-3 Ranking of paired fellow passengers in respective passengers’ preference lists anddegree distribution of the number of shareable trips . . . . . . . . . . . . . . . . . . . 31

2-4 Distribution of increase in travel time for all matched passengers under differentmatching methods and empirical cumulative distribution of the percentage of increasein travel time in the travel time without sharing . . . . . . . . . . . . . . . . . . . . . 31

2-5 Stability analysis of matching results . . . . . . . . . . . . . . . . . . . . . . . . . . . 322-6 Average ranking of paired fellow passengers with regard to the number of shareable

trips for ∆t = 100–600 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332-7 Average ranking of paired fellow passenger under two-group scenarios . . . . . . . . . 34

3-1 Geographic distribution of respondents . . . . . . . . . . . . . . . . . . . . . . . . . . 393-2 Parallel analysis for social interaction variables . . . . . . . . . . . . . . . . . . . . . 433-3 Diagram of structural equation model . . . . . . . . . . . . . . . . . . . . . . . . . . 463-4 Average ranking of paired fellow passenger with gender preference compared with

four two-group scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4-1 Diagram of the major components in this chapter . . . . . . . . . . . . . . . . . . . . 554-2 Dependency diagram of the optimization problem. . . . . . . . . . . . . . . . . . . . 654-3 Sioux Falls network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664-4 Polynomial smoothing of travel demand in different time intervals . . . . . . . . . . . 674-5 TR decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684-6 Preliminary results for the TR-decomposition algorithm with approximation prepro-

cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734-7 Total vehicle travel time vs. percentage of shareable trips with different time flexi-

bility values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774-8 Percentage of actually matched trips vs. percentage of shareable trips with different

time flexibility values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784-9 Reduction in total travel time vs. percentage of shareable trips and amount of time

flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5-1 A random generation of initial locations of all agents and a sample of locations ofagents after 200 ticks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5-2 Flow chart of the reverse Schelling model . . . . . . . . . . . . . . . . . . . . . . . . 89

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5-3 Indexes of dissimilarity and Gini coefficients with difference block sizes in one run ofthe reverse Schelling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5-4 Convergences with three algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955-5 Average detour, average trip distance without sharing, and average trip distance with

sharing under different matching algorithms . . . . . . . . . . . . . . . . . . . . . . . 965-6 Convergences of integration with densities from 50% to 90% . . . . . . . . . . . . . . 975-7 Convergences of integration with group shares from 12.5% to 50% . . . . . . . . . . . 985-8 Small clusters of Group 2 agents and slow decay of preference threshold of Group 1

agents with 12.5% share of Group 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985-9 Estimating the time of tipping point . . . . . . . . . . . . . . . . . . . . . . . . . . . 995-10 Convergences of integration with convergence rate β = 0.01, . . . , 0.05, and 0.1; branch-

ing dissimilarity index κ = 0.2, 0.5, and 0.8 . . . . . . . . . . . . . . . . . . . . . . . 1025-11 Oscillation of dissimilarity index in 10 random runs of the model . . . . . . . . . . . 1035-12 Convergence of average preference threshold and number of adversarial agents . . . . 103

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List of Tables

2.1 Efficiency measures for efficiency-based models and preference-based model . . . . . 302.2 Matching results for two-group scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Survey questions and variable names . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Result summary of exploratory factor analysis . . . . . . . . . . . . . . . . . . . . . . 443.4 Result summary of confirmative factor analysis . . . . . . . . . . . . . . . . . . . . . 453.5 Result summary of structural equation modeling . . . . . . . . . . . . . . . . . . . . 473.6 Preference of Women and Men for the Gender of Fellow Passengers . . . . . . . . . . 493.7 Matching results with gender preference compared with four two-group Scenarios . . 49

4.1 Notations used in Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Value table of σ, π, ,א ב . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1 Notations used in Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B.1 SEM results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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1

Introduction

“Civilized men have gained notable mastery over energy, matter... But, by contrast, weappear to be living in the Stone Age so far as our handling of human relations is concerned.”

– Gordon W. Allport, The Nature of Prejudice, 1954

1.1 Background and Motivation

Traffic congestion, dominated by single-occupancy vehicles, reflects not only transportation systeminefficiency and externality, but also a sociological state of human isolation. Advances in informationand communications technologies are enabling the growth of real-time ride sharing—whereby driversand passengers or fellow passengers are paired up on car trips with similar origin-destinations andproximate time windows—to improve system efficiency by moving more people in fewer cars. Lesserknown, however, are the opportunities of shared mobility as a tool to foster and strengthen humaninteractions. In contrast to typical social interactions in public or private space (meeting rooms,streets, pubic squares, living rooms, etc.), the nature of shared car rides is impromptu, captivefor a considerable duration, and remarkably more intimate, representing a unique juxtaposition ofspontaneity and intensity. It is also distinct from mass transit modes such as buses and trains,where most passengers refrain from engaging each other.

The existing literature on the large-scale on-demand mobility sharing mostly focuses on theefficiency side, including how to increase the number of trips matched, how to reduce the totalvehicle kilometers traveled, how to reduce the number of cars needed to meet the same traveldemand, etc. [1–4]. However, the discussion on the social interaction in mobility sharing services isvery limited.

On the social interaction of mobility sharing, scholars since the 1970s have investigated thedemographic factors which impact the mode choice and acceptance of traditional carpooling [5–7],and recent research has targeted the driver to passenger, and passenger to passenger discriminationin mobility sharing [8, 9]. Nonetheless, a systematic understanding of the social interaction in thenew form of on-demand large-scale mobility sharing, and how to actively utilize the co-ridership inthis mobility sharing setting as opportunities to generate more positive social interaction, have notbeen fully developed.

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From the sharing economy perspective, being a passenger of mobility sharing is not only aconsumption of this service. Each passenger, while being a consumer, is also acting as part of theexperience of his/her fellow passengers. In this perspective, each passenger of mobility sharing isco-producing and co-consuming the service [10]. This duality of the role of passengers triggers thethought in this dissertation of how we could actively engage mobility sharing passengers into morepositive social interactions.

In this dissertation, I used preference as a lens to investigate the social interaction in mobilitysharing, including how the interpersonal preference in mobility sharing can be understood, utilizedand reshaped. More specifically, I answered the questions of how preference could be used tomatch fellow passengers and to improve trip experiences; how gender, one of the key factors maycontribute to this preference; and in the reverse direction, if there are factors in the preference whichare unrespectable and need to be changed, whether mobility sharing can be used as a tool to changeit, and improve the integration of cities. Besides, I also studied how time flexibility of trips can beincorporated into mobility sharing models to reduce congestion.

1.2 Conceptual Framework

In this section, I introduce two conceptual frameworks of this dissertation. The first one emphasizesthe epistemological structure of this dissertation, i.e., how I approach preference of fellow passen-gers in mobility sharing through the process of understanding, utilizing, and shaping. The secondframework explains how mobility sharing, as a center piece, is connected to other major componentsof this dissertation.

1.2.1 Procedure-wise framework

From an epistemological perspective, the dissertation can be divided into three parts: understanding,utilizing, and reshaping fellow passenger preference in mobility sharing. While not all the questionswere fully answered in this dissertation, I listed the questions to complete the conceptual framework.The unanswered ones were discussed in the discussion sections of the corresponding chapters, andin the last chapter as questions for future research.

• On the understanding of fellow passenger preference, I would like to answer the followingquestions: 1) What is the overall structure of the fellow passenger preference in mobilitysharing? 2) What factors are included in this preference? What are the relationships of thosefactors, and can we categorize or frame the factors? 3) Among the factors, which ones arerespectable, which ones are not, and which ones are debatable? 4) Regarding the specificpreference factors, e.g., gender, race, ethnicity, etc., how does each of them impact the socialinteraction of mobility sharing, and how to understand the process of impact?

• On the utilization of fellow passenger preference, the following questions were investigated: 1)Can we use people’s preference for fellow passengers to match trips? 2) If preference is usedfor fellow passenger matching, what is the efficiency trade-off? 3) How does the structure ofpreference impact the matching outcomes? 4) How can the preference be elicited to provideinput for the preference matching algorithm? 5) What is the possible misuse of preference-based matching, and how to avoid the use of unrespectable preference in the algorithm?

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• On the reshaping of fellow passenger preference, this dissertation targets the following ques-tions: 1) If there is unrespectable preference in the society, can mobility sharing be used as atool to change it? 2) What factors may impact the effect and speed of this change? 3) Whatassumptions need to be made in order to realize the shaping process? Are they realistic? 4)How do we approach the preference shaping by system design?

1.2.2 Substantive framework

This dissertation consists of many components in the mobility sharing systems, including match-ing, social interaction, time flexibility, and social inclusion. From a substantive perspective, thedissertation can also be divided by those components and their roles in mobility sharing systems.

• Matching: several matching algorithms in mobility sharing were investigated, includingpreference-based matching, matching incorporating time flexibility of trips, and matchingfor social integration;

• Social Interaction: while acknowledging the social interaction of mobility sharing is differentfrom in all previous forms in other social contexts, this dissertation specifically targeted tworesearch questions—how gender changes mobility sharing usage and satisfaction via socialinteraction, and how to increase cross-group social interaction with mobility sharing to improvesocial integration;

• Time Flexibility: how can time flexibility be used in mobility sharing to reduce traffic, andhow the length of time flexibility impacts the significance of the reduction?

• Social Integration: can mobility sharing be used as a tool to improve social integration?What factors may impact the effectiveness of the tool? How to design the mobility sharingsystem to achieve better social integration.

1.3 Data and Approach Overview

In this dissertation, I combined inductive and deductive approaches. On the inductive side, in theunderstanding of the role of gender preference, I used empirical data from a survey to conduct thestructural equation models. On the deductive side, I designed and implemented matching models,optimization models, and agent-based models to test the consequences of different scenarios inmobility sharing. By combining deductive and inductive approaches, I would like to not only analyzethe existing conditions of the social side of mobility sharing, but also how the social interaction willevolve under various assumptions.

The data used in this dissertation is also a combination of real-world data and synthetic data.The real-world data include the trips and road networks of Manhattan, New York City and SiouxFalls, South Dakota, the survey on fellow passenger preference in U.S. cities where Uber or Lyft isavailable, and trip departure time data from the National Household Travel Survey. The syntheticdata include preference for fellow passengers, time flexibility intervals of trips, and the agent-basedworld of the social integration model. The real-world data helped both to establish the foundation forunderstanding the existing condition of mobility sharing, and provided a test base for the deductive

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methods, while the synthetic data filled the gap when real-world data were not available, and alsoacted as the system design frameworks and parameters for the scenarios to be tested.

The specific data and methods used for each research question will be discussed in detail inrespective chapters.

1.4 Dissertation Organization

The overall structure of the dissertation is shown in Figure 1-1. The two major branches of thegraph are how to understand the structure of preference for fellow passengers, and how to use thispreference to match trips. The three colors, blue, green, and red refer to the topics of understanding,utilizing, and reshaping in the corresponding parts of the dissertation. The dashed components arethe research questions that are beyond the scope of this dissertation, but added to make the structurecomplete.

The structure ofpreference for fel-low passengers

Duality of efficiencyand preference (sc. 6.2)

Social group, e.g.,race/ethnicity

Gender, socialinteraction and

mobility sharing (ch. 3)

Mobility sharing as apreference matching

problem (ch. 2)

Mobility sharing withtime flexibility (ch. 4)

Mobility sharingas a tool for socialintegration (ch. 5)

Mobility sharingwith gender pref-erence (sc. 3.5)

Figure 1-1: Structure of dissertation

In Chapter 2, I proposed a mobility sharing matching model which uses passengers’ preferencefor their fellow passengers, and optimize matching with the stability of preference as the objective.While most existing mobility sharing algorithms optimize fellow passenger matching based on ef-ficiency criteria (maximum number of paired trips, minimum total vehicle-time or vehicle-distancetraveled), very few explicitly consider passengers’ preference for their peers as the matching objec-tive. Existing literature either considers the bipartite driver–passenger matching problem, whichis structurally different from the monopartite passenger–passenger matching, or only considers thepassenger–passenger problem in a simplified one-origin–multiple-destination setting. Therefore, Iformulated a general monopartite passenger matching model in a road network, and illustrated themodel by pairing 301,430 taxi trips in Manhattan in two scenarios: one considering 1,000 randomlygenerated preference orders, and the other considering four sets of group-based preference orders.In both scenarios, compared with efficiency-based matching models, preference-based matching im-proves the average ranking of paired fellow passenger to the near-top position of people’s preferenceorders with only a small efficiency loss at the individual level, and a moderate loss at the aggre-gate level. The near-top-ranking results fall in a narrow range even with the random variance ofpassenger preference as inputs.

In Chapter 3, I focused on gender, one key aspect of the preference for fellow passengers toanalyze how it impacts the usage and satisfaction level of mobility sharing services via social inter-

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action. In this chapter, I answered three questions: 1) Does social interaction in mobility sharingimpact the usage and satisfaction level with it? 2) Is there any gender difference in consideringsocial interaction as motivation or deterrent for mobility sharing? 3) Is there gender difference inthe usage and satisfaction with mobility sharing services? With a survey (n = 997) in the U.S. citieswhere Uber or Lyft is available, the data of sociodemographics, social interaction indicators, andthe usage and satisfaction with mobility sharing services were collected. Using structural equationmodels I revealed the underlying structure between gender, social interaction, and the usage andsatisfaction of mobility sharing. The findings of this chapter include: 1) positive and negative socialinteractions both have significant impacts on the usage and satisfaction levels of mobility sharingservices, with the former one increasing usage and satisfaction, and the latter one reducing them;2) there is significant gender difference in the agreement on considering social interaction and lackof fellow passenger information as motivation or deterrents for using mobility sharing; and 3) basedon the previous two findings, there is significant indirect gender effect on the usage and satisfactionwith mobility sharing. Nonetheless, this effect is offset by the direct effect of gender on mobilitysharing usage and satisfaction, resulting in an insignificant total effect. I also examined the effectof user experiences on the three relationships, and found that being users significantly reduced thegender difference in considering positive social interaction and fellow passenger information as themotivation for mobility sharing, but did not significantly affect the gender difference on the usageand satisfaction of mobility sharing, either directly or indirectly.

At the end of Chapter 3, I combined the data of preference for the gender of fellow passengerswith the preference-based matching model in Chapter 2. By comparing the matching results fivetwo-group scenarios, I would like to use gender as an example to answer the following question. Ifin a two-group preference structure, both group prefer to be paired with members of one of thetwo groups, how will the efficiency and preference trade-offs be different from with only same-grouppreference or indifferent preference with group affiliation. I found that compared to the latter twopreference structures, if both groups prefer one of the two groups to be fellow passengers, there isnot only efficiency trade-off of lower matching rate, on the preference side, the average ranking ofpaired fellow passenger will also be worse due to the “competition” effect.

Chapter 4 is focused on the time flexibility of trips, as a temporary egress from the discussionof fellow passenger preference. By observing that mobility sharing combined with time flexibilitycan reduce traffic congestion, which then frees more space for further sharing opportunities andforms a positive feedback loop, I formulated ridesharing matching, driver designation, departuretime assignment, and traffic assignment as a mixed integer nonlinear optimization problem. Con-sidering the difficulty of solution due to the nonlinearity, I then introduced an approximation of theproblem—an iterative combination of traffic assignment steps (T-step), and ridesharing–matching,driver-designation, and departure-time-assignment steps (R-step). I also proved that the R-step canbe simplified as a mixed integer linear programming (MILP) problem. Due to the non-convergenceof the TR-decomposition approach, I further adjusted the method to be a quick converging onefollowing a greedy strategy. I implemented the quick converging method on the transportationnetwork of Sioux Falls, South Dakota, and found that by even including a small time flexibility ineach trip, the total vehicle travel time can be significantly reduced. However, when time flexibilitykeeps increasing, the additional total vehicle travel time reduction declines.

In Chapter 5, reflected on the Schelling segregation model, I proposed a model to explore howmobility sharing can be used to change people’s preference for outgroup members, and then improvespatial integration and reverse the Schelling segregation process. In this chapter, based on inter-group contact theory, I assumed that to actively pairing people from different social groups, more

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opportunities for intergroup contact can be created, and thus the exclusionary attitudes towardoutgroup members can be reduced. This reduction of exclusionary attitudes will then be expressedin the location choice of the destinations of other activities. With the preference change for tripdestination choices, spatial segregation can be reduced and evolve into more integrated equilibria. Iused the model to test the impact of a series of system parameters, including density, group share,maximum detour allowed, matching algorithm, on the equilibria of convergence of integration. Fromthe results of the model, both uneven group shares and high densities will lead to slower conver-gence of integration. Further, based on the literature of negative social contact, three scenarios ofthe relationship between the integrating force and the segregating force were also explored.

In the last chapter, I summarized the dissertation, reflected on the findings, and elaborated thefuture research topics which might validate or falsify many of the assumptions in the dissertation,and thus further complete the discussion of preference in mobility sharing. Two future researchquestions are notated in dedicated sections—the boundary of respectable preference, and the dualityof efficiency and preference in the matching of mobility sharing. The two sections could furtherextend discussions on the models, findings, and inferences in this dissertation, and are connected tothe two branches of Figure 1-1, respectively.

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2

Mobility Sharing as a PreferenceMatching Problem∗

2.1 Introduction

Online mobility on demand (MoD) services emerging with the advance of modern information tech-nology enrich peoples’ transportation mode choices. Ridesharing is an MoD service that connectsthe trips of passengers who can combine their trips with only small increases in travel time, suchthat a single vehicle can accommodate more than one passenger at a time. Examples of ridesharingservices include Lyft Line, UberPool, and GrabHitch. By increasing the occupancy of vehicles,ridesharing has the potential to reduce the number of cars on roads, leading to reduction in trafficcongestion and pollution. Lyft reported in 2014 that over fifty percent of their trips in San Francisco,and thirty percent in New York City were shared trips [11].

The current paradigm of mobility sharing is focused on optimizing system efficiency. The under-lying assumption is that travelers place no value on the characteristics of their fellow passengers, orif they do, the relative importance is negligible compared to travel time and cost. Researchers haveproposed various algorithms to maximize these efficiency benefits with different optimization objec-tives under different assumptions [1–3, 12–14], and have carried out simulations with real-world data[15]. In the proposed algorithms, trip pairing is modeled as a graph-matching problem, and withdifferent matching strategies, system optimization can be achieved. In a recent study, Alonso-Moraet al. investigated the minimum number of vehicles that would be needed by New York City ifsharing possibility was optimized [3].

Whereas the efficiency benefits of ridesharing have been extensively studied, ridesharing is lessunderstood as a sociological phenomenon [16]. In contrast to typical social interactions in pub-lic or private spaces (meeting rooms, streets, public squares, living rooms, etc.), the nature ofshared car rides is impromptu, captive for a considerable duration, and remarkably more intimate—representing a unique juxtaposition of spontaneity and intensity. It is also distinct from mass transitmodes such as buses and trains, in which most passengers refrain from engaging each other. Forexample, Uber promotes their UberPool service as a potential platform for business and job oppor-tunities by meeting new people [17].

∗© 2018 IEEE. Reprinted, with permission, from Hongmou Zhang and Jinhua Zhao, Mobility Sharing as aPreference Matching Problem, IEEE Transactions on Intelligent Transportation Systems, October 2018.

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In previous studies, researchers have used preference to match drivers and passengers [18, 19],and formulated ridesharing as a bipartite matching problem, taking drivers and passengers as twoseparate sets of nodes. The mathematical structures of monopartite and bipartite matchings aredifferent [20, 21]. For example, there is guaranteed to be stable solutions for bipartite matchingproblems, but there may not exist any stable solution for a monopartite matching problem. There-fore, the driver–passenger matching problem and the fellow passenger matching problem are notonly operationally different, but also observe different system structures. Furthermore, these studiesonly considered travel distances and financial costs of trips to represent preference and may thusunderestimate the variation and complexity of user preference resulting from social factors.

Thaithatkul et al. conducted the first research on using preference for fellow passengers matching[22, 23], and considered trip distance and randomly-generated personal utility for fellow passengerpreference. They implemented the model in a one-origin–multiple-destination setting, using artifi-cially generated trip features and synthetic preferences. The one-origin–multiple-destination settingis applicable to real-world scenarios such as trips from an airport, but does not capture the share-ability of the general multiple-origin–multiple-destination situations. In this paper, we propose amore generic preference-based fellow passenger matching model, and examine the model propertieswith empirical taxi trips and the road network in Manhattan.

Berlingerio et al. quantified the enjoyability of passengers based on people’s interests, sociallinks, and tendency to connect to people with similar or dissimilar interests, mined from socialnetwork data, and formulated enjoyability together with the number of cars as a multi-objectivelinear programming problem [24]. Enjoyability is closely related to but different from preference,and the linear additivity of enjoyability of people does not fit in our paper, where we use Paretocomparison instead of quantified enjoyability.

Preference matching originated from, and has been widely studied in the context of collegeadmission, marriage, roommate assignment, doctor residency assignment, and kidney exchange [20,25–27]. Each of these problems is one case of matching people to people, or people to institution—students to colleges, doctors to hospitals, or kidney donors to receivers. Furthermore, in each of thesettings, each individual—either person or institute—is assumed to have, explicitly or latently, aranking of objects or people representing their preference. Within such a system, a stable outcomeis expected, and can be achieved regarding the preference of all individuals in the system. A stablematching is a matching with which no individual could be better off by changing the match withoutmaking at least one other individual worse off. In other words, a stable matching is an optimalmatching in a socially Pareto sense.

In our paper, we explicitly consider passengers’ preference for their fellow passengers as thetrip matching objective, formulate ridesharing as a monopartite preference matching problem, andcompare the matching outcome with those of efficiency-based methods. We use 301,430 taxi tripsin Manhattan [28] (Fig. 2-1) on a randomly selected day in 2011 (April 24) to illustrate the model:a single set of passengers are paired with each other, and a maximum stable matching as the pairingobjective is identified with regard to all passengers’ preferences for their peers. We quantify thetrade-off between efficiency-based ridesharing methods, and our proposed preference-based matchingmethod.

We note that matching people with preferences and the elicitation of preferences are two distincttasks. In this paper, we take preferences for fellow passengers as given in the form of rank orders,but do not address the issues of the preference elicitation. The matching algorithm we propose canoperate with any complete or partial preference orders. At the end of this paper, we discuss the

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Origin1-29

29-72

72-119

119-174

174-239

239-318

318-427

427-604

604-1,166

1,166-1,983

Destination1-29

29-72

72-123

123-184

184-272

272-377

377-499

499-910

910-1,812

1,812-2,337

Legend

0 5 10 15 20 25 30 35 40

Duration of trips (min)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fre

qu

en

cy

104

Figure 2-1: Distribution of the origins and destinations of taxi trips in Manhattan and distributionof trip duration without sharing

complexity in the preference for fellow passengers and its difference from the preference for othertrip attributes. The complexity of preference for fellow passengers exists in its structure—beingheterogeneous, dynamic, and more about compatibility than similarity—and in its elicitation. Inthe discussion section we propose possible preference elicitation methods with associated challenges.

We also emphasize that not all preferences are respectable. [8] and [29] demonstrated the ethicalconcerns surrounding ridesharing. Although an ethical discussion of preference in ridesharing isbeyond the scope of this paper, we want to point out that it is critical for the society as a wholeto draw the boundary between acceptable and unacceptable articulations of preferences. A positiveunderstanding of the role of preference in ridesharing is an important prerequisite for addressingthose concerns.

2.2 Data and Methods

Formally, we consider the vehicle trips in a city to be in a shareability network [2]. A shareabilitynetwork is an undirected graph G(V, E) where each node in the node set V represents a trip, andE = {{vi, vj}|vi, vj ∈ V} is the edge set indicating whether two trips are shareable. For example, fortrips vi, vj ∈ V , if vi and vj can be shared, we have {vi, vj} ∈ E . The criterion used to determinewhether two trips are shareable in this paper follows the “cap of maximal detour” rule [2]. Two trips

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are shareable if there exists a new route that can connect the origins and destinations of the twotrips, and the additional travel time for either trip does not increase more than ∆t, a predeterminedparameter set by the system designer. In other words, in a shareability network with parameter∆t, if a trip would take a person time t to travel individually, the travel time would never be morethan t + ∆t if the trip is shared with another allowed trip. In the scope of this paper, we limit ourdiscussion to the ridesharing of two parties of passengers.

The departure and arrival time constraints are also considered when building the shareabilitynetwork. For any trip pairs, if the actual departure time of any of the two when sharing is laterthan the original departure time without sharing plus ∆t, the trip pair is considered non-shareable,and similarly for arrival time [2].

Therefore, detour time and waiting time, or the difference between the original departure timewithout sharing and the pick-up time with sharing are unified as ∆t, or the “passenger discomfortparameter”, in the configuration of shareability networks. In other words, if two trips are shareable,neither the detour time of each party, nor the difference between the actual pick-up time of theshared trip and the original departure time can be greater than ∆t.

Fig. 2-2 illustrates the distribution of node degree of the shareability network built on the taxitrips in Manhattan in one day. For example, when ∆t is 300 seconds, the majority of trips haveapproximately 100 shareable trips, and the maximum number of shareable trips that a trip hasis approximately 900. Both numbers increase with ∆t as sharing possibility increases with longerallowed detours.

0 500 1,000 1,500 2,000 2,500

Number of shareable trips

100

101

102

103

104

105

Fre

quency o

f tr

ips

t = 100 s

t = 200 s

t = 300 s

t = 400 s

t = 500 s

t = 600 s

Figure 2-2: Frequency of the number of shareable trips with a trip in shareability networks with ∆t= 100–600 s (each point represents the frequency of trips with a certain number of shareable trips)

Upon the shareability network, we define a weight function ω : E → R indicating the efficiencybenefits of sharing two trips. In this paper, we use the savings of vehicle-minutes (veh-min) andvehicle-kilometers-traveled (VKT) as the measures ω of efficiency. The travel time on each road linkis estimated using the real-world taxi travel time data, and the travel route of each trip is inferredas the one with the closest total travel time to the actual time [5]. In this paper, we assume that thesharing of trips does not change the overall underlying traffic condition, i.e., we do not consider thefeedback between traffic congestion and ridesharing, which is an important future research directionwhen the shared trips start to contribute to a high proportion of the overall traffic.

We then consider the preference of the passenger in each trip for all other passengers. Asdiscussed by [30] and [31], the reasons for a passenger preferring a certain fellow passenger toanother vary. Here, we take the preferences as exogenous inputs to the problem, and the matching

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algorithm we develop can operate given any preference rank order.

If a shared trip consists only of individual passengers, the preference is individual preference. Inother cases where there are more than one passenger in a party, we assume that all the passengersin the party are able to consent on an order; thus, we can consider each party equivalent to asingle passenger. In the remainder of this paper, we treat trips and passengers as equal. Hence, forpassenger vi in V , his/her preference can be denoted as an ordered list vi : vk1 � vk2 � . . . � vkm ,where k1, k2, . . . , km is the permutation of a sublist of {1, 2, . . . , n}\i, and n is the number of passengersin the system.

Note that the preference list for a passenger does not have to be a complete list containingall other passengers in the system, but only needs to contain all the shareable passengers, or the“neighbors” in G(V, E). Nevertheless, we need to require that the preference lists be symmetricallycompatible—if passenger vi is on passenger vj ’s list, passenger vj also needs to be on passenger vi’slist.

A matching M is a subset of E, and it requires that ∀ei, ej ∈ M, ei∩ej = � such that no passengeris sharing with more than one other passenger. We denote the set of all feasible matchings of ashareability network G as M(G). A stable matching is a special matching M ′ such that there areno two passengers in the system who both prefer each other to their paired fellow passengers, orformally �ei = {vi1, vi2}, ej = {vj1, vj2} ∈ M ′, vi1 : vj1 � vi2 and vj1 : vi1 � vj2 . The set of stablematchings is denoted as M ′(G) ⊂ M(G).

Gale and Shapley found that for such a system, referred to as the “stable roommate problem,”a stable matching consisting of all passengers does not always exist [20]. Irving proposed a methodthat finds a stable matching if one exists [27]. When there is no stable matching for all peoplein the system, an alternative objective is to find a stable matching on a maximum subset of V—a “maximum stable matching” [21]—and Tan proposed an algorithm to find such a solution [32].Therefore, we can look for the matching that is stable on the subset of V with maximal cardinality:

Mpref = argmaxM ∈M′(G)

{|M |} . (2.1)

Because there may be more than one such matching, Mpref can be a subset instead of a singleelement in M.

Further, we wish to determine the efficiency trade-off arising from the use of stable-preferencematching; therefore, we compare it with two efficiency-based matching methods, maximum cardi-nality matching (MC), which maximizes the number of shared trips (n(shared trips)), and maximumweight matching (MW), which minimizes total system veh-min or VKT:

Mmc = argmaxM ∈M(G)

{|M |} , (2.2)

Mmw = argmaxM ∈M(G)

{∑e∈M

ω(e)

}. (2.3)

As M ′ ⊂ M, ∀M ′ ∈ Mpref and M ∈ Mmc, there always is |M ′ | 6 |M |.

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2.2.1 Synthetic preference orders

We use two types of preference inputs in this paper. 1) Random preference orders: for each nodein the shareability network, we generated a random permutation of all the neighboring nodes andused the permutation as the preference order. We repeated the random permutation assignment1,000 times to generate sufficient randomness in people’s preference for others, and delineated therange of possible results, which turned out to be very narrow. 2) Group-based preference orders: weacknowledge that various factors can be related to people’s preference for fellow passengers, rangingfrom gender, age, and income level, to personal interest, political affiliation, hobbies, and talkative-ness; therefore, we implemented group-based preference assuming that each passenger belongs toone of two groups based on one binary characteristic. S1–S4 are four scenarios that test differentgroup shares, and different preference symmetricity assumptions between groups.

S0. One group: randomly assigned preference orders;

S1. Even, symmetric: 50% of passengers in Group 1, 50% in Group 2; people in both groups preferfellow passengers from their own group to those from the other;

S2. Even, asymmetric: 50% of passengers in Group 1, 50% in Group 2; people in Group 1 preferfellow passengers from their own group to those from Group 2, whereas people in Group 2 areindifferent;

S3. Uneven, symmetric: 20% of passengers in Group 1, 80% in Group 2; people in both groupsprefer fellow passengers from their own group to those from the other;

S4. Uneven, asymmetric: 20% of passengers in Group 1, 80% in Group 2; people in Group 1 preferfellow passengers from their own group to those from Group 2, whereas people in Group 2 areindifferent;

We first randomly assigned trips into two groups based on the group share hyperparamter.We assumed that both groups are evenly distributed in space. Then, for each group, we assignedpreference based on the group type. For groups in which people prefer same-group fellow passengers,we first determined whether the neighboring nodes were in the same group or the other group,and carried out random permutation on each of the two neighbor sets. We concatenated the twopermuted lists as the preference lists. In Appendix A, the order of magnitude of the number ofpreference possibilities and the reduction of possibilities when groups exist are discussed.

We acknowledge that in a real-world ridesharing platform it is impossible to ask users to rankall the possible fellow passengers. This paper focuses on the matching process but not the thepreference elicitation process. We develop a general matching algorithm which can operate givenany complete or partial preference rank orders. In addition the complete or partial rank orders forfellow passengers do not have to be explicitly given by the users, but may also be derived based onuser behaviors, travel history, or relevant information that the users provided to the service provider.We will comment on both the potential technical and ethical issues in the discussion section.

2.2.2 Irving–Tan algorithm

We build our stable-preference matching model following the Irving–Tan algorithm [27, 32]. Thereare two steps in the algorithm. In the first step (Algorithm 3), each passenger proposes to the top

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choice in their preference lists, and the proposed passenger could either hold the proposal if there isno better choice, or reject the proposal if a better choice is already present. If a passenger acceptsa proposal while holding another proposal, the passenger rejects the originally held proposal. Therejected passenger will then have to re-propose. The process is repeated until each passenger isaccepted by another passenger, or the preference list has been exhausted.

Algorithm 1 Preference list reduction: proposal and rejection1: for each passenger vi ∈ V do2: propose to the first one in the preference list3: if the proposed passenger does not have a better option then4: hold the proposal5: else6: reject the proposal and delete each other in the preference lists7: while rejected or preference list is not empty do8: propose to the next one9: for each passenger vi ∈ V do

10: delete all the choices with rankings larger than the proposal it holds

After the preference list reduction, the preference lists are significantly reduced. By symmetricityof each operation in step one, if there is only one fellow passenger vj left in passenger vi’s preferencelist, vi should also be the only one left in vj ’s preference list. Therefore, for the passengers withreduced preference lists of length zero or one, the matching is complete. For passengers withpreference lists of length at least two, there must be “rotations” that include at least three passengers.In the second step of the algorithm (Algorithm 4), one rotation is identified in each iteration. Arotation corresponds to a subset of the shareability network, in which the matching solution will belocal to the nodes within it and independent of the rest of the graph. In other words, as long as arotation R is identified from the network, the problem can be divided into a matching m1 on R anda matching m2 on G\R, and the overall matching will be m = m1 ∪ m2.

There are two types of rotations, odd rotation and non-odd rotation. An odd rotation is arotation with an odd number of elements. For example, consider the following case, in which nostable matching exists for all four elements.

1: 2 � 3 � 42: 3 � 1 � 43: 1 � 2 � 44: 1 � 2 � 3

After step one, the preference lists will be reduced to the following table, and {1, 2, 3} can befound as an odd rotation.

1: 2 � 32: 3 � 23: 1 � 24: �

To find a maximum stable matching for an odd rotation, one element needs to be eliminatedfrom the list randomly [32].

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In the other type of rotation, the non-odd rotation, there is an even number of elements. Forexample, in the following case, {1, 2, 4, 3} is a non-odd rotation. In this case, the rotation can befurther removed from the preference lists as described by [27] and [32]. After all rotations have been

1: 2 � 4 � 32: 4 � 13: 1 � 44: 3 � 1 � 2

Algorithm 2 Matching with the reduced preference lists1: while there is rotation in the preference lists do2: if it is an odd rotation then3: Randomly delete an element from the odd rotation4: m1 = matching(the odd rotation \ the deleted element)5: m2 = matching(the rest of the preference lists)6: return m1 + m2

7: else8: delete the rotation from the preference lists

removed from the preference lists, each person should hold at most one choice in the correspondingpreference list, which gives the maximum stable matching. The time complexity of this algorithmis O(n2) [32].

For the efficiency-based matching methods, we used the Python package NetworkX [33] for themaximum cardinality matching, and simple greedy algorithm for the maximum weight matching.The algorithm keeps pairing nodes with the highest weighted edges until no further matches can bemade.

2.3 Results

First, we conducted the efficiency-based and preference-based matchings for the shareability networkof taxi trips in one day in Manhattan with ∆t = 300 s, and repeated the test for 1,000 randompreference assignments to show the variability of the results. The solution time for each run wasabout eight minutes. Second, in order to estimate the impact of ∆t on the convergence of preferencematching, we performed sensitivity analysis for preference-based matching with a range of ∆t =100–600 s for 1,000 runs each. Third, for each of the four two-group scenarios, we conductedpreference-based matching for 100 structured random assignments of preference.

2.3.1 Efficiency vs. preference

Table 2.1 summarizes the efficiency measures under different matchings. The table shows thatpreference-based matching results in only a marginally lower matching rate than MC matching.Further, the rate is 3.8% higher than those of MW matchings. For other efficiency measures,system-wise preference-based matching behaves similar to MC matching and has longer distancesand travel times for each vehicle trip on average. However, for each individual passenger, the

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difference in travel time under preference-based matching and MW matchings is only about 40seconds. Fig. 2-4 shows more details of the efficiency trade-off from a passenger’s perspective.Fig. 2-4a shows the distribution of increase in travel time (detour); preference-based matching hasmore long detours than MW matchings, with all detours still capped by ∆t. Fig. 2-4b depicts thecumulative distribution of the detour time as a proportion of the total travel time if the trip is notshared. MW matchings have 96.8% trips with detour of less than 10% of the non-sharing traveltime, whereas for preference-based matching, the number is still as high as 92.8%.

Fig. 2-3 shows the performance of the matchings in terms of preference. The y-axis on the left,in logarithmic scale, shows the ranking of the actually paired fellow passenger in the correspondingpassenger’s preference list, averaged across 1,000 runs, with regard to the x-axis—the number ofshareable trips that the passenger has. For example, under both MC and MW matchings, forall passengers with 100 shareable fellow passengers, the average ranking of matched passengersare their 50th preferred choices, whereas under preference-based matching, the average ranking isslightly more than 10. The curves of both MC and MW approximate k/2, where k is the number ofshareable fellow passengers—on average efficiency-based matchings pair a passenger to an averagepreferred fellow passenger as we assume preference is independent of space and time. Preference-based matching asymptotically approaches the tenth most favorable choice even with a k as largeas several hundreds.

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Tab

le2.

1:E

ffici

ency

mea

sure

sfo

reffi

cien

cy-b

ased

mod

els

and

pref

eren

ce-b

ased

mod

el(∆

t=

300

s)N

osh

arin

g†M

ax.n(s

hare

dtr

ips)

mat

chin

g(M

C)

Min

.ve

h-m

inm

atch

ing

(MW

1)M

in.

VK

Tm

atch

ing

(MW

2)P

refe

renc

e-ba

sed

mat

chin

gAvg

.nu

mbe

rof

mat

ched

trip

s0

297,

818

(98.

8%)

285,

856

(94.

8%)

285,

902

(94.

8%)

297,

049.

5(9

8.6%

)Tot

alsy

stem

veh-

min

2,68

6,93

32,

391,

910

1,68

9,48

41,

689,

465

2,34

1,19

9Tot

alsy

stem

VK

T88

5,53

678

8,13

255

8,41

555

8,38

877

1,94

2Veh

-min

per

trip

8.91

7.94

5.60

5.60

7.77

VK

Tpe

rtr

ip2.

942.

611.

851.

852.

56Pas

seng

ertr

avel

tim

epe

rtr

ip(m

in)

8.91

12.1

511

.42

11.4

212

.14

Tot

alnu

mbe

rof

trip

s30

1,43

0Tri

psw

ith

atle

ast

one

shar

eabl

etr

ips

300,

776

(99.

8%)

†A

ssum

ing

ther

eis

only

one

pass

enge

rin

each

trip

.

Tab

le2.

2:M

atch

ing

resu

lts

for

two-

grou

psc

enar

ios

Scen

ario

S0S1

S2S3

S4

Avg

.nu

mbe

rof

mat

ched

trip

s29

7,05

0(9

8.6%

)29

4,49

4(9

7.7%

)29

6,04

0(9

8.2%

)29

4,62

9(9

7.7%

)29

6,53

0(9

8.4%

)Veh

-min

per

trip

7.77

7.77

7.77

7.77

7.77

Pai

red

with

pass

enge

rin

the

sam

egr

oup

n.a.

G1:

97.6

%G

2:97

.6%

G1:

97.6

%G

2:96

.1%

G1:

94.4

%G

2:98

.5%

G1:

94.3

%G

2:97

.4%

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0 100 200 300 400 500 600 700 800 900

Number of shareable trips

100

101

102

103

Rank o

f paired p

assenger

in p

refe

rence lis

t

0

200

400

600

800

1,000

1,200

1,400

Fre

quency

Degree distributionMax n(shared trips)

Mean 5th-95th percentileLeft y-axis

Min veh-min

Min VKT

Preference-based

Right y-axis

Figure 2-3: Ranking of paired fellow passengers in respective passengers’ preference lists (left y-axis). The darker dots show the average values of results, and the lighter dots are the 5th and95th-percentile values; The three efficiency-based matchings almost overlap; Degree distribution:the frequency of the number of shareable trips of a given trip (right y-axis), a replicate of the ∆t =300 s curve of Fig. 2-2 for reference.

In the mean values and the gap between the 5th and 95th percentiles in Fig. 2-3, the range ofresults is very narrow across 1,000 repeat tests with randomly generated preferences, especially whenthe number of shareable trips is not sufficiently large. For example, for passengers with 300 shareabletrips, the average rankings of their mean, 5th, and 95th-percentile-matched fellow passengers are13.8, 12.6, and 15, respectively. The narrow range indicates that this near-top pairing performanceis not dependent on specific preference orders.

a. b. ,

50 100 150 200 250 300

Increase in travel time

0

0.5

1

1.5

2

Fre

qu

en

cy

104

Max n(shared trips)

Min veh-min

Min VKT

Preference-based

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Increase in travel time / travel time if not shared

0

0.2

0.4

0.6

0.8

1

Em

piric

al cu

mu

lative

dis

trib

utio

n

Max n(shared trips)

Min veh-min

Min VKT

Preference-based

Figure 2-4: Distribution of increase in travel time for all matched passengers under different match-ing methods (a) and empirical cumulative distribution of the percentage of increase in travel timein the travel time without sharing (b)

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0 100 200 300 400 500

Run index

0

2

4

6

8

10

12

Avg

. ra

nk o

f p

aire

d f

ello

w p

asse

ng

er

2.55

2.552

2.554

2.556

2.558

2.56

2.562

2.564

2.566

2.568

2.57

VK

T p

er

trip

Trips with 10 shareable trips

Trips with 50 shareable trips

Trips with 100 shareable trips

Trips with 500 shareable trips

VKT per trip

Figure 2-5: Stability analysis of matching results

2.3.2 Result stability analysis

Since there is randomness in the preference–matching algorithm, we need to understand the stabilityof matching results in both the efficiency aspect and the preference aspect. To this end we ran thematching algorithm with fixed inputs of preference orders for 500 times, and eliminated odd-rotationelements randomly in each run. As shown in Fig. 5 both the efficiency performance of the algorithm,such as the VKT saving per trip, and preference performance, i.e., the average ranking of pairedpassengers for those with the same number of shareable trips are very stable across runs (fourrepresentative results are shown—the paired passenger’s ranking for trips with 10, 50, 100, 500shareable trips), and the ranges of variance of results are small. For VKT per trip, the rangebetween the maximum value and the minimum value of results in the 500 runs is 2.4× 10−4, and forrankings of paired fellow passengers, the variance across runs is much smaller than one rank step inpreference.

2.3.3 Sensitivity analysis for ∆t

While considering five minutes (300 s) as a meaningful detour cap for shared trips, we are alsointerested in the impact of ∆t on the performance of the preference-based matching model.

Fig. 2-6 shows the results of preference-based matching with ∆t from 100 s to 600 s. In Fig. 2-6a,the shapes of the curves are similar but the curves have different values. In the shareability networkwith a larger ∆t, the average ranking of paired fellow passenger is worse for trips with a givennumber of shareable trips. For example, for trips with 200 shareable trips, with ∆t = 200 s, theaverage ranking of paired fellow passenger is 9.4, whereas for ∆t = 400 s, the ranking is 16.8, andfor ∆t = 600 s it is 20.3. However, it is worth noting that the trips with 200 shareable trips are nolonger the same in the three cases. For any given trip, the larger ∆t is, the more shareable trips itwill have. This explains why the curves stretch to the right as ∆t increases.

Further, as ∆t keeps increasing, the gap between curves becomes increasingly smaller—leading

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us to expect a limiting curve when ∆t → ∞. This is because when ∆t is sufficiently large, i.e.,when very long detours are allowed, most trips are shareable with each other, and the shareabilitynetwork becomes very dense, which eventually becomes a complete graph. In other words, all tripsare shareable if no detour limit is applied. Increasing ∆t at this time does not add any shareability,thus the curve reaches the limit position. Therefore, while increasing ∆t makes the ranking worse,it has a limit.

At the same time, although in an absolute sense the ranking gets worse as ∆t increases, theranking still gets better relatively, as shown in Fig. 2-6b. In this graph, the ranking is normalized bythe number of shareable passengers, and the number of shareable trips is normalized by the highestnumber of shareable trips in the graph. For example, if we denote the largest number of shareabletrips that a passenger has as K, when ∆t = 100 s, the people who have 20%K shareable trips willbe paired with their top 21.6% preferred fellow passengers, whereas for ∆t = 300 s, the rankingimproves to the top 8%, and for ∆t = 600 s, it further improves to the top 4.5%.

a. b. ,

0 500 1,000 1,500 2,000 2,500

Number of shareable trips

0

5

10

15

20

25

Avg. ra

nk o

f paired fello

w p

assenger

t = 100 s

t = 200 s

t = 300 s

t = 400 s

t = 500 s

t = 600 s

0 0.2 0.4 0.6 0.8 1

Number of shareable trips / max. number of shareable trips in the network

0

0.2

0.4

0.6

0.8

1

Avg

. ra

nk o

f p

aire

d f

ello

w p

asse

ng

er

/

nu

mb

er

of

sh

are

ab

le t

rip

s

t = 100 s

t = 200 s

t = 300 s

t = 400 s

t = 500 s

t = 600 s

Figure 2-6: Average ranking of paired fellow passengers with regard to the number of shareable tripsfor ∆t = 100–600 s (a: absolute values; b: normalized values)

2.3.4 Two-group scenarios

Table 2.2 and Fig. 2-7 summarize the results for preference-based matching under the four two-groupscenarios with the one-group scenario S0 for reference. For scenarios S1–S4, the efficiency measuresshow that the preference for a specific subgroup of fellow passengers leads to fewer shared trips.The comparison between the symmetric scenarios (S1, S3) and asymmetric scenarios (S2, S4) showthat the more people there holding group-based preferences, the lower the probability of sharingwill be. However, the greatest difference between any two scenarios is only 1.1%.

Comparing the same-group pairing rates, in S2 we find that the groups that are indifferent (G2)have slightly lower chances of being paired with same-group passengers. Moreover, in S3 and S4,the chance of being paired with passengers from the same group is lower for the “minority” group(G1), even if the majority group is indifferent, although the difference is only 3–4%, and consideringthe group splits, 20% and 80%.

From Fig. 2-7, it is clear that the average ranking of paired fellow passenger is worse when bothgroups prefer to be paired with passengers from the same group (S1, S3), compared to if only onegroup does (S2, S4). However, this difference diminishes when the group split is uneven, as the gapbetween the curves of S3 and S4 is smaller than that of S1 and S2. The best-performing preferencematching is observed in the even-group symmetric scenario (S1).

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0 100 200 300 400 500 600 700 800 900

Number of shareable trips

0

2

4

6

8

10

12

14

16

Ra

nk o

f p

aire

d p

asse

ng

er

in p

refe

ren

ce

lis

t

Scenario 0

Scenario 1

Scenario 2

Scenario 3

Scenario 4

Figure 2-7: Average ranking of paired fellow passenger under two-group scenarios

2.4 Discussion

This paper formulated the passenger matching in a mobility sharing system as a monopartitematching problem, and examined the trade-off between this matching model with that arising fromefficiency-based matching models. The results show that with only a small efficiency loss at theindividual level, and a moderate one at the aggregate level, the improvement in preference rankingis substantial—from the average to the near-top. We also found that increases in the detour cap ∆tlead to slightly—and boundedly—worse preference rankings in absolute sense, but to better rank-ings in relative sense. Based on the actual context and system design objective, ridesharing systemdesigners can make the decision of which matching strategy to use, or how to combine multiplestrategies.

We also modeled a two-group preference structure for fellow passengers in addition to the simplerandom case. The preference for a specific group of fellow passengers leads to lower pairing rates—the more people holding it, the lower the pairing rate will be. The same-group pairing rate is lowerfor the minority group.

It is important to distinguish the question of obtaining the ridesharing preferences from thequestion of developing the matching algorithm that utilizes such preferences. This paper focusedon the latter question and developed the algorithm that can generate the optimal matching outputwith any set of input preference rank orders. The algorithm is independent of the input ridesharingpreferences. We now comment on the former question. The challenge of obtaining the ridesharingpreferences of the users is twofold: the complexity in the structure of preference for fellow passengers,and the complexity of eliciting such preferences.

We have identified at least three aspects of the structure of preference for fellow passengersin ridesharing that are different from the preference for typical travel attributes such as traveltime and travel cost: 1) Heterogeneity—there is a higher degree of heterogeneity across individualsin the preference for fellow passengers: some people really enjoy the shared ride with a fellowpassenger while others strongly prefer riding alone; 2) Dynamism—e.g. even for the same personhe or she may want to be silent in the morning ride while hope to engage a conversation with afellow passenger in the afternoon. The preference for fellow passengers is dynamic and transient;3) Compatibility—since the pleasure or displeasure of a shared ride is a result of the co-productionby fellow passengers, it is more about compatibility between both passengers than the absolutequality of each individual. An extreme example is smoking: a smoker may like to be paired with

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another smoker, even though smoking per se is a bad behavior for most people. (Of course smokingis typically banned in ridesharing in most cities for this example to be relevant.) Furthermore,compatibility does not necessarily require similarity. Some people prefer to be paired with peoplesimilar to them, but others may well like being paired with people different from them.

The elicitation of preference represents another layer of difficulty. There are several possibleapproaches to eliciting the preference for fellow passengers but each has its challenges: 1) directlyask users to identify the characteristics they prefer to see in their fellow passengers, such as profes-sion, hobby, personal interests, etc. 2) train a machine learning model with the characteristics ofpaired passengers in previous trips and their post-trip ratings, and 3) start by giving users optionsof possible fellow passengers, and train models based on their choices in each trip and their charac-teristics. There are two different types of challenges: first, passengers may not be able to articulatethe preferences, the descriptions may be ambiguous, and disparity may exist between what peoplesay and what they actually do, i.e., between stated preference and revealed preference (passengersmay not want to express their actual preference for various reasons); second, a deeper challengeis that certain preferences for fellow passengers may not be appropriate or respectable, such asdiscriminatory attitude and behavior. Such preferences can either be explicitly expressed in the ap-proach 1 or implicitly embedded in the past behavior and codified into machine learning algorithmsin approaches 2 and 3. There are important questions to be addressed. For example, what arethe boundaries between acceptable and unacceptable preferences? People may see less controversywhen gender is used as a preference factor for security reasons, but race as a preference factor isdefinitely unacceptable. Often there are factors that are acceptable by some but not by others. Who(or which institutions) shall have the authority to determine which preference or preference factoris respectable and which is not? Since the transportation network companies are the designers ofthe ridesharing platforms, whether and how they shall be regulated in this regard? There remainsa major gap in the development of both the social norms and the regulatory frameworks that canguide the ridesharing behavior and the associated system and policy design.

The complexity of preference structure and the process of eliciting preference for fellow passen-gers demand a thorough discussion on the behavioral, ethical, and institutional aspects, and arebeyond the scope of this paper. We identify this as a critical direction for future research.

Further research may also examine 1) the spatial and social heterogeneity of sharing preferences;2) the interaction between ridesharing and congestion; 3) the pricing of ridesharing services whenpreference is incorporated; and 4) the vehicle routing for preference-based ridesharing.

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3

Gender, Social Interaction, and MobilitySharing

3.1 Introduction

As an emerging mode of transportation, mobility sharing enriches people’s choices of traveling. Oneform of mobility sharing is ridesharing—–in a shared vehicle trip, a driver picks up and delivers morethan one passenger from their origins to their destinations based on route and time proximities.

Mobility sharing platforms like UberPool or Lyft Line implement this type of service with smartphones to collect trip requests and use real-time matching algorithms to pair fellow passengers.Based on the on-demand and real-time nature of the service, the matched passengers are mostlystrangers. However, the experience of mobility sharing with these strangers will impact people’ssatisfaction and further usage of mobility sharing.

With mobility sharing as a very unique social setting, the social interaction in this service be-tween fellow passengers is very different from that in other social settings, such as in public or privatespaces (meeting rooms, streets, public squares, living rooms, etc.) The nature of shared car ridesis impromptu, captive for a considerable duration, and remarkably more intimate—representing aunique juxtaposition of spontaneity and intensity [34].

While the social side of mobility sharing is not yet fully understood [16], there has been researchon the social interaction in other transportation modes. For example, the opportunities of being inthe same vehicle of public transit are considered as an invasion into personal spaces [35], leading tonegative utilities. In mobility sharing trips, we can expect more positive use of social interaction.[36] identified the important aspects of social interaction in mobility sharing services, including theclarity of the purpose of trip sharing, richness of user profiles, and visibility of other passengers’information, and advocated for better service design. However, empirical studies on the relationshipbetween social interaction, and mobility sharing usage and satisfaction are still lacking.

The sociodemographic characteristics of passengers, including gender, age, income, may impacttheir usage and satisfaction of mobility sharing services because these characteristics are not onlyassociated with their perception of the social interaction in the shared rides, but act as factors forco-producing the sharing experience together with the fellow passengers’ characteristics [10].

In this chapter, we will also combine the preference for the gender of fellow passengers with the

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preference-based matching model in Chapter 2. By comparing the matching results, we would liketo use gender as an example to understand the following question. If in a two-group preferencestructure, both group prefer to be paired with members of one of the two groups, how will theefficiency and preference trade-offs be different from with only same-group preference or indifferentpreference with group affiliation.

3.2 Literature Review

3.2.1 Gender and mobility sharing

Among the sociodemographic characteristics related to mobility sharing, gender has been recognizedas an important factor and is widely discussed [1, 37–39]. Nonetheless, with most research on thegender impact in mobility sharing relies on description and simple statistics without mechanismexplanation, the understanding is still limited.

[40] argued that “work on how mobility shapes gender has emphasized gender∗, to the neglectof mobility, whereas research on how gender shapes mobility has dealt with mobility in great detailand paid much less attention to gender.” In this chapter, we are going to focus on how genderimpacts mobility sharing, by investigating not only the gender difference, but trying to decomposethe gender difference through social interaction.

Many scholars have studied the impacts of gender on the usage, satisfaction, and experiencewith mobility sharing. [5] and [7] studied how gender impacts the mode choice of mobility sharing.[6] used a survey in California to show the gender difference in commuting stress, and the buffereffect of ridesharing on the commuting stress. [8] investigated the gender discrimination in mobilitysharing by requesting trips with Uber, Lyft, and Flywheel. [41] used a survey in Canada to showthe gender difference in the motivation of carpooling, but did not include social interaction to bea factor of motivation. To summarize, the prior research only surveyed and modeled the directimpact of gender on the usage and satisfaction with mobility sharing, but did not consider theindirect effect through social interaction in this relationship. Our paper is the first to conduct amediation analysis on the role of social interaction in the path between gender, and the usage andsatisfaction of mobility sharing.

3.2.2 Gender difference in social interaction

Moreover, scholars of various disciplines have explained the gender difference in social interactionof broader settings. [42] summarized that the gender difference in the social interaction in sharingeconomy is due to the conflicts regarding intimacy, privacy, comfort, and safety in these services.Psychologists have studied the processes of how gender difference evolves in social interaction. [43]studied the relationship between gender difference, leadership, and social interaction in four-persongroups by observation. They found that although all-male and all-female groups are equally likelyto develop hierarchical patterns of leadership, in mixed-gender groups males are five times morelikely to exercise leadership, showing a significant gender difference in social interaction styles.[44] studied the group interaction and influence styles of different genders in two-person dyads—very similar to a typical ridesharing composition—and found that people are less certain about

∗Later in the paper Hanson further illustrated this shaping as that ‘enforced immobility or denial of mobility isused to keep women in a subordinate position and to sustain traditional gender relations.’

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appropriate behaviors in mixed-gender interactions than interacting with people of their own gender.To summarize, in the current social context, while females and males are both less certain about theappropriate behaviors in mixed-gender interaction, they also have imbalanced influence styles andhierarchical patterns in social interaction—leading to possible gender difference in the perception ofsocial interactions in mobility sharing. However, how these differences change their mobility sharingusage and satisfaction is still to be investigated.

In this chapter, we focus on the role of fellow passenger interaction in shared rides, and investigatethe gender difference in it. We would like to answer the following three questions regarding therelationship of gender, social interaction, and usage and satisfaction with mobility sharing. 1) Doessocial interaction in mobility sharing impact the usage and satisfaction level with it? 2) Is theregender difference in considering social interaction as motivation or deterrent for mobility sharing?3) Is there gender difference in the usage and satisfaction with mobility sharing services? Besides,we will also examine the impact of user experiences—having user UberPool or Lyft Line before—onthe three sets of relationship. We use factor analysis and structural equation models to formulatethe triad of gender, social interaction, and mobility sharing.

Figure 3-1: Geographic distribution of respondents

3.3 Data and Methods

3.3.1 Survey summary

We conducted a survey on Mechanical Turk with the respondent selection criterion to be living inU.S. cities where Uber or Lyft is available, and collected 997 valid responses. The geographicaldistribution of respondents is shown in Figure 3-1—showing a similar pattern to the geographicdistribution of Mechanical Turks user [45]—and the distribution of the sociodemographic character-

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istics of the sample population is shown in Table 3.1. Most survey respondents were younger than 35(78%), were male (57%), and held an undergraduate or graduate degree (65%). Most respondentswere white (70%); Asian, black, and Hispanic respondents made up 9%, 8%, and 7% of the sample,respectively. The survey results were presented in a previous paper [29].

For this chapter, we used three sets of questions in the survey regarding sociodemographic char-acteristics, attitudes toward social interaction in mobility sharing services, and self-estimated usageand satisfaction level with the services of the respondents (Table 3.2). The variable notations used

Table 3.1: Descriptive statisticsVariable Count PercentageSample Size (N) 997Having used UP/LL Yes 752 75.4%

No 245 24.6%Age 17 or under 0 0.0%

18–25 265 26.6%26–30 307 30.8%31–35 204 20.5%36–45 160 16.0%46–55 43 4.3%56–64 18 1.8%60 or over 0 0.0%

Gender Male 565 56.7%Female 427 42.8%Other 5 0.5%

Education level Less than High School 0 0.0%High School / GED† 62 6.2%Some College 278 27.9%College Degree 493 49.4%Graduate Degree or Higher 164 16.4%

Income level Less than $30,000 215 21.6%$30,000–49,999 242 24.3%$50,000–74,999 244 24.5%$75,000–99,999 128 12.8%$100,000–149,999 107 10.7%$150,000–199,999 34 3.4%$199,000 or more 27 2.7%

Employment status Employed 795 79.7%Student 135 13.5%Unemployed 67 6.72%

Race/ethnicity White or Caucasian 702 70.4%Black or African American 88 8.8%Hispanic or Latino 74 7.4%Asian 91 9.1%Native American 10 1.0%Pacific Islander 4 0.4%Middle Eastern 2 0.2%Other 20 2.0%Prefer not to answer 6 0.6%

Marriage status Single 407 40.8%In a relationship 294 29.5%Married or in a domestic partnership 293 29.4%Other 3 0.3%

Having children living with you Yes 263 26.4%No 733 73.5%NA 1 0.1%

† General Educational Diploma

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Table 3.2: Survey questions and variable namesSociodemographicvariables

Indicators for social interaction asmotivations of mobility sharing

Usage andsatisfaction

Having usedUP/LL?AgeGenderEducation levelIncome levelEmploymentstatusRace/ethnicityMarriage/relationship statusHaving childrenliving with you

user

agegendereduincomeem.*†

re.*mr.*

havchd

When I choose to use / if I chose touse UP/LL‡ instead of other modes,it is...◦ ...because I enjoy meeting peoplefrom different social circles◦ ...because of the potentialnetworking opportunities withanother passenger◦ ...because of the potential to meetsomeone I am attracted to◦ ...because of the potential to makenew friends◦ ...because I enjoy making small talkwith new people◦ ...because I want to meet peopleheading to/coming from the sameevent as me

One of the reasons I DO NOT useUP/LL (more often) is that...◦ ...I cannot rate and see ratings ofother passengers◦ ...I cannot see the name, gender,and age of the other passenger◦ ...I cannot see a picture of the otherpassenger◦ ...I cannot indicate a preference notto interact with the other passenger◦ ...there are no clear norms ofinteraction◦ ...I am afraid to be paired with anunpleasant passenger◦ ...I prefer privacy in the back seat ofthe car

circl

netwk

atrct

frnds

smtlk

smevt

rtngs

pdemo

pictr

prefi

normi

unpls

prvcy

Number ofU/L trips inthe last 30days.Overall, howsatisfied areyou with U/L?

ntrips

satisf

†Employment status: em.employed, em.student, em.unemployed; Race/ethnicity: re.white, re.black, re.asian, re.hispanic;Marriage/relationship status: mr.single, mr.relation, mr.married‡U/L = Uber or Lyft; UP/LL = UberPool or Lyft Line

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in the later statistical models in this chapter are also listed besides each question for convenience.

The sociodemographic variables are binary (user history, gender, and having children living withyou), categorical (employment status, race/ethnicity, and marriage/relationship status), or ordinal(age, education level, and income level). The answers to the the questions regarding the attitudestoward social interaction and fellow passenger information (the second column of Table 3.2) are inseven–point Likert scales, from “strongly disagree” to “strongly agree.” Later in this chapter, wewill first examine the structure within these indicator questions for social interaction in mobilitysharing.

The self-estimated usage of mobility sharing service is measured as the number of Uber or Lyfttrips in the past 30 days, and the overall satisfaction levels with Uber and Lyft Services are measuredin ten–point Likert scales.

3.3.2 Model formulation

We would like to investigate whether there is gender difference in the usage and satisfaction withUber or Lyft, and whether having used the pooled service of them (user history), i.e., UberPool orLyft Line, exaggerates or diminishes this gender difference. Moreover, we would like to answer howgender impacts usage and satisfaction via the mediation of attitudes toward the social interactionin mobility sharing.

For convenience of discussion, we denoted the three sets of variables as X for sociodemographics,including gender and the interaction between gender and user history; Y for the usage (Y1) and sat-isfaction level (Y2) with mobility sharing services; and M for the attitudes toward social interactionin mobility sharing.

In our model, we first conducted factor analysis (FA) to reveal the structure of the social inter-action attitude indicators, from the responses to the social interaction questions in the survey, andthen combined them into more robust latent social interaction motivation factors. Factor analysisdoes not only reduce the number of variables by combining indicators into factors, but by groupingthe indicators containing relevant information into factors, increases the interpretability of eachfactor.

Structural equation modeling (SEM) is a modeling technique that can construct a system fora large number of observed and unobserved (latent) variables, and the direct and indirect effectsamong them. It has been used in travel behavior research since 1980s [46].

Using structural equation models, we tested the direct effect of gender on the usage and satisfac-tion with mobility sharing, and also the indirect effect between the two through social interactionfactors (Mj). Our hypothesized model is as shown in Equations (1–2), with the conjecture that theusage (Y1) and satisfaction (Y2) are at least partially impacted by the gender of passengers throughthe difference in the attitudes toward social interaction in mobility sharing, and both the direct andindirect effects of gender interact with user history.

Yi = di · gender + si · user + fi · gender · user +n∑j=1

bi jMj +

m∑k=1

lk · controlk + ϕi (3.1)

Mj = aj · gender + tj · user + ej · gender · user +m∑k=1

hk · controlk + ψj (3.2)

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In Equation (1), the direct effect of gender is indicated by di, and the impacts of social interactionattitudes are indicated by bi j , where j is the index for social interaction factors. In Equation (2), thegender difference in the attitudes toward social interaction is indicated by aj . Therefore, the indirecteffect of gender on the usage and satisfaction is indicated by the product of aj and bi j , denoted asci =

∑nj=1 ajbi j . To test how user history changes gender difference, we include the interaction term

between gender and user history. Similarly, we denote the coefficient of the interaction term as fi inthe direct effect, and gi =

∑nj=1 ejbi j for the indirect effect. Then, the total effect of gender on usage

and satisfaction of mobility sharing is ti = ci + di for the linear gender term, and ui = gi + fi for theinteraction term. In the equations, n is the number of social interaction factors, and ϕi and ψj arethe error terms. For clarity, in the equations we do not specify other sociodemographic variablesbut denote all of them as control variables (control1 to controlm), which we use to isolate the effectsof gender.

3.4 Findings

3.4.1 Factor analysis

2 4 6 8 10 12

01

23

45

Factor Number

eig

en

va

lue

s o

f p

rin

cip

al f

acto

rs

FA Actual Data FA Resampled Data

Figure 3-2: Parallel analysis for social interaction variables

To combine the social interaction indicators to more robust social interaction factors, we firstconducted exploratory factor analysis (EFA) with the responses to the social interaction questionswith oblique rotations. In the parallel analysis shown in Figure 3-2, the eigenvalue line for therandomly resampled data crosses the eivenvalue line of the actual data at the third factor. Wetested from two to four factors, balanced between the cumulative variance, and the explainabilityof factors, and determined to use three factors. The EFA results are summarized in Table 3.3.

Based on the EFA results we found that the social interaction indicators can be grouped intothree interpretable factors: Factor 1 is associated with the indicators on the agreement of positivesocial interactions as the motivation for using mobility sharing, including meeting people from dif-ferent social circles, potential networking opportunities, making new friends, having small talks,and meeting people coming from or going to same events; Factor 2 is associated with the indica-tors on considering the lack of information of fellow passengers as deterrents for using mobilitysharing, including not being able to see ratings, pictures, and social demographic information of

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Table 3.3: EFA result summaryFactor Loadings (only |values | > 0.1 are shown)Variable Factor1 Factor2 Factor3circl 0.874netwk 0.891atrct 0.714 0.130frnds 0.923smtlk 0.867smevt 0.850rtngs 0.616 0.205pdemo 0.975pictr 0.833prefi 0.203 0.624normi 0.729unpls 0.724prvcy -0.104 0.672SS loadings 4.411 2.082 1.980Proportion variance 0.339 0.160 0.152Cumulative variance 0.339 0.499 0.652Factor structure of social interaction

Factor1 Factor2 Factor3Factor1 1.000 0.150 -0.002Factor2 0.150 1.000 0.694Factor3 -0.002 0.694 1.000Test of the hypothesis that 3 factors are sufficient.χ2: 329.98 (df=42)p-value: 2.31 × 10−46

fellow passengers; Factor 3 is associated with indicators on the agreement of considering negativesocial interactions being deterrents for using mobility sharing, including not being able to indicatepreference, no clear form of social interaction, the possibility to be paired with unpleasant fellowpassengers, and preferring privacy in the back seat. In later models in this chapter, we denote thethree social interaction factors as possoc, psginf, and negsoc. The three factors accumulate for 65%of the variance in the social interaction indicators.

To validate the structure of social interaction factors, we then conducted a confirmatory factoranalysis (CFA), with results shown in Table 3.4. From the CFA fit indices, we can confirm thisstructure of the three social interaction factors. The covariances between positive social interactionand fellow passenger information, and between negative social interaction and fellow passenger in-formation are significant, indicating the non-independence for these latent factor pairs. Intuitively,before taking shared vehicle trips, fellow passenger information may imply either potential posi-tive social interactions, or potential negative social interactions. Therefore, in the later structuralequation model, we will keep this covariance structure among latent social interaction factors.

3.4.2 Structural equation modeling

Based on our hypothesized model in Equations (1–2) and factor analysis results, we propose thestructural equation model illustrated in Figure 3-3. In the model, the social interaction factorsare measured by the respective indicators; the social interaction factors; the usage and satisfactionvariables are regressed on sociodemographic variables; and the usage and satisfaction variables areregressed on both the sociodemographic variables and the social interaction factors. We keep thecovariances between positive social interaction and fellow passenger information, and between fellowpassenger information and negative social interaction based on the CFA results. We conducted the

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Table 3.4: CFA result summaryEstimate Std. Err. z-value p-value

Factor Loadingspossoc

circl 1.00+

netwk 1.03 0.02 41.84 .000atrct 0.79 0.03 27.26 .000frnds 1.06 0.02 43.68 .000smtlk 1.02 0.03 38.92 .000smevt 1.00 0.03 37.04 .000

psginfrtngs 1.00+

pdemo 1.24 0.04 31.63 .000pictr 1.13 0.04 30.49 .000

negsocprefi 1.00+

normi 0.93 0.04 24.87 .000unpls 0.90 0.04 21.66 .000prvcy 0.73 0.04 18.29 .000

Residual Variancescircl 0.63 0.04 18.03 .000

netwk 0.59 0.03 17.48 .000atrct 1.40 0.07 21.14 .000frnds 0.51 0.03 16.36 .000smtlk 0.79 0.04 18.76 .000smevt 0.92 0.05 19.36 .000rtngs 1.16 0.06 19.22 .000

pdemo 0.48 0.05 10.53 .000pictr 0.65 0.05 14.28 .000prefi 1.05 0.07 15.21 .000

normi 1.06 0.07 16.23 .000unpls 1.74 0.09 18.93 .000prvcy 1.93 0.09 20.32 .000

Latent Variancespossoc 2.28 0.13 17.67 .000psginf 1.78 0.13 14.24 .000negsoc 1.95 0.14 14.37 .000

Latent Covariancespossoc∼psginf 0.29 0.07 4.22 .000possoc∼negsoc 0.05 0.08 0.71 .476psginf∼negsoc 1.43 0.09 15.17 .000Fit Indices

χ2 492.73 (df=62) .000CFI 0.95TLI 0.94

RMSEA 0.08+Fixed parameter

model using R package lavaan [47], and summarized the results in Table 3.5. The fit indices at theend of the table (CFI=0.94, TLI=0.91, and RMSEA=0.05) suggest good validity of the model.

Direct effect

From the SEM results, we can find that after controlling for other sociodemographic variables, onaverage females reported 0.3 more trip in 30 days than males (d1), and were more satisfied withmobility sharing than males (d2), but the usage and satisfaction differences are not statistically

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SociodemographicVariables (X)◦ . . .◦ gender◦ gender·user◦ . . .

Positive SocialInteraction (M1)◦ possoc

Positive Social Interac-tion Indicators◦ circl ◦ netwk◦ atrct ◦ frnds◦ smtlk ◦ smevt

Fellow Passen-ger Information(M2)◦ psginf

Fellow Passenger Infor-mation Indicators◦ rtngs◦ pdemo◦ pictr

Negative SocialInteraction (M3)◦ negsoc

Negative Social Inter-action Indicators◦ prefi ◦ normi◦ unpls◦ prvcy

Usage (Y1): ntripsSatisfaction (Y2): sat-isf

a1, e1a2, e2 a3, e3 bi1

bi2 bi3

di, fi

Social Interaction Factors

Figure 3-3: SEM diagram (arrows for residual variances and covariances between exogenous variablesare omitted)

significant.

Indirect effect

The indirect effects of gender on usage and satisfaction are significant. There are significant genderdifferences in all three social interaction factors (a1–a3), indicating that females and males havesignificantly different agreement levels on considering the three factors being their motivation ordeterrents for using mobility sharing services.

Specifically, females agree less on that the potential of positive social interaction is the reasonfor them to use mobility sharing (a1), and agree more on that lack of fellow passenger information(a2) and potential negative social interaction (a3) are deterrents for them to use mobility sharing.

Two of the three social interaction factors—positive social interaction and negative social interaction—are significantly associated with the usage and satisfaction levels (b11, b13, b21, and b23). To sum-marize, the more people agree on that potential positive social interaction is their motivation forusing mobility sharing, the more they use Uber or Lyft, and they are more satisfied with the service.Consistently, the more people agree on that potential negative social interaction is a deterrent forthem to use mobility sharing, the less they use it, and the less satisfied they are with the service. Incontrast to the attitudes toward social interactions, the lack of fellow passenger information has nosignificant relationship with usage or satisfaction (b12 and b22). However, as the fellow passengerinformation factor is correlated with both the positive social interaction factor, and the negativesocial interaction factor, this insignificance may be a result of the internal offset between the positiveand the negative implications of fellow passenger information.

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Table 3.5: SEM result summary

Estimate p-valueDirect effect (X → Y)ntripsgender (d1) 0.30 .594user 1.50 .001gender.user ( f1) 0.27 .677satisfgender (d2) 0.18 .370user -0.08 .621gender.user ( f2) 0.04 .866

Indirect effect part I (M → Y)ntrippossoc (b11) 0.36 .000psginf (b12) 0.12 .559negsoc (b13) -0.23 .270satisfpossoc (b21) 0.12 .001psginf (b22) 0.01 .847negsoc (b23) -0.18 .013Part II (X → M)possocgender (a1) -0.69 .000user -0.34 .025gender.user (e1) 0.45 .040psginfgender (a2) 0.59 .001user 0.18 .193gender.user (e2) -0.37 .073negsocgender (a3) 0.53 .008user -0.10 .502gender.user (e3) -0.17 .440Combined (X → M → Y)ntripc1 -0.30 .023g1 0.16 .190satisfc2 -0.17 .002g2 0.08 .133

Total effect (X → Y)ntript1 -0.01 .990u1 0.43 .509satisft2 -0.01 .973u2 0.12 .597Fit Indicesχ2 844.33 .000

(df=233)CFI 0.94TLI 0.91RMSEA 0.05

As products of the gender difference in the social interaction factors in mobility sharing (aj),and the relationship between social interaction factors and usage and satisfaction (bi j), the indirecteffects of gender on usage and satisfaction are significant (c1 and c2). Indirectly, after controllingfor other sociodemographic variables, females on average took 0.32 less Uber or Lyft trips, and wereless satisfied with the services.

Total effect

The total effect is the addition of direct and indirect effects. There is no significant gender differencein the usage and satisfaction of mobility sharing services, and the coefficients are almost zero afteradding the coefficients of direct and indirect effects (t1 and t2).

The impact of user history

We then examined the interaction terms between gender and user history on the aforementionedrelationship for the impact of user history on the gender differences.

For the direct effect, the interaction term between gender and user history (gender.user: binaryindicating whether female users), user history did not significantly change the gender differencein usage and satisfaction levels, with additional 0.27 trip and 0.04–level (out of a ten–point Likertscale) satisfaction improvement on average ( f1 and f2). It is worth noting that the user history is for

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UberPool and Lyft Line, while the number of trips and satisfaction are for all Uber and Lyft servicesincluding also regular Uber and Lyft trips. Therefore, the direct effect of having used UberPool andLyft Line on the overall usage and satisfaction of Uber and Lyft is positive but not significant.

For considering social interactions as motivations, the gender differences are mitigated by userhistory (e1 and e2)—after using UberPool or Lyft Line, the gender differences in the attitudes towardsocial interaction in mobility sharing were smaller. Consequently, the user history factor also showedindirect mitigation effects on the gender difference in usage and satisfaction (g1 and g2). Speficically,having used UberPool or Lyft Line slightly reduced the indirect gender effect on the usage andsatisfaction with Uber or Lyft. Together, the total effects of the gender–user history interaction arealso insignificant (u1 and u2).

To summarize, being users of UberPool or Lyft Line did not directly impact the gender differenceof mobility sharing usage and satisfaction. Nonetheless, it significantly reduced the gender differencein considering social interaction as motivations for mobility sharing, and thus mitigated the indirectgender effect on usage and satisfaction. However, after summarizing the indirect and indirect effects,being users did not significantly change the total gender difference in usage and satisfaction.

The impact of other sociodemographic variables

In the model, we included other sociodemographic variables as control variables to isolate theeffects of gender. In Table B.1 of the Appendix we can find several other sociodemographic factorssignificantly associated with the social interaction factors, and the usage and satisfaction withmobility sharing.

For considering positive social interaction to be a motivation for using mobility sharing, peoplewho are more educated, and people who have children living with them disagree more on thispoint. In addition, people who have used UberPool or Lyft Line significantly agree more on thatthe potential positive social interaction is a motivation for them to use it. Although this could bedue to self-selection bias—people enjoying social interaction would be more likely to have tried theshared services, this result could also imply that after taking the shared service, people are moreoptimistic about the potential of positive social interactions. The causality needs further studieswith experiments.

3.5 Matching with Gender Preference

In this section, the preference-based matching algorithm proposed in Chapter 2, and gender pref-erence for fellow passengers from the survey are combined to understand if gender is to be used forfellow passenger matching, what are the efficiency and preference trade-offs.

3.5.1 Combining trip data and gender preference

The preferences of both males and females for the gender of fellow passengers are shown in Table3.6. Based on the results, the majority of males and females are indifferent with the gender of fellowpassengers. However, for the people how are not indifferent, both males and females have higherproportions of respondents prefer to be paired with females. This asymmetry is different from allof the four scenarios used in Section 2.3.4. In the asymmetric scenarios in the two-group scenarios

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in Section 2.3.4, a group either prefers their ingroup members or are indifferent, while in the surveyresults both groups prefer to be matched one of the two.

Table 3.6: Preference of Women and Men for the Gender of Fellow PassengersPreference Women MenOnly agree to be paired with women 17.8% 5.6%Prefer to be paired with women but agree to be paired with men 34.4% 18.4%Be indifferent to the gender of fellow passenger 46.5% 73.4%Prefer to be paired with men but agree to be paired with women 1.3% 2.4%Only agree to be paired with men 0% 0.2%

Based on the survey results, another scenario (S5) was added to compare with the results of thefour two-group scenarios. Since the locations of respondents were not known to us, we assumed that1) females and males are both evenly distributed spatially, 2) the preference for the gender of fellowpassenger is also independent with spatial locations. Thus, for each trip in the Manhattan datasetused in Chapter 2, the traveler of each trip was first randomly assigned a gender, and then giventhe gender, the preference for the gender of fellow passenger was also randomly drawn according tothe distribution in Table 3.6.

When constructing the shareability network, for the people who “only agree” to be paired witha specific gender, the edges between the only-agree node and the nodes that are not assignedthe corresponding gender were removed. Then, the generation of preference orders was conductedsimilarly to as in the preference generation for the two-group scenarios in Chapter 2. For example, ifa traveler is assigned to be “prefer to be paired with women but agree to be paired with men,” all thenodes connected to it which have been assigned as females will be ranked before all nodes assignedas males. The preference orders within male and female neighboring nodes are also randomlygenerated.

3.5.2 Matching results

Table 3.7: Matching results with gender preference compared with four two-group ScenariosScenario S0 S1 S2 S3 S4 S5Avg. number ofmatched trips

297,050(98.6%)

294,494(97.7%)

296,040(98.2%)

294,629(97.7%)

296,530(98.4%)

293,810(97.5%)

Veh-min per trip 7.77 7.77 7.77 7.77 7.77 7.78Paired with passengerin the same group

n.a.G1: 97.6%G2: 97.6%

G1: 97.6%G2: 96.1%

G1: 94.4%G2: 98.5%

G1: 94.3%G2: 97.4%

Female: 66.2%Male: 63.9%

Table 3.7 and Figure 3-4 summarize the matching results of efficiency measures and preferencemeasures for all the five two-group scenarios and the one-group random preference scenario asa benchmark. With preference based-matching, the preference structures in five scenarios lead toroughly the same average trip distance. However, because the numbers of shareable trips are smallerfor a subgroup of people—the ones who only accept being paired with one gender—the number ofactually matched trips are smaller. The is can also be observed from the shape of curve in Figure3-4. Compared to other curves, the curve of S5 stretches shorter, indicating the number of shareabletrips being smaller in the gender preference scenario. The low rates of same-group pairs in the genderscenario (66.2%, and 63.9%) also suggests how the symmetry of preference can impact the matchingoutcome. Compared with in other two-group scenarios, where people either prefer fellow passenger

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0 100 200 300 400 500 600 700 800 900

Number of shareable trips

0

2

4

6

8

10

12

14

16

Ra

nk o

f p

aire

d p

asse

ng

er

in p

refe

ren

ce

lis

t

Scenario 0

Scenario 1

Scenario 2

Scenario 3

Scenario 4

Scenario 5

Figure 3-4: Average ranking of paired fellow passenger with gender preference compared with fourtwo-group scenarios

from their own groups or are indifferent, this result exemplifies that if two groups both prefer to bepaired with fellow passengers from one group, the “competing” for the members in this group willresult in the low ingroup matching rates for both groups. Another result of this “competition” forfemale fellow passengers is that, as shown in Figure 3-4, the average fellow passenger ranking in thegender preference scenario is the worst among all scenarios.

This section uses gender as an example to show if in a two-group scenario, both groups preferone of the two, compared to with only same-group preference or indifference with groups, there isnot only efficiency trade-offs of lower matching rate, on the preference side, the average rankingof paired fellow passenger will also be worse. With more empirical analysis, this finding may begeneralized to other two-group factors that has similar asymmetry.

3.6 Conclusion

In this chapter, we answered three questions regarding the relationship between gender, socialinteraction, and the usage and satisfaction of mobility sharing services.

Our first question is whether social interaction impacts the usage and satisfaction of mobilitysharing. We found that positive social interaction significantly increased the usage and satisfactionof mobility sharing services, and negative social interaction significantly reduced the two variables.In contrast to social interaction factors, the lack of fellow passenger information before trips is notconsidered significantly as a deterrent to use mobility sharing.

Second, we tested whether there is significant gender difference in the social interaction motiva-tion factors, and found that the gender differences were significant for the agreement on consideringpositive social interaction being motivation for using mobility sharing, and negative social interac-tion and lack of fellow passenger information being deterrents for using it.

Third, we conducted mediation analysis with social interaction factors on the relationship be-tween gender, and the usage and satisfaction of mobility sharing. Our finding is that throughmediation of social interaction, gender had a significant indirect effect on usage and satisfaction.However, this indirect effect was canceled out by the direct effect, and the total gender effect on

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mobility sharing usage and satisfaction was not significant. Therefore, social interaction factorsplayed a full mediation role between gender, and usage and satisfaction.

We also tested the effect of user history on the three sets of relationships, and found insignif-icance in direct effect; user history reducing gender difference in considering social interaction asmotivations, and thus mitigating indirect effects; and insignificance in the total effect.

Moreover, we also combined the gender preference for fellow passengers with the preference-based matching algorithms in Chapter 2 to understand how this asymmetry of preference changesthe matching outcome. We found that due to the “competition” effect of both groups preferring oneof the two, both the matching rate and rankings of matched fellow passengers were worse comparedwith other two-group scenarios.

Our results provide a framework for understanding the relationship between gender, social in-teraction, and mobility sharing, and further provide policy and practice implications. As socialinteraction factors are significantly associated with mobility sharing usage and satisfaction, theimprovement on the social interaction experience through better service design or instruction onsocial interaction styles will likely increase the usage and satisfaction of mobility sharing services.Moreover, considering the different agreement on social interaction being motivation or deterrentsfor mobility sharing, especially the different agreement level between genders, the fellow passengermatching algorithm could match people with similar expectations of interaction during shared rides.

There are many critical questions untestable due the limitation of data and methods in thischapter. We list them here for future research. 1) The mediation effect of social interaction issignificant to the usage and satisfaction of mobility sharing, but is offset by the direct effect in thetotal effect. As an inference, if the social interaction experience is improved in mobility sharing,there is a potential for increasing the usage and satisfaction of it. However, this conjecture needsfurther experiments for validation. For example, by matching people by their preference for fellowpassengers, we can improve the experience of social interaction. If we can observe stronger indirecteffect after the preference-based matching, the causal relationship can be better proved. 2) Theimpact of fellow passenger information is not significant on usage and satisfaction, but it is highlycorrelated with the social interaction factors. As fellow passenger information may imply bothpositive and negative social interaction, the insignificance may just be a result of the duality. Furtherresearch may investigate in what fellow passenger information may help identify potential positivesocial interaction, and what for negative social interaction. 3) As we only conducted ex postfacto analysis, we cannot eliminate the effects of self-selection, i.e., whether the gender differenceamong users is smaller simply because the people who take mobility sharing services are the oneswho have smaller gender differences in the social interaction considerations. In future research, byexperiments of asking randomly selected people to take mobility sharing trips, we can have a betterunderstanding of whether having used the service can actually change people’s attitudes toward thesocial interaction in mobility sharing.

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4

Congestion–Sensitive Mobility Sharingwith Time Flexibility†

4.1 Introduction

The objective of mobility sharing is to reduce the number of circulating vehicles while satisfyingthe same level of travel demand. When two or more individuals decide to share a car to traveltogether, either for short or long distances, mobility sharing is called ridesharing, and it is moreeconomical and ecological compared to driving alone [48, 49]. Researchers have proposed differentstrategies to match vehicle trips, based on both efficiency and social objectives. Efficiency objectivesinclude minimizing total vehicle-miles, maximizing total number of shared trips, and minimizingtotal vehicle travel time or total delay [1–3, 12]. Social objectives include stable preference for fellowpassengers and maximizing total mingling time of different social groups [22, 23, 34, 50].

An individual vehicle trip has both spatial and temporal constraints. Spatially, a trip is con-strained by an origin and a destination—a traveler may choose different routes, but has to startfrom the origin and end at the destination.

The temporal dimension is often constrained in a more flexible way—it is very unlikely thata trip needs to start or end at a very precise time point. Thus, the temporal constraint is moreoften represented as a time interval, with an earliest possible departure time and a latest possiblearrival time. Temporal flexibility can have many different causes, such as the uncertainty of trafficconditions, parking availability, and the time flexibility of the activity linked to a trip [51]. Forexample, work schedule flexibility can make the departure time of commute trips flexible for asmuch as a few hours, while non-work trips can be even more flexible if a trip does not have to bemade during a certain time period. Hendrickson and Plank considered the departure time changein transportation infrastructure disruption as an indicator for time flexibility, and found that worktrips were 19 minutes earlier during road construction in Pittsburgh [51], reflecting a time flexibilityof at least the same level due to the flexibility coming from travel time uncertainty. Erdoğanet al. conducted a survey among the students and faculty/staff of the University of Maryland, andfound that only for 9.6% of students and 8.5% of faculty/staff “constrained and irregular schedules,”i.e., inflexible travel time, are barriers for them to share rides. In a travel survey conducted in

†This chapter is joint work with Xiaotong Guo, Han Qiu, and M.Elena Renda. X.Guo and H.Qiu helped withthe programming of optimization; M.E.Renda helped with writing the last section and commented on the chapter.

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Pisa, Italy, it was found that among 1,948 people who reported their time flexibility for morningcommute trips, 15% reported that their morning commute time was “completely flexible”—mostof them are university professors, researchers, managers, and freelancers [53]. Besides, 70% ofrespondents reported more detailed time flexibility in travel time, for which the average flexibilitywas 38 minutes.

The time flexibility of vehicle trips, especially driving-alone trips, provides a bigger potential formobility sharing. In a city, if by using the flexibility information more originally driving-alone tripscould be matched into ridesharing trips, there is a potential to reduce congestion. This could beachieved by a ridesharing matching system collecting the trip flexibility time windows rather thanprecise departure times. For example, if a traveler can submit to the system that “I am going todrive/be a passenger from point A to point B between 8:00 a.m. to 9:00 a.m.,” the trip has a muchhigher probability to be paired than if the traveler only submits “I am leaving at 8:00 a.m.” In otherwords, by combining mobility sharing and time flexibility, there is a bigger chance of making moreridesharing matches, and reducing more cars on the road than with mobility sharing alone.

Looking from the other direction, in a road network, when more vehicle trips are shared, thetraffic is lighter. The improved traffic condition will then free more space for ridesharing opportuni-ties. Using the same example as in the last paragraph, if the driver cannot even finish his/her owntrip between 8:00 a.m. and 9:00 a.m. due to congestion, the sharing becomes impossible. If trafficcondition improves, there will be opportunities for him/her to pick up and drop off a passenger witha little detour time but still be on time before 9:00 a.m.—more ridesharing opportunities becomeavailable when traffic condition improves. Therefore, ridesharing and improving traffic conditionsform a positive feedback loop. Nonetheless, the traffic improvement will only be non-trivial if thereis a sufficient proportion of trips to be shareable, or willing to share.

To summarize, incorporating time flexibility in mobility sharing will increase the possibility ofvehicle trips being matched, and further reduce the number of cars on the road compared to thespecific-time sharing. The improved traffic will in turn free up more opportunities for mobilitysharing—forming a positive feedback loop, especially when there is a sufficient proportion of peoplewho currently drive but would like to share their trips as passengers.

In this chapter, to understand how time flexibility can be included in mobility sharing to improvethe traffic in a city, we propose an optimization model which minimizes the total vehicle travel time,with variables including ridesharing matching, departure time assignment within each traveler’s timeflexibility, and route assignment. In this model, we also include the proportion of shareable trips ina city as a system parameter to understand how the the impact differs when a city can encouragemore people to share rides. We also experiment with two different route assignment strategies, userequilibrium (UE) and system optimum (SO), to understand the difference between two scenarios:1) when drivers only consider the shortest-time routes for themselves, and 2) when drivers canfollow the system routing instruction, which can lead to the minimum total vehicle time of the city.While the second scenario is non-realistic by itself, it provides a speculation for the future when theautonomous vehicles are deployed, and comply with system route assignment.

To give a real-world example, this model can be used by a city to improve the morning commutetraffic. With a platform to collect the morning commute trip information in advance, including thetrip origins and destinations, and the earliest departure time and latest arrival time, our model canprovide a matching, driver/passenger role, and departure time assignment. From our simulation,the traffic improvement is significant even when only 10% of the trips are shareable, as shown laterin the results.

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Time flexibility is related to but different from detour time in a shared vehicle trip. Detourtime is the additional travel time that a traveler has to take as a result of taking ridesharing—inmost cases it is a positive number but can also be zero if one trip is completely on the same routewith the other; while time flexibility of a trip is the range between its earliest departure time, andthe latest arrival time, and is associated with both the travel time uncertainty, and the scheduleflexibility of activities at the trip’s origin and destination. In this chapter, we are only focusing onthe time flexibility of shared vehicle trips, but it is worth noting that time flexibility is an intrinsicproperty of all types of trips, including non-shared trips and even non-motorized trips.

4.2 Literature Review

As discussed in the Introduction, the relationship between the three components of our model—time flexibility, ridesharing matching and traffic models are not unidirectional, and there has beenliterature on the relationship between any two of the three components. Before starting formulatingour model, we would like to review the existing models that discuss the nodes and the three edgesof the triangular graph (Figure 4-1), which sketches the structure of our model in this chapter.

RidesharingMatching

TimeFlexibility

TrafficAssignment

Figure 4-1: Diagram of the major components in this chapter

4.2.1 Time flexibility

The time flexibility of trips has been defined in many ways including at least two connotations: 1)the extra trip duration time planned to buffer the uncertainty of traffic [51]; 2) the time flexibilityof departure and arrival time associated with the schedule flexibility of the events at the trip originor destination [54], which may be related to the properties of trips, including departure time andtrip purposes, and also travelers’ sociodemographic information. The first definition is related topeople’s incapability of predicting travel time before the trips actually happen—although the traveltime estimates are much more precise nowadays with the help of smart phone map services—theyeither leave earlier, or can accept longer duration of travel time and thus arrive later. The seconddefinition is more discussed in the activity-based travel demand models [54], and employs the ‘time–space prisms’ from time geography [55]. In the time–space prism model, a person only needs to beat a certain place during a certain time interval to conduct an activity, and the trip is flexible aslong as it can connect the people between two subsequent activities.

To combine the two definitions, we can use an earliest departure time, and a latest arrival timeto delineate the feasible time range of a trip. In this chapter, we consider this range to be a hardconstraint, i.e., a trip can only start at its origin after the earliest departure time, and needs toarrive at its destination before the latest arrival time.

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Scholars have also studied time flexibility as a “soft constraint”—a traveler can start a trip atany time point, but the actual arrival time will result in different utilities, by comparing with amost “preferred” arrival time under a non-linear loss-averse function [51, 56]. In this time flexibilityframework, both early arrivals and late arrivals result in negative utility. However, in theory itis difficult to combine the utility-based time flexibility model with ridesharing matching as 1) theinter-personal utility comparison is difficult, and 2) the exchange rate between the loss of utilityand the gain in vehicle travel time saving is hard to estimate. Therefore, in this chapter we will stilluse time flexibility as a hard constraint, and only use it to delineate the feasibility of whether tworidesharing trips can be matched.

4.2.2 Time flexibility and mobility sharing

Before the on-demand mobility sharing, time flexibility had been used as an optimization constraintin the Dial-a-Ride Problem (DARP) [57]—a model in the setting of which a fleet of on-call vehiclespick up and deliver passengers as a means of public transit. In the DARP model, the time flexibilityis represented as the desired, earliest, and latest pickup/delivery time, with an objective to minimizethe deviation of the actual pickup/delivery times from the desired ones, but still within the earliestand latest range. The DARP does not assume some travelers to act as drivers and to pick up others,but computes “matching” strictly between the vehicles of the service provider, and the people whorequest only to be passengers. Therefore, the structure of the model is different from the setting inour model—pairing within one group of travelers, who can either be drivers or passengers and thusreduce the number of cars needed.

Time flexibility has also been used in the on-demand mobility sharing models. In a recent paper,Štiglic et al. proposed a model to include time flexibility with ridesharing to maximize the totalnumber of shared trips [58]. In their model, several assumptions were made: 1) the travel speeds ofall vehicles are assumed to be uniform and constant, and are not a function of local traffic; 2) driversand passengers are assumed to be from two distinct groups, and the roles are fixed throughout thematching; 3) only two simplified scenarios—a one-dimensional corridor and a rectangle plain witha highway in the middle were considered. In our model, we would like to relax all these threeassumptions. For the first assumption, as discussed in the Introduction, traffic and mobility sharingcan form a positive feedback loop. Therefore, to consider traffic as static may underestimate thepotential for sharing. For the second assumption, to fix the driver/passenger role at the beginningmay also underestimate the vehicle travel time reduction since in some occasions the reversion ofthe driver and passenger could increase the overall sharing opportunities and, in a shared trip, maylead to bigger vehicle time saving. For the model application, we will apply our model to a roadnetwork to illustrate the improvement of overall traffic with time flexibility in the mobility sharingmodel.

Long et al. considered the time uncertainty in the ridesharing model [59], which is in two aspectsrelated to our model: 1) the authors also relaxed the assumption of fixed travel time, but assumedit to be stochastic, following given distributions; 2) there is also associated disutility related to earlyor late arrivals, as framed in [51, 56]. However, the authors did not discuss the interaction betweenridesharing and travel time reduction. Besides, as discussed, time uncertainty is only part of thecauses of time flexibility, and we would like to enlarge the discussion to include all types of timeflexibility.

To summarize, in existing literature, when time flexibility is used in mobility sharing matching,the travel time is either assumed to be static, or from a given distribution. In this chapter, we

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will formulate the relationship between ridesharing matching and the aggregated travel demand oneach road link, and thus bridge the travel time estimation, from traffic assignment models, intoridesharing matching models.

4.2.3 Mobility sharing and traffic

The relationship between mobility sharing and traffic has also been widely studied. Alexander andGonzález used the Call Detailed Records (CDR) of the Greater Boston Area to estimate the traveldemand, implemented the aggregate travel demand into a traffic assignment model, and analyzedthe impact of ridesharing on congestion in different mode share scenarios, represented as the differentadoption rates of ridesharing. They found that under moderate to high adoption rates of ridesharing,there are noticeable decreases on congested travel times [60]. Xu et al. also studied the relationshipbetween ridesharing and traffic congestion by constructing the traffic assignment model with threeinterchangeable groups of travelers—solo drivers, ridesharing drivers, and ridesharing passengers[61, 62]. The roles are interchangeable by price incentives. They modeled the relationship betweenridesharing price and congestion, and solved the optimal price for ridesharing.

In addtion to simulation and optimization models, scholars also emprically examined the impactof ridesharing on traffic congestion. Li et al. combined urban mobility reports and the entry timeof Uber in different cities to test the impact of on-demand ridesharing service on the congestion ofcities, and found that after entering an urban area, ridesharing services significantly reduce trafficcongestion time, congestion costs, and excessive fuel consumption [63].

While the scholars have both in theory and empirically found that mobility sharing can reducecongestion, in existing research when combining traffic assignment models with mobility sharing,the trips are aggregated and the number of matched passengers is also estimated at the aggregatelevel. Therefore, the individualized traveler matching cannot be directly output from the models.In our model, we will propose an approach to combine the individualized matching and the trafficassignment model, and provide an operationalizable output.

4.2.4 Time flexibility and traffic

Time flexibility of trips has been used to improve traffic by the means of congestion charging, whichhas been implemented in Singapore, London, and Milan. Scholars have empirically investigatedthe impact of congestion charging on urban traffic [64–66]. Saleh and Farrell used discrete choicemodels to investigate the relationship between departure time flexibility, congestion charging price,trip purposes, and the sociodemographics of travelers. They found that travel time flexibility isassociated with the commitment of the activity, even with congestion charging. With this result, itis more straightforward to directly operate with people’s time flexibility in order to improve traffic,and thus to understanding the relationship between the two becomes a key question.

The research on the relationship between time flexibility and traffic are mostly empirical in-vestigations at the aggregate level, and an active assignment of departure time based on the timeflexibility of each individual has not yet been combined with traffic assignment models, especiallya model that combines ridesharing matching, driver/passenger role designation, departure timeassignment, and traffic assignment.

In this section we have summarized the development of the models considering any two of

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the three aspects among time flexibility, ridesharing matching, and traffic assignment. In [59]the authors categorized the mobility sharing models to be flow-based and individual-based. Thecategorization can be used to explain the lack of discussion on a comprehensive model on all thethree components. The flow-based mobility sharing models have a natural capacity to include atraffic assignment component. But as the trips are aggregated into flows, it is then difficult toincorporate the individualized matching, as well as individual time flexibility. As for the individual-based mobility sharing model, the individual time flexibility is easy to include. Nonetheless, it isthen difficult to bundle the individual trips into flows for a systematic traffic assignment.

In the next section, we will propose a model including all three components as shown in Figure4-1, and formulate the relationship among them.

4.3 Model Formulation

We will start by discussing the traffic assignment model, how the traffic assignment model connectsto travel demand, and how the demand is characterized by ridesharing matching, driver/ passengerrole designation, and trip departure time assignment. Our overall objective is to minimize the totalvehicle travel time in the system in a certain time period. The notation used in this chapter is listedin Table 4.1.

For our model, to simplify problem formulation, we only consider ridesharing of two trips. Wealso assume that there is no induced demand if traffic condition changes, including both new traveldemand, or mode shift from public transit, walking, or biking.

4.3.1 Traffic assignment

Consider a road network given as a graph G(L ,A ), where L is the set of locations, and A the setof links (road segments connecting locations). Empirically scholars have developed volume–traveltime functions sa(va) for any given link a ∈ A , which defines how much delay will be added to eachvehicle as traffic volume increases [68]. For example, a commonly used function is the BPR (Bureauof Public Roads) function [62, 68, 69]

sa(va) = Ba

(1 + γ

(va

Ca

)β), (4.1)

where sa(va) is the travel time on link a with traffic volume va, Ba is the free-flow travel time, Ca

is the capacity of the link, and γ and β are empirical coefficients. Because s is a given function, thetravel time of a trip is determined only by its route choice and the traffic volume v on all the linksof this route. Let Ri j represent the set of all possible routes that connect node i and j. If we dividethe study time period T into smaller periods, we can denote the minimal travel time between i andj during time interval t as

δti j(vt )∆= min

r ∈Ri j

∑a∈r

sa(vta). (4.2)

It is worth noting that with both UE and SO traffic assignments, the travel time is only determinedby the traffic volume of each link. Therefore, we can always write the relationship as δ = δ(v).

In existing literature, given travel demand within a network, [70] considered two possible traffic

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Table 4.1: Notations used in this chapterFormulationG(L ,A ) Road network G with node set L representing locations, and link set A

representing road segmentsi, j ∈ L , a ∈ A Any two nodes in node set, and any link in link setsa Travel time on link aBa,Ca Free-flow travel time and capacity of link at ∈ T Time period index in the set of time period indexesvta, v

t, v Traffic volume of link a during time period t, volume vector of all linksduring time period t, and volume matrix of all links during all time periods

r ∈ Ri j Any route in the route set between nodes i and jδti j, δ

t, δ Equilibrium travel time from node i to node j during time period t, matrixof equilibrium travel time between all node pairs during time period t, andthree-dimensional matrix of equilibrium travel time between all node pairsduring all time periods

δ̃, δ̂ The non-sharing travel time and the expected travel timexi j, x Travel demand between node i and j and the travel demand matrixp, q ∈ X Any two trips in trip setOp,Dp The origin and destination of trip pOi,Di The set of trips whose origin or destination is ifpi, gpi Indicator functions for whether node i is the origin ( f ) or destination (g) of

trip pσpq,σ Binary variable for whether trips p and q are shared, and the sharing matrixπpq, π Binary variable indicating who is the driver between p and q, and the driver

assignment matrixτp, τ Departure time of trip p, and the departure time vector of all tripsξp Drop-off time of trip p if p is a passengerup, lp Earliest departure time, and latest arrival time of trip pData Synthesis∆ f The flexibility length parameter in data generation` Travel time discount ratioTd The set of possible departure time after time discretizationApproximationW Maximum delay time for driverstminpq Minimum travel time for trip pair (p, q) giving fixed travel timeτ∗pq ∈ G The best departure time of drivers for any trip pairs (p, q)M ,M ∗, M̃ The set of feasible trip pairs, the set of optimal trip pairs and the set of

fixed trip pairs

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assignment criteria: a system optimal condition (SO), and a user-equilibrium condition (UE) todetermine the volume on each link [70]. Under the SO condition total vehicle travel time is mini-mized, while under the UE condition, no user could be better off by switching route, meaning thattravel time is the same for any route between two nodes. For Eq. (4.2), if traffic volumes are derivedusing the UE criterion—a user will switch route if there exists another route with shorter traveltime—users traveling along the same origin-destination pair (OD) will end up with the same traveltime on different routes. In this case, the min sign could be canceled with user-equilibrium trafficflows v.

Because the SO assignment requires all vehicles to follow the routing instruction from a central-ized system, it is not realistic a real-world case including human drivers and passengers who onlyconsider their own travel times. However, if we look into the future and think about a scenariowhere all vehicles are autonomous and receive routing instruction to minimize a city’s total vehicletravel time, the SO assignment would be worthy of investigating and comparing the performancewith UE.

4.3.2 Objective function

We would like to minimize the total vehicle travel time in a city during a time period, e.g., themorning peak hours of a certain day, and we have divided the time period into a series of timeintervals labeled as t ∈ T so that we can spread travel demand among these time periods bymaking use of time flexibility information, which we will discuss in a later section.

We limit our discussion to driving-alone trips, so we consider trips and travelers to be equivalent.Let p and q be any two vehicle trips in the trip set X , we denote σpq ∈ {0, 1} as whether p and qshare or not, πpq ∈ {0, 1} as a binary variable denoting whether p or q is designated as the driver topick up the other person, and τp, τq as the departure time of p and q. For example, σpq = 1, πpq = 1indicates p and q share, and p is the driver.

Formally, we would like to minimize the total travel time within the city during the study timeperiod

Z = minσ,π,τ,v

∑t

∑a

vta · sa(vta). (4.3)

This is a quasi-dynamic traffic assignment, in which for each time period a static traffic assignmentis solved. Here we do not use a dynamic traffic assignment because we will need the travel time asa function of travel demand, while with the dynamic assignment the function is implicit and canonly be output with simulation [71]. We can treat the whole summation as a function of v, denotedas ϕ(v), and write the optimization problem as Z = min

σ,π,τ,vϕ(v).

4.3.3 Collecting travel demand

The previous discussion is based on the assumption that the vehicle travel demand is known to us.However, even if we know the origins and destinations of all travelers, vehicle travel demand is stillnot fixed when sharing is incorporated.

In general, the vehicle travel demand is lower when more trips are shared. However, the specifictravel demand between two nodes in the network depends not only on the trips that start or end atthe two locations, but also needs to include the trips that connect the two locations after sharing.

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For example, if a driver at node i needs to pick up a passenger at node j, another vehicle trip willbe added between the two nodes. In total, there are three cases considering the sharing and drivingcombinations.

1. Driving alone: for a trip p ∈ X , if ∀q , p, σpq = 0, let σpp = 1. Then, there is still only onevehicle trip for p, Op → Dp, i.e., the person is still driving alone.

2. Sharing, p as driver: when σpq = 1 and πpq = 1, i.e., when p is the driver to pick up q, therewill be three trips, Op → Oq, Oq → Dq, and Dq → Dp. In this case, p needs to make a detourfor q, and two original trips become three. However, this may not increase total vehicle traveltime as there are fewer cars on the road.

3. Sharing, q as driver: the case will be similar for σpq = 1 and πpq = 0.

Therefore, we can calculate the vehicle travel demand between any two nodes from the persontravel demand, and ridesharing matching. Let Oi and Di denote the sets of person trips whoseorigin or destination is i, and fpi

∆= 1[p ∈ Oi], gpi

∆= 1[p ∈ Di] indicating whether i is the origin and

destination of trip p. Then, we can sum up the vehicle travel demand between nodes i, j from f , g,σ, and π

xi j =∑p∈X

∑q∈X

{σpqπpq( fpi fqj + fqigqj + gqigpj) + σpq(1 − πpq)( fqi fpj + fpigpj + gpigqj)}. (4.4)

When the initial person travel demand is given, f and g are constant numbers. We can denotevehicle travel demand as a function of sharing matching σ, and driver designation π, x = x(σ, π).

4.3.4 Travel demand and traffic

Even given vehicle travel demand, the traffic volume on a link can also vary as there are many routechoices. However, the flow for all routes r that connect two nodes need to sum up to the totaldemand ∑

r ∈Ri j

xri j = xi j, (4.5)

which can also be further split to xrt and xt among time periods. Besides the traffic volume of alink is the passing traffic from all OD pairs as long as it is part of the route

vta =∑i

∑j

∑r

αari j xrti j , (4.6)

in which the binary coefficient αar indicates whether a link a is part of a route r [72]. We canrewrite the constraint as T (v, x(σ, π)) = 0.

4.3.5 Natural constraints about sharing and pairing

Based on the previous discussion, there is an enormous number of possible combination of ridesharingmatches if no constraints are applied. However, several constraints are necessary to limit the number

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of possible values of σ and π, either naturally or for convenience of our formulation. First, we limitour discussion to the matching of two parties so that∑

q∈X

σpq = 1, ∀p ∈ X, (4.7)

or simply σ1 = 1. Specifically, we require σpp = 1 if trip p is not shared with other trips.

If p and q do share, i.e., σpq , 0, one of the two needs to be the driver, indicated by πpq. Asonly one driver is needed, we require πpq = 1 − πqp. Formally, for any pair of trips, there always is

σpq(πpq + πqp − 1) = 0, ∀p, q ∈ X . (4.8)

We can write it as element-wise product, σ ◦ (π + πT − 1) = 0.

4.3.6 Generic time flexibility constraints

Time flexibility further constraints the possible combinations of ridesharing matching. Each travelercan only start a trip after the earliest possible departure time from the origin, and end before thelatest possible arrival time at the destination. For shared trips, the driver also needs to accommodatethe time constraints of the passenger, while satisfying his or her own time constraints.

Denoting the earliest departure time of trip p as up and latest arrival time as lp, we need torequire all departure times to be bounded,

τp > up, ∀p ∈ X, (4.9)

or with vector notation τ > u. If p is not sharing

σpp

(τp + δ

t3τpOpDp

)6 lp, ∀p ∈ X, (4.10)

in which the superscript t 3 τp indicates the time period t that τp falls in. Note that if p is indeedshared (σpp = 0), the left hand side will be 0. Thus, no matter what the value of σpp is, theconstraint always needs to be satisfied. We can rewrite Eq. (4.10) by defining a function L to bethe same as the left hand side and let

L(diag(σ), τ, δ(v)) − l 6 0, (4.11)

or simply write it as L(σ, τ, δ(v)) 6 0, if we incorporate the constant vector l into the definition ofL.

4.3.7 Time flexibility constraints for shared trips

If two trips are shared, when the departure time of the driver (for example p) is given, and if weknow the travel time on any link in the system, in principle, the departure time of the passenger qis determined. It needs to be later than the earliest departure time of the passenger, i.e.,

τq = τp + δt3τpOpOq

> uq . (4.12)

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Symmetrically, if q is the driver we need to have

τp = τq + δt3τqOqOp

> up . (4.13)

Using the symmetry, and noticing that the inequalities are always guaranteed by Eq. (4.11), we onlyneed to add constraints for the equality parts of Eq. (4.12) and (4.13). Formally, we can combinethe equations as

σpq

[πpq

(τq − τp − δ

t3τpOpOq

)+ (1 − πpq)

(τp − τq − δ

t3τqOqOp

)]= 0, ∀p, q ∈ X . (4.14)

The left hand side is a function of σ, π, τ and δ, which is a function of v. Denote it as U and wehave U(σ, π, τ, δ(v)) = 0.

For the constraint of latest arrival time, when the departure time of the driver is set, it willalso be determined given the travel time. For simplicity of notation, we introduce an intermediatevariable ξ as the drop-off time of the passenger. Therefore, we need to require that both thepassenger and the driver can end their trips before their latest arrival times. For example, if q isthe passenger, we need to have

ξq∆= τq + δ

t3τqOqDq

6 lq, (4.15)

ξq + δt3ξqDqDp

6 lp . (4.16)

Symmetrically, when q is the driver and p is the passenger to be picked up and dropped off, weneed to have

ξp∆= τp + δ

t3τpOpDp

6 lp, (4.17)

ξp + δt3ξpDpDq

6 lq . (4.18)

Similar to Eq. (4.14), we can combine the constraints with the ridesharing matching variable σ, andrequire ∀p, q ∈ X

σpq

[πpq

(ξq − lq

)+ (1 − πpq)

(ξp − lp

) ]6 0, (4.19)

σpq

[πpq

(ξq + δ

t3ξqDqDp

− lp)+ (1 − πpq)

(ξp + δ

t3ξpDpDq

− lq)]6 0. (4.20)

We can also define L1 and L2 to represent the left sides of Eq. (4.19) and (4.20) as functions of theoptimization variable such that

L1(σ, π, ξ(τ, δ(v))) 6 0, (4.21)L2(σ, π, ξ(τ, δ(v)), δ(v)) 6 0. (4.22)

4.3.8 Formulation summary

Now we can summarize the optimization problem. Our objective is to minimize the total systemtravel time. Given the origins and destinations of all the current vehicle trips, we can change thepairing of trips with σ, driver designation among the travelers with π, departure time of trips withτ, and the traffic volume on each road link with v. We have constraints that traffic flow sum up totravel demand, one trip is shared with only one other trip, there is only one driver in each trip, and

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all travelers are bounded by the corresponding time window. By combining the formulation above,we have the following optimization problem

Z = minσ,π,τ,v

ϕ(v), (4.23)

subject to

σ1 − 1 = 0

σ ◦ (π + πT − 1) = 0

T (v, x(σ, π)) = 0

U(σ, π, τ, δ(v)) = 0

u − τ 6 0

L(σ, τ, δ(v)) 6 0

L1(σ, π, ξ(τ, δ(v))) 6 0

L2(σ, π, ξ(τ, δ(v)), δ(v)) 6 0

. (4.24)

4.3.9 Dependency diagram

To illustrate the dependency relationship between the variables, we can define eight dummy indi-cators z1, . . . , z8 corresponding to the eight constraints

z1 = 1{σ1−1=0}

z2 = 1{σ◦(π+πT−1)=0}

z3 = 1{T=0}z4 = 1{U=0}z5 = 1{u−τ60}z6 = 1{L60}z7 = 1{L160}

z8 = 1{L260}

(4.25)

and factor the constraints into the objective function

Z ′ = maxσ,π,τ,v

ψ(σ, π, τ, v) = maxσ,π,τ,v

[(M − ϕ(v))

∏k

zk(σ, π, τ, v)

], (4.26)

where M is a large constant. With the help of the dummy variables, we can visualize the depen-dency relationship of this optimization problem, shown in Figure 4-2. In this graph, for conceptualconvenience, the traffic volume of each link v is represented as a direct function of route choicesof all travelers r . However, in calculation, the traffic volume is the actual decision variable of theoptimization system, as in the traffic assignment models, and the route choices can be reverselycalculated when all traffic volumes are determined.

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Z ′

ψ

z1 z2 z3 z4 z5 z6 z7 z8ϕ

T U L L1 L2

ξ

x δ

σ π τ v

r

Decision variables

Constraints

Figure 4-2: Dependency diagram of the optimization problem.

4.4 Data

4.4.1 Sioux Falls network

We use the transportation network of Sioux Falls, as shown in Figure 4-3, provided by the Trans-portation Networks for Research Core Team [73]. The network has been widely used in transporta-tion research, including network design, traffic assignment, and ridesharing [61, 74–76].

Since there is no trip departure time in the Sioux Falls dataset, we used the existing traveldemand in this dataset as a probability matrix for trip origin–destination (OD) generation for afour-hour morning peak from 6 a.m. to 10 a.m. totaling 6,724 trips, and used the departure timedistribution from the National Travel Household Survey [77] during the same time period as theseed for a random generation of the departure times of these trips.

Based on the generated departure time, and the travel time with a traffic assignment withoutsharing, we estimated the arrival time of each individual trip. Then, with the departure and arrivaltime, we added time flexibility buffers to both ends of each trip as the generated earliest departuretime and latest arrival time. In later sections, we use the length of this time flexibility buffer tounderstand its impact on the total vehicle travel time.

We would like to point out that here the travel time plus buffer approach is only for datasynthesis in this chapter, and is independent with our definition of time flexibility. In a real-worldapplication of the model, the earliest departure and latest arrival time, as the definition of time

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flexibility in our model, can be gathered from the travelers when they submit trip sharing requestsalong with their origins and destinations.

4.4.2 Trip generation

As there is only one combined OD matrix for a day in the Sioux Falls dataset, we first convertedthe matrix into a probability matrix by dividing each entry by the sum of all entries. Then, foreach trip p in the morning peak we generated an origin Op and a destination Dp by applying thisOD probability matrix.

For each trip, we also generated its departure time τp by using the data from the NationalHousehold Travel Survey (NHTS) [77] to estimate the distribution of departure time between 6a.m. and 10 a.m., and sampled departure time τp for each trip p within this time range T .

In the NHTS trip data, we first counted the number of trips departed in each time periodt ∈ T (15 min each and 16 time periods in total). There is a rounding bias in the self-reporting ofdeparture time, thus the counts are higher for integral or half-integral hours, and lower for othertime periods. To reduce the bias, we smoothed the data across time periods T by a third degreepolynomial regression. Figure 4-4 shows the percentage of trips in each time period according toNHTS, and the smoothed departure time distribution.

Figure 4-3: Sioux Falls network [73] (edge labels indicate link IDs; generated by John Li; retrievedfrom http://jlitraffic.appspot.com/tap.html)

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Then, within each 15-min time interval, we assumed the departure time to be evenly distributed.Using the OD probability and departure time distribution, we generated the OD and departure timesfor all trips.

6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m.Time period

0%

2%

4%

6%

8%

10%

Shar

e of

trip

s in

the

stud

y tim

e pe

riods

Figure 4-4: Polynomial smoothing of travel demand in different time intervals. (The dots representthe percentage of travel demand in 15-minute time intervals from 6 a.m. to 10 a.m.; The curveshows a third-degree least-squares polynomial fit of the dots.

4.4.3 Time flexibility generation

In a real-world application, the earliest departure time up and the latest arrival time lp can bedirectedly collected from users of shared mobility service. While for the Sioux Falls network, thisinformation is not in the dataset, so we synthesized time flexibility for these morning-peak trips.By the definition of time flexibility in this chapter as discussed in the Introduction and LiteratureReview, we need an earliest departure time up and a latest arrival time lp for each trip p.

Based on our decomposition of time flexibility—travel time uncertainty and activity flexibility,we assume the time flexibility [up, lp] for traveler p depends on the expected travel time and extendtwo buffer time intervals on both ends—at the origin and the destination.

We estimated the expected travel time with the original non-sharing demand as a benchmark.However, the expected travel time will be over-estimated since the traffic condition will be improvedwhen more people share and the number of cars is reduced. At the same time, at the beginning ofsimulation we would not be able to estimate the with-sharing travel time before the system actuallymatches the rides and get a with-sharing travel demand. Therefore, as a starting point, we applieda discount ratio ` combined with the non-sharing travel time δ̃ to approximate the expected traveltime δ̂, where δ̂ = `δ̃. In the case study, the discount ratio ` was a system parameter satisfying` ≤ 1, and we tested different values of it. Because the value of ` is only used for the estimationof initial travel time, and then travel time will be updated once matchings start to be made in thefirst iteration, the actual value of ` should only be related a ‘’starting point” of the optimization,and will not impact the final converging point of the optimization.

We obtained the non-sharing travel time δ̃ from Transportation Networks for Research CoreTeam [73]. The team conducted the UE traffic assignment with the average-hour demand for theSioux Falls network and generated the benchmark traffic assignment results.

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For the flexibility of the activity schedule at both origins and destinations, we assumed it tobe a uniform parameter across all trips and it was symmetric at the origin and the destination,which implied having the same length at both ends. Let ∆ f represent the flexibility of the activityschedules for all trips, the time flexibility for any trip p is [up, lp] = [τp −

∆ f

2 , τp + δEp +

∆ f

2 ].

In this chapter, we only applied a uniform time flexibility ∆ f to understand the relationshipbetween this parameter and the total vehicle time. We acknowledge that in the real world, theflexibility interval varies for different people, different trip purposes, and at different times of day.An comprehensive investigation of the distribution of flexibility values is beyond the scope of thischapter, and will be left for future research.

4.5 Approximations

4.5.1 TR-decomposition

The overall optimization problem is highly nonlinear in both the objective function and the con-straints, which indicates computational challenges. We notice that in many constraints, like Eq. (4.18),if travel time δ can be pre-calculated, many of the constraints can be linearized. However, as wehave discussed in the introduction, ridesharing matching and departure time spreading both changetravel demand patterns, and further reduce travel time, we cannot simply assume travel time to befixed.

In this section, we propose an iterative approximation of the problem, which divides the opti-mization process into traffic assignment steps (T-step), and ridesharing matching, driver designation,and departure time assignment steps (R-step). The optimization starts from a T-step, and uses thetravel time generated from the initial traffic assignment as the travel time δ(0) in the first R-step,which outputs the new vehicle travel demand x(1) for a new iteration of traffic assignment, and gen-erates the new travel time δ(1), as shown in Figure 4-5. We will prove that R-step can be simplifiedas a mixed integer linear programming (MILP) problem. This TR decomposition is enlightened bythe expectation–maximization algorithm (EM algorithm) in statistics [78]. The EM algorithm canbe used in statistical models to estimate parameters with the maximum likelihood when there areunobserved variables that the estimation depends on. The algorithm iterates between calculatingvalues of the unobserved variable given the current estimate of parameter, and estimating the pa-rameter based on the calculated values of the unobserved variable. The analogy in our model isthat δ is the unobserved variable, and σ, π, and τ are the parameters that need to be estimatedsuch that total vehicle travel time is minimized—in analogy to that likelihood is maximized.

x(0) · · · x(k) v(k) δ(k) σ(k), π(k), τ(k) x(k+1) · · ·

T-step

R-step

Figure 4-5: TR decomposition.

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4.5.2 Traffic Assignment (T-step)

As in T-step, travel time is calculated with given travel demand, in theory any traffic assignmentmethod could be used. The traffic assignment model takes the origin-destination matrix (OD) fromx, and generates travel time between node pairs. However, different traffic assignment methodsrely on different assumptions. [72] discussed the assignment methods under both system-optimalcriterion, and user-equilibrium criterion. In the specific implementation of the traffic model, thesystem designer needs to make the decision about which method to use. For example, if theassumption is that there is a centralized route recommendation system for all drivers, the system-optimal assignment could be used. If assuming each driver will choose the route with shortest traveltime in their individual choice set, the equilibrium assignment method could be used.

4.5.3 Linearization of R-step

In this part, we can separate the travel time variables used in optimization problem (4.23) asconstants. Then, we can simplify the nonlinear terms in the optimization objective, and in theconstraints.

Noticing that σ and π always appear as products—either element-wise, or anti-element-wise(multiplying one with the transpose of the other)—we denote two new variables pqא = σpqπpqand pqב = σpqπqp to represent the products. The relationship between the four variables, and thetranspose of π, is listed in Table 4.2.

Table 4.2: Value table of σ, π, ,א בσpq πpq πqp pqא pqב

0 0 1 0 0 p and q are not sharing0 1 0 0 0 p and q are not sharing1 0 1 0 1 Sharing, and q is the driver1 1 0 1 0 Sharing, and p is the driver

By definition, we always have pqא = σpqπpq = σqpπpq = .qpב In other words, in matrix formatא = ,Tב so we only need one of the two.

Also by definition, we require sharing to be symmetric, i.e., σpq = σqp. As pqא + qpא =

σpqπpq + σqpπqp = σpq(πpq + πqp), and πpq + πqp = 1, we have σpq = pqא + qp—theא right side isnaturally symmetric. Besides, for constraints (4.7) and (4.8), we can combine them as∑

q

pqא + qpא = 1, ∀p ∈ X , (4.27)

and eliminate all the products of σ and π in the model.

The reason that we can simplify the product of σ and π is that by definition σ is symmetric, andπ is anti-symmetric. Therefore, there is duplicate information in both matrices—the upper or lowertriangle of each matrix would suffice to determine pairing or driver assignment. After replacing theproduct with ,א we can represent the sharing and driving designation between p and q by checkingthe values of pqא and .qpא Table 4.2 explains the specific correspondence.

Besides, we notice the product of σ, π, and τ, now א and τ, in the conditions (4.14), (4.19), and

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(4.20). Noting that א is binary, and τ has an upper bound provided by l, we denote the productsas pqל = pqτpא and pqש = .pqτqא Equivalently, we need to guarantee in the conditions

pqל 6 lpאpq

pqל 6 τppqל > τq − (1 − pq)lqאpqל > 0

. (4.28)

Therefore, when pqא = 0, pqל = 0 by the first and the fourth conditions, and when pqא = 1,pqל = τp by the second and the third conditions. Similarly, we also need to have

pqש 6 lqאpq

pqש 6 τqpqש > τq − (1 − pq)lqאpqש > 0

. (4.29)

We can then rewrite the conditions (4.9), (4.10), (4.14), (4.19), and (4.20) into the followingform

pqל > pqupא

ppל + ap 6 lp (L)

pqש + qpש − pqל − qpל − bpqאpq − bqpאqp = 0 (U)

pqש + qpש + (aq − lq)אpq + (ap − lp)אqp 6 0 (L1)

pqש + qpש + (aq + cqp − lp)אpq + (ap + cpq − lq)אpq 6 0 (L2)

, (4.30)

where ap, bpq, cpq are constants that come from travel time δ if there is only one time period(|T | = 1). They represent the travel time for Op → Dp, Op → Oq, and Dp → Dq.

When there is more than one time period, travel time will not be fixed as a constant but willdepend on trip departure times, in the form of step functions of ל and .ש In this case, the Big-Mmethod can be used to linearize the step functions. For example, for the second condition in (4.30),if there are two time periods split at time point T , we can write the condition as{

ppל + a 〈1〉p 6 lp, ppל > Tppל + a 〈2〉p 6 lp, ppל < T

, (4.31)

where a 〈1〉p and a 〈2〉p are the fixed travel times corresponding to the two time periods. In thiscase, we can treat ap as a variable constrained by the following conditions, with the introduction ofa new binary variable צ indicating the time period

ap = paצ 〈2〉p + (1 − p)aצ

〈1〉p

ppל − T > −M(1 − (pצ

ppל − T 6 Mצp

, (4.32)

in which M is a sufficiently large positive constant. Therefore, when ppל is in the later time

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period (> T), pצ can only be one, and ap = a 〈2〉p . When ppל is in the earlier time period, pצ canonly be zero, and ap = a 〈1〉p .

The objective function can also be written into linear form as the element-wise product of fixedtravel time di j (or step functions, as discussed above) between node i, j, and travel demand from ,א

Z ′′ = minש,ל,א

∑i, j

di j∑p,q

[pqא

(fpi fqj + fqigqj + gqigpj

)+ qpא

(fqi fpj + fpigpj + gpigqj

) ]. (4.33)

We can change the order of summation

minש,ל,א

∑p,q

pqא

[∑i, j

di j(fpi fqj + fqigqj + gqigpj

) ]+

∑p,q

qpא

[∑i, j

di j(fqi fpj + fpigpj + gpigqj

) ], (4.34)

and let

epq =∑i, j

di j(fpi fqj + fqigqj + gqigpj

), (4.35)

which is constant. Then, the optimization problem becomes an MILP problem.

Z ′′ = minש,ל,א

∑p,q

epqאpq + eqpאqp . (4.36)

The linearization of R-steps with travel time of multiple time periods can use a similar approachas described above.

4.5.4 Pruning infeasible trip pairs and discretizing departure time in the R-steps

The TR-decomposition algorithm iterates over T-steps and R-steps until convergence, and we haveshown that the R-steps can be simplified as mixed integer programming problems (MILP). WhileMILP is simple in the form, the algorithm for solving it is still NP-hard. To solve the problem moreefficiently, we need to further reduce the search space. Noticing that the ridesharing trips will notbe feasible to individuals when the detour time is large, we introduce a “cap” of detour time anduse it to reduce the infeasible trip pairs for sharing, and thus reduce the search space of the MILP.This is similar to the detour cap used in the formulation of shareability networks in [2].

However, to guarantee the feasibility of the sharing of a trip pair, the travel time of the tripsneeds to be used, which depends on the the departure times τ of the trips. Since τ is a continuousvariable, it is impossible to examine all possible values of it to evaluate the feasibility of sharingtwo trips. Therefore, we discretize the whole study period by a resolution ϑ, and generate a finitediscrete set Td with time points to represent the discretized departure time of trips. In this chapter,we set the departure time resolution ϑ to be one minute. While theoretically by discretizing time

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to be only integer points we may lose the ultimate optimality of solutions to the original problem,practically this approximation can accelerate calculation by significantly reducing the search space.

It is worth noting that, after discretizing the departure time, now if two trips start at the sametime, it means they depart within the same minute, e.g., between 9:00:00 and 9:00:59 a.m. However,as less than one minute is a negligible time interval for most people and can be considered not evenas a waiting, we will still use 0-min flexibility to represent this shorter-than-one-minute situation.

By combining the detour cap pruning and the departure time discretization we can furtherreduce the R-steps to be Integer Linear Programming (ILP) problems, as specified below.

Noticing that we have assumed travel time to be fixed in the R-steps, to make the total traveltime smaller, for a given trip pair which is feasible to share, the best (discretized) departure timewill be the one which minimizes the total travel time of the shared trip pair. In other words, withinthe scope of the R-steps, each local minimal travel time to this given trip pair—only depending onthe departure time choice—will lead to global optimality.

Formally, for any trip pair (p, q) ∈ X 2, driver p will depart at the time τp which minimizes theirtravel time if pqא = 1 (p shares with q; p is the driver and q is the passenger). Let M ⊂ X 2 denotethe set of feasible trip pairs satisfying the time flexibility constraints for sharing. As discussedabove, we need to find the “best” departure time, and thus the shortest travel time for each feasibletrip pair. Then, with the approximation, the problem becomes to find the best trip pairs—with theshortest travel time of each—that leads to the total minimum vehicle time. We denote tmin

pq for theminimum travel time for the trip pair (p, q), and the set for “best” departure time of all trip pairsas G .

Algorithm 3 gives the details of finding the best departure time and the shortest travel time forall feasible trip pairs as preprocessing for R-steps. With the set of feasible trip pair M generatedby the prepossessing algorithm, we can formulate the following simplified ILP

min∑

(p,q)∈M

tminpq ,pqא (4.37)

s.t.∑q∈X

pqא +∑

q,p∈X

qpא = 1, ∀p ∈ X , (4.38)

pqא ∈ {0, 1}, ∀(p, q) ∈M , (4.39)

which can be solved more efficiently.

It is worth noting that when the departure time resolution gets finer, i.e., ϑ → 0, the optimalsolution of the approximated R-steps will also improve, and converge to the solution of the original R-steps. Therefore, the trade-off can be evaluated based on the computation capacity and requirementof the stringency of optimality of solution. The assessment of the trade-off will be based on theactual problem and availability of computation resources, and is beyond the scope of this chapter.

4.5.5 Making TR-decomposition converge

In order to assess the feasibility and performance of the TR-decomposition with approximation pre-processing, we conducted a preliminary simulation in the Sioux Falls network. For the experiment,we set the proportion of shareable trips, i.e., the trips of travelers who are willing to share theirvehicle trips, to be 10%. For the T-steps we implemented the modified Frank–Wolfe algorithm [79]

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Algorithm 3 Preprocessing for R-step. Input: the trip set X , the set of discretized departuretimes Td, the time flexibility {(up, lp)|∀p ∈ X } and the fixed travel time δ

1: function Prepossessing(X ,Td, {(up, lp)|∀p ∈ X }, δ)2: M ← � . Initialize the set of feasible trip pairs3: for (p, q) ∈ X 2 do4: tmin

pq ← +∞, τ∗pq ← null . Initialize the min. travel time, and the “best” departure time5: for τp ∈ Td do6: τq ← τp + δ

t3τpOpOq

. Pick-up time of trip q

7: ξq ← τq + δt3τqOqDq

. Arrival time of trip q

8: ξp ← ξq + δt3ξqDqDp

. Arrival time of trip p9: if τp > up and τq > uq then . Check the earliest departure time constraints

10: if ξq 6 lq and ξp 6 lp then . Check the latest arrival time constraints11: if ξp − τp 6 W then . Check the detour cap constraint for driver p12: if ξp − τp < tmin

pq then13: tmin

pq ← ξp − τp, τ∗pq ← τp . Update the min. travel time and best departure time

14: if tminpq < +∞ then

15: M ←M ∪ {(p, q)} . Add the feasible trip pair to M

16: G ← {τ∗pq |∀(p, q) ∈M }17: return M ,G

for the system optimum traffic assignment. The simulation results are shown in Figure 4-6.

20 40 60 80 100 120

Iteration index

2.06

2.065

2.07

2.075

2.08

2.085

2.09

Tota

l vehic

le tim

e

105

Total vehicle time

5-step average

10-step average

Figure 4-6: Preliminary results for the TR-decomposition algorithm with approximation preprocess-ing after 130 iterations (10% trips shareable; 15 minutes each time period; system optimum (SO)traffic assignment). Each iteration includes a T-step and an R-step. 5-step and 10-step movingaverages of total vehicle travel time are shown.

The simulation results revealed non-convergence of the TR-decomposition. This is due to thelack of immediate feedback between the change of vehicle travel demand in each time period, whichonly happens in the R-steps, and the change of travel time, which only happens in the T-steps.

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More specifically, after a T-step when travel times are fixed, in the next R-step no matter howmany trips change their departure time, the travel time for a time period will not be affected evenwhile the demand is now changed. Therefore, there is an excessive demand change in the R-stepswhich should have led to travel time change.

Essentially, the number of trips that can change matchings, driver/passenger roles, and departuretimes in the R-steps is the “step size” in this optimization problem. In optimization, a too large stepsize will induce the solution to jump over the optimal point back and forth in each iteration withoutbeing able to asymptotically approach it. However, due to the binary nature of the matchingproblem, the solution is also constrained by the feasibility of trip pair matching, so a randomsubsetting of mutable trips is not feasible either. Therefore, we propose another approximationapproach to reduce the number of trips that can change departure times in each iteration in orderto guarantee convergence of the algorithm.

The thought for designing such an approximation approach is to make the solution converge byfixing a proportion of trip pairs in the R-step in each iteration. We denote the set of fixed trip pairsas M̃ .

We start by a T-step, which generates the initial travel time without sharing, and an R-step,which matches a set of trips based on the initial travel time. Then, in the next iterations, thealready-matched trip pairs will be fixed. Only the trips that have not been matched will be changed.

We denote RStep(δ,X ,M , M̃ ) to be the process of solving the simplified ILP (Eqs. 4.37–4.39),which returns the locally (with regard to the current R-step) optimal trip matching M ∗ and thecorresponding vehicle travel demand x. The function RStep(·) takes the inputs of travel time δfrom the previous T-step, the set of shareable trips X , the set of feasible trip pairs M , and the setof trip pairs that are fixed M̃ . The trips in the set M̃ are the trips that are matched in previousR-steps, and will not be changed for matching in the current R-step.

Similarly, we denote TStep(x) to represent the traffic assignment process which returns traveltime δ with the input of vehicle travel demand x.

Algorithm 4 describes the details of the converging TR-decomposition (C-TR). In the i-th it-eration, an R-step is performed with the input of travel time δ(i−1) from the (i-1)-th iteration andoutput the optimal trip-matching with regard to the current R-step, which are then fixed in theset M ∗(i). Given the optimal matching, we then update the total travel demand x(i) and perform aT-step to generate the new travel time δ(i).

After the travel time updates, the existing trip pairs may be infeasible. Therefore, we need tocheck it with the new travel time. Formally, we check the time flexibility constraints of all trip pairs(p, q) ∈M ∗(i) with the new travel time δ(i) and only keep the feasible pairs (p, q) while putting theinfeasible ones back to be unmatched.

Since the fixed trip pairs will not be included for matching in the new iteration, the number offixed trips |M̃ | will monotonically increase. The algorithm will terminate if no new trip pairs arefixed after one iteration, i.e., |M̃ | stays unchanged, and the remaining unmatched trips will be leftwithout sharing.

As shown by the experiment results, the convergence of the algorithm is fast, and most of thetrips are fixed after the first iteration. Intuitively, in the initial traffic assignment no trip is shared,so the travel time with some shared trips δ(1) should be better than δ(0). Therefore, the optimaltrip pairs calculated with δ(0), which leads to higher stringency in the time flexibility constraints,

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Algorithm 4 Converging TR-decomposition (C-TR) algorithm. Input: total vehicle travel demandx, the total trip set X , the set of discretized departure time Td, and the time flexibility {[up, lp]|∀p ∈X }.1: function ConvergingTR(x,X ,Td, {[up, lp]|∀p ∈ X })2: M̃ ← � . Initialize the set of fixed trip pairs3: X (0) ←X , x(0) ← x, δ(0) ← TStep(x(0)) . Perform the first T-step4: i ← 15: while true do6: M (i−1),G (i−1) ← Prepossessing(X (i−1),Td, {[up, lp]|∀p ∈ X (i−1)}, δ(i−1))

7: x(i),M ∗(i) ← RStep(δ(i−1),X (i−1),M (i−1), M̃ ) . Perform one ILP R-step8: δ(i) ← TStep(x(i)) . Perform one traffic assignment (T-step) with updated demand9: M ∗

+(i) ← RecheckFeasibility(M ∗(i),G (i−1), δ(i), {[up, lp]|∀p ∈ X (i−1)})

10: if M ∗+(i) , � then

11: M̃ ← M̃ ∪M ∗+(i) . Add fixed trips

12: X (i) ←X (i−1) \ {{p, q}|(p, q) ∈M ∗+(i)} . Eliminate fixed trips for the next iteration

13: i ← i + 114: else15: break . Stop if no new trip pairs can be fixed16: for p ∈ X (i−1) do . Remaining trips will stay non-shared17: M̃ ← M̃ ∪ {(p, p)}18: return M̃19: function RecheckFeasibility(M ∗,G , δ,{[up, lp]|∀p ∈ X })20: M ∗

+ ← � . Initialize the set of infeasible trip pairs21: for (p, q) ∈M ∗ do22: if p = q then . Only consider shared trips23: continue24: τp ← τ∗pq, τq ← τp + δ

t3τpOpDq

, ξq ← τq + δt3τqOqDq

, ξp ← ξq + δt3ξqDqDp

25: if τp > up and τq > uq then26: if ξq 6 lq and ξp 6 lp then27: if ξp − τp 6 W then28: M ∗

+ ←M ∗+ ∪ {(p, q)} . Check the time flexibility and detour constraints

29: return M ∗+

should all be feasible under the travel time δ(1), and trips in the optimal pairs will be fixed afterthe first iteration.

Moreover, our experiment results showed that the C-TR algorithm converged quickly after nomore than ten iterations for all parameter combinations in the Sioux Falls network.

To summarize, in this section we first introduced the TR-decomposition, which can reduce theoriginal non-linear optimization problem to a combination of traffic assignment problems (T-steps),and MILPs (R-steps), which both have existing solving methods. Although the solution techniquesof the MILPs exist, the problem is by nature NP-hard and intractable for large instances. Therefore,we further proposed another layer of approximation, by pruning and discretizing, to simplify theR-steps to be ILPs. After testing the iteration between traffic assignment and ILP R-steps, we foundthe results to be non-convergent as the step size of the R-steps was too large. To address the non-

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convergence issue for the original TR-decomposition algorithm, we proposed the C-TR algorithm,by asymptotically matching the remaining trips, and only iterating over the unmatched ones.

Given the approximations we have made, the solution to the C-TR will not be as good as thesolution to the original problem, in terms of reducing the total vehicle travel time. However, inthe experimental results, we will show that given small increase in the time flexibility, the C-TRalgorithm can already significantly reduce total vehicle travel time.

We would like to point out that although this approximation method is not guaranteed to outputthe optimal solution to the original problem, it does provide an operationalizable solution includingmatching, driver/passenger designation, and departure time assignment. At the same time, basedon the results the reduction of total vehicle travel time is significant. Therefore, while an ultimatesolution is of interest to us for further research, the approximation method could be implementedin a real world application.

4.6 Computation

After introducing the prepossessing and quick-convergence decomposition algorithms, the time com-plexity of C-TR is simply the number of iterations times the time complexity for each iteration—aT-step, an R-step, and a feasibility checking process.

The number of iterations is bounded by |M | since the number of fixed trips monotonicallyincreases, and the algorithm will terminate when |M |’s size stops increasing.

For the T-steps, the time complexity is that of the traffic assignment method used, which wedid not find literature to have explicitly given. However, for some well-known traffic assignmentalgorithms, the practical performance has been shown to be good. For example, Fukushima demon-strated the Frank–Wolfe to show quick convergence in a few iterations with good approximation ofsolutions [79].

In this chapter, we used the iTAPAS algorithm [80] for the UE traffic assignment, and themodified Frank–Wolfe algorithm [79] for the SO traffic assignment. In both implementations, thePython package graph-tool is used for generating and managing road networks [81].

It is also hard to explicitly derive the time complexity of the R-steps since ILP problems areNP-hard. The number of variables in the ILP problem is O(|M |) = O(|X |2), and the number ofconstraints is O(|X |). Therefore, the dimension of the ILP is O(|X |3).

Although a theoretical proof of time complexity of the ILP in this chapter is hard to derive, inpractice there are efficient off-the-shelf ILP solvers, such as CPLEX and Gurobi. We used Gurobi8.0.1 with its interface on Python 3.6.0 [82] to solve the ILPs for the R-steps.

Noticing that in the R-step preprocessing the time flexibility constraints are independent foreach trip pair, we can parallelize this constraint checking process. Given k computation threads,we can divide the trip pairs into k sub-lists, and reduce the computation time of the preprocessingpart in each iteration to be approximately 1/k of the original time (i.e., without parallelization).

We implemented our algorithm on the computing cluster provided by Massachusetts Green HighPerformance Computing Center (MGHPCC), using the Intel Xeon CPU E5-2650 v3 @ 2.30GHz withsixteen cores, and 64 GB memory. We ran simulation for the Sioux Falls network (|X | = 6, 724)with various parameter combinations—one CPU core for each parameter combination. The longest

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execution time with the UE traffic assignment was 57 minutes with 100% trips shareable and 60-min time flexibility. The longest execution time with the SO traffic assignment method is with thesame parameter combination, and the time was 38 minutes. With shorter time flexibility and fewershareable trips, the computation time was shorter. For example, with 30% trips to be shareableand 20-min time flexibility, the computation times were no longer than 5 minutes with both UEand SO traffic assignments.

4.7 Result Analysis

To understand the relationship between total vehicle travel time, percentage of shareable trips ina city, and time flexibility values, we implemented the C-TR algorithm with different parametercombinations on the Sioux Falls network using the four-hour travel demand. We used time flexibilityvalues 0 min, 5 min, 10 min, 20 min, . . ., up to 60 min. It is worth noting, as in C-TR we havediscretized departure time, 0-min time difference actually means two time points fall within thesame nominal minute, e.g., both happen after 9:01 a.m. and before 9:02 a.m. For the proportion ofshareable trips in the city, we used 10%, 20%, . . ., 100%. The results are shown in Figures 4-7 and4-9.

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Figure 4-7: Total vehicle travel time vs. percentage of shareable trips with different time flexibilityvalues, using the user equilibrium (left) and the system optimum traffic assignment models (right).

4.7.1 Impact of proportion of shareable trips

When the proportion of shareable trips increases, the number of cars on the road is smaller. There-fore, the proportion of shareable trips impacts the total vehicle travel demand even given the sameperson travel demand. Although locally the detour of some drivers may increase the travel demandto some locations, still being within the individual time flexibility constraints, the overall vehicletravel demand is smaller, and thus the total vehicle travel time is significantly smaller with higherproportions of shareable trips.

For example, in Figure 4-7, even with 0-min time flexibility (or flexibility shorter than 1 min)

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20 40 60 80 100

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Figure 4-8: Percentage of actually matched trips vs. percentage of shareable trips with differenttime flexibility values, using the user equilibrium (left) and the system optimum traffic assignmentmodels (right).

and with UE traffic assignment, by increasing the proportion of shareable trips from 10% to 100%,the total vehicle travel time reduced from 1.92 × 105 min to 0.83 × 105 min, corresponding to a56% reduction (with the non-sharing no-time-flexibility total travel time being 1.95 × 105 min asa benchmark). With SO traffic assignment the improvement is minor, but still a 52% reductionhas been achieved with 0-min flexibility, since the total vehicle travel time is also optimized for thewhole network during traffic assignment.

The improvement is more significant when we combine the percentage of shareable trips withtime flexibility. With 5 min time flexibility, the total vehicle travel time reduction are 77% and 76%,with UE and SO assignments, respectively (Figure 4-9). This is what we expected at the beginningof this chapter—when we can change the departure time of trips, even with a small time flexibility,the number of actually shared trips will be higher.

4.7.2 Impact of time flexibility

Comparing the different curves in Figure 4-7 for each of the traffic models we used, the mostsignificant change of total vehicle travel time occurs when increasing time flexibility from 0 min to5 min. If we further increase time flexibility from 10 up to 60 minutes, we still observe total traveltime reduction, but smaller. For example, with UE traffic assignment and 50% of shareable trips,we have a 34% reduction of total travel time when time flexibility changes from 0 to 5 minutes, butonly 20% when time flexibility changes from 5 to 60 minutes.

These findings are quite intuitive: by simply allowing small amount of increased time flexibilityfor the trips to be performed and matched, the probability that they could be matched will signifi-cantly increase. The number of actually matched trips (Figure 4-8) also confirms this finding—thelonger time flexibility is allowed, not only more trips are shared, but there is also a higher pro-portion of actually matched trips in shareable trips. However, the percentage of shareable tripsultimately put a lower bound of the number of cars on the road. When time flexibility is beyond 10min, more than 80–90% of the people who are willing to share can be successfully matched. Then,keep allowing longer flexibility will not pair much more trips. Furthermore, time flexibility cannot

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overcome intrinsic unbalanced flow distribution in space, due to the fact that there is not a uniformdistribution of origins and destinations.

The implications of these findings are quite interesting though: there is no need to ask travelersto drastically change their habits and/or modify theirs departure time by tens of minutes. Havinga non-trivial proportion of people who are willing to share their vehicle trips allowing for 5 minutestime flexibility is enough to attain a significant reduction in the total vehicle travel time.

The surface reported in Figure 4-9 shows the interpolation of the total vehicle travel time reduc-tion given the combination of the percentage of shareable trips and the amount of time flexibility.As explained, the largest gradient is along the time flexibility axis, when it changes from 0 to asmall number. The four corners of the surface show the four benchmark cases:

- Bottom corner: no flexibility allowed and 10% of allowed shareable trips result in only 2% oftotal vehicle travel time reduction (comparing with non-sharing no-time-flexibility; same forbelow);

- Left corner: maximum allowed time flexibility (60 min) plus 10% of shareable trips achieve a17% total travel time reduction;

- Right corner: no flexibility allowed and allowing all the trips to be shared reduce the totaltravel time of 57%;

- Top corner: maximum allowed time flexibility (60 min) combined with 100% of shareable tripsachieve 78% of total travel time reduction.

Therefore, although the C-TR algorithm is not guaranteed to provide the optimal solution tothe original problem but only a lower bound to actual savings, the demonstrated improvement inthe total travel time is already sufficiently large.

4.7.3 User equilibrium vs. system optimum traffic assignment

Comparing the results obtained with the two traffic assignment models (Figure 4-7), the systemoptimum model is always outperforming the other. As already said, this is due to the fact that SOtraffic assignment also optimizes total vehicle travel time during the T-steps.

Nonetheless, the differences between the corresponding curves get smaller when the proportion ofshareable trips increases. When it is allowed to share 100% of the trips, there is almost no differencebetween the total vehicle travel times attained with UE and SO traffic assignments. When onlya small portion of trips is allowed to be shareable, the number of vehicles in the network almostremains the same, which results in local traffic congestion, and the SO assignment could globallyredistribute the traffic to the network, while UE could not.

4.8 Discussion

The main contribution of this chapter is proposing, for the first time in the literature to the bestof our knowledge, a theoretical model that incorporates travelers’ time flexibility into an optimiza-tion model that comprises ridesharing matching, driver/passenger role designation, departure time

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assignment, and route assignment. We also provide an approximation of the model, presenting apractical and efficient algorithm to easily resolve the problem of total travel time reduction evenif with sub-optimal solutions. The model allows studying for the first time how riders’ time flex-ibility can be included in mobility sharing to improve the traffic in an urban context, leveragingthe positive feedbacks that occur between ridesharing, travel departure time spreading, and trafficconditions.

Considering the difficulty of finding an exact solution due to the non-linearity in the optimizationobjective and constraints, we introduce some approximations of the problem: first, we decompose itinto an iterative process composed of so-called T-steps, i.e., the traffic assignment ones, and R-steps,i.e., the ridesharing-matching, driver/passenger–designation, and departure-time–assignment ones;secondly, after proving that the R-step can be simplified into a mixed integer linear programming(MILP) problem, we further simplify it into an ILP by pruning and discretizing the input; lastly,to address the non-convergence issue in the original TR-decomposition algorithm caused by thedimension of the R-step step length, we proposed the C-TR algorithm, a greedy approach that, byasymptotically matching the remaining trips, only iterates over the unmatched ones. Even thoughall these approximations could lead to a solution relatively less good than that of the original exactproblem, our framework nevertheless provides a constructive methodology that shows that, evenwith small values of time flexibility, very significant reduction of total travel time are indeed possible.

To implement the quick converging model given OD points and background traffic, we usedthe Sioux Falls transportation network, a well-known and widely used network in transportationresearch, applied to different scenarios that include ridesharing. In order to understand the inter-actions between total vehicle travel time, ridesharing, and time flexibility values, we implementedthe C-TR algorithm with different parameter combinations, varying both time flexibility values andpercentage of allowed shareable rides. We also used two different strategies for route assignment, tohighlight the differences between the scenario where drivers are only concerned about the shortest-time routes (user equilibrium), and the one where drivers can follow the system routing instruction(system optimum). The system optimum scenario, while not reflective of current human driving

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strategies, indeed provides clear insights onto a future scenario with autonomous vehicles followingsystem route assignment.

The results reported in this chapter show that even relatively low levels of flexibility inducegreat savings in terms of total travel time: with a non-trivial proportion of people who would like toshare their vehicle trips, the reduction of total vehicle travel time is significant by simply allowing 5min of time flexibility of trip departure/arrival time. However, by further increasing time flexibility,the total vehicle travel time does not significantly reduce, highlighting that time flexibility cannotovercome by itself the intrinsic unbalanced flow distribution within the network. If time flexibilityalone cannot reduce the number of traveling vehicles, increasing the percentage of shareable tripscould since it allows to reduce the number of traveling vehicles. Even though the C-TR algorithmonly provides a lower bound to actual travel time savings, the demonstrated improvement in thetotal saved travel time is very significant.

As per the two analyzed route assignment strategies, system optimum always outperforms userequilibrium, mainly because SO traffic assignment optimizes total vehicle travel time during theT-steps too, redistributing traffic and avoiding local congestion phenomena.

We would like to acknowledge the limitations of this chapter, and also the future works thatcould extend the current results. 1) The significance of the improvement of total vehicle travel timemay depend on the road network properties, origin–destination distribution, and the departuretime distribution of specific cities. In this chapter we used the Sioux Falls network to exemplify thealgorithm, while a more systematic assessment of the model performance on different types of citiescould be further conducted; 2) The total travel time reduction is significant comparing with the non-sharing no-time-flexibility benchmark. However, the optimality of results could not be guaranteedwith approximation in the model. Future research could target at improving the approximationapproach, and increasing the potential of further improving traffic; 3) In this chapter, we assumed nomode switch from public transit, walking, cycling to cars. Nonetheless, in reality, as traffic improvesthere may be people switching from transportation modes to driving or ridesharing. Moreover, thetime flexibility of a trip by itself may also play a role in people’s mode choice. The relationshipbetween time flexibility and mode choice also needs further studies. As we have discussed at thebeginning of this chapter, time flexibility is an intrinsic property of trips, not only for vehicle trips,but for any type of trips. Moreover, since the time flexibility allows for both redistribution ofdeparture time, and the detour for sharing, which effect is dominating with what temporal andspatial distribution of trips is also worth of further investigation.

In this chapter, we have assumed that in the current ridesharing matching, as the baselinepeople do not have time flexibility, or have no more than one minute of flexibility. However, due tothe nature of uncertainty incorporated in people’s decision making process, even with the currentmobility sharing services, people can still accommodate time flexibility by waiting and reopeningthe app interface. How have people already implicitly or explicitly included time flexibility in theirridesharing decisions also needs further empirical studies.

Nonetheless, the model presented herein could be used both by cities wishing to reduce commut-ing traffic, and private ridesharing services wishing to reduce costs and optimize vehicle utilization.However, it is worth noticing that the direct application of the model, by reducing total travel timeand congestion, could end up in reduced revenues for the mobility sharing services. In that case,municipality should study policies to guarantee a balanced outcome for all the players in a sharedmobility market.

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5

Mobility Sharing as a Tool for SocialIntegration: A Reverse Schelling Model

5.1 Introduction

In 1971, Professor Thomas Schelling published the paper Dynamic Models of Segregation. In thispaper he proposed an abstract model to explain that even without organized discrimination, if eachindividual simply does not want to be outnumbered in the neighborhood and otherwise relocate,complete spatial segregation could happen [83]. The Schelling model is a classic economic modelthat exemplifies how individual behaviors can result in collective results. Nonetheless, although thepaper discussed how the individual variance of tolerance levels to the outnumber neighbors may leadto different system equilibria, the individual choices were all considered to be fixed throughout themodel. Therefore, the different configurations of the Schelling model will all evolve into segregation,and the only difference is the extent of segregation.

The assumption that each individual would need to have a certain proportion of their neighborsto be from their own groups seems to natural. This may be induced from the historical segregationin social connections, e.g., connections built at work places or churches. However, this minimumpreference threshold for ingroup members in the neighborhood can also be dynamically changed.Allport proposed the intergroup contact theory, which states that intergroup contact may lead tomore positive attitudes toward outgroup members [84], and many empirical studies have confirmedthis theory [85].

Hence, if contacting outgroup members can dynamically reduce the minimum preference thresh-old in the Schelling model, a non-segregation outcome could be anticipated. The question thenbecomes how to encourage more contact between social groups.

In this chapter, I propose a reverse Schelling model, to integrate mobility sharing into Schelling’smodel as a tool for creating more opportunities for intergroup contact, and thus lead the system toequilibria of integration. Specifically, the model will match people from different social groups intheir trips, and then more contact time can be potentially encouraged.

Travel is a frequent human behavior—everyone may take one to many trips in a day. In additionto being the means of getting to the place for other activities, scholars have found that there areintrinsic values of travel other than being merely a period of time with only negative utility [86, 87].

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The reduction of prejudice through intergroup contact during shared trips can further extend theintrinsic value of travel time from the perspective of the society.

In the Schelling model, the location of agents were interpreted as their homes. In this chapter,while integrating the travel of agents, I would like to generalize the interpretation of location tobe the places of all activities, including going to work, restaurants, shopping, or any other leisureactivities. With this generalization, the sequence of relocation can be interpreted as a chain oftrips, and the higher frequency of travels than only home relocations can generate much moreopportunities of intergroup contact.

Transportation has been regarded as a domain where social contact or intergroup conflict arises,either passively or actively. The Transcontinental Railroad in the U.S. has been considered notonly as a tool to foster economic growth, but also a tool for creating social connections across thecountry [88]. The Bus Boycott in the Civil Rights Movement also exemplified how transportationwas used as a tool for expression of the disagreement to unrespectable social preference.

On the intergroup contact theory in the transportation context, empirical study is very limited.Enos conducted an experiment by asking pairs of Spanish-speaking confederates to show up on theplatforms of commuter rail stations in the Greater Boston Area, and surveyed routine riders atthese stations on their attitudes toward immigrants. He found that the train riders’ attitudes weremore exclusionary after a 3-day indirect contact with the confederates. However, after 10 days,the exclusionary attitudes reduced. Since he did not extend the experiment to longer time periods,a further reduction in the exclusionary attitudes could be possible. In addition, this experimentalso hinted that there may be two counteracting forces, one leading to more positive attitudestoward outgroup members while the other leading to more exclusionary attitudes. Without anempirical evidence on the relationship between the two forces, in this chapter I will include bothinto the model, and explore different assumptions on their relationship—including either one beingdominant, or the two competing with each other—to illustrate the potential outcomes.

The concept of social groups in this paper can also be generalized to any sociodemographicfactor that could lead to the exclusion and segregation in any spatial context. This may includebut is not limited to race, gender, age, occupation, languages spoken, etc. However, the discussionis limited to only two-group case in this chapter, leaving the modeling of multigroup interaction forfuture research.

While being similar to the Schelling paper that in this chapter the reverse Schelling model willonly be be discussed and experimented abstractly, I acknowledge that in each of the assumptionsin the model, there are many more rich theories further beyond the capacity that can be exhaustedin this single chapter, and there are more richness and nuances about the social integration beyondsimple uniform assumptions about all the people. With further empirical and theoretical under-standing of the assumptions in the model, all formulation used in this chapter could be improved.However, this chapter can act as the first model which combines mobility sharing with the evolutionof social integration, and the implications and inferences derived from the model could be helpfulto planners and policy makers.

Moreover, beyond the records of the short human history, there are many other possibilitiesregarding the contact of different social groups that may have not revealed in the past but mayevolve in the future. Besides, as speculated under the term cosmic sociology by Liu and Liu [90],for a broader sense of intergroup contact in the interplanetary communications, the human beingsmay have only experienced a small proportion of the possible scenarios. Therefore, in this chapter,I hope to keep the model to be deductive and leave for the flexibility of extension.

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The structure of the chapter is as follows. I will first review the relevant literature in Section5.2. Then, I will introduce the overall structure of the reverse Schelling model, and the specificationof multiple key components. In Section 5.4 I will investigate how the system parameters, such asdensity, maximum detour, and group share, impact the outcome of the model. In this section, onlyintergroup contact theory is assumed. In Section 5.5, I will relax the assumption of the positiveimpact of intergroup contact on the attitudes toward outgroup members to include both integrationforce and segregation force, and discuss how the relative strengths of the two forces change theintegration equilibria. In the last section, I will summarize this chapter, and discuss future researchquestions.

5.2 Literature Review

5.2.1 The Schelling model

As introduced at the beginning of this chapter, Schelling proposed the model to illustrate howindividual choice of locations may result in segregation [83]. In this model, Schelling specificallydifferentiated between “organized” action and “unorganized” individual behavior. “Orgnized” dis-crimination is illegal in most countries of the world today. However, based on the Schelling model,even unorganized individual behavior can also lead to segregation, based a simple assumption thateach person would need a minimum number of neighbors to be their ingroup members.

In the development of the model, Schelling also investigated several parameters on their im-pacts on the segregation equilibria, including density, group shares, preference threshold for ingroupneighbors, and different compositions of the preferences within groups. In this chapter, I will alsodiscuss the impact of the parameters of the Schelling paper on the reverse model, and also discussother parameters that the mobility sharing model needs, including the detour cap and matchingalgorithms.

In the Schelling paper, the measure of segregation is mostly based on the observation of patterns,with only simple statistic like the average number of neighboring outgroup members. For a moresystematic recording of the evolution of integration, in this chapter I will use Gini coefficient andindex of dissimilarity.

Further, in the Schelling model, the composition of people holding different preference thresholdsfor outgroup neighbors were used to discuss the possible equilibria of segregation. He consideredthe agents with lower preference threshold as “integrationists” and those with higher preferencethreshold as “segregationists,” and discussed how the different compositions of the two will lead todifferent equilibria. However, the composition parameter is static is his model. In this chapter, Iwill relax the assumption, and assume the composition of the two to be unfixed but depends on thecurrent integration status. The specific discussion will be given in Section 5.5.

There has been tremendous research since the Schelling paper extending the model by modify-ing the assumptions but still confirming the segregation outcome [91], or demonstrating analogiesbetween Schelling’s model and in other systems [92]. Prior literature on how to reverse the Schellingprocess from segregation to integration has not been found.

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5.2.2 Social contact theory: positive and negative intergroup contact

Scholars have been studying the reasons for segregation for decades. Allport analyzed how over-categorization in human minds might turn into prejudice [84], and Haidt explained how humannature inherited from ancient times shapes the instinct of grouping [93]. In a more concrete case ofthe segregation in residence, Krysan and Crowder pointed out that incomplete information aboutlocations and dependence on social networks contribute to segregation [94].

To resolve the prejudice and segregation existing in the society, Allport was one of the earliestresearchers to propose the intergroup contact theory in the 1950s, stating that the prejudice may bereduced by interacting with outgroup members. Pettigrew et al. conducted a meta-analysis on over500 empirical studies on the intergroup contact theory, confirmed the validity, and summarized thatthe positive attitude after interacting with one outgroup member can generalize to other membersin the same group, and the positive attitude can also generalize to other outgroups [85, 95]. For thischapter, the generalizability of preference change can support the assumption that the trip sharingdoes not have to keep the pairs fixed in order to change preference. Random matching betweengroups can also improve the acceptance of both groups to each other.

In addition to the positive social contact, as shown in both Allport’s original work, and in re-cent studies [96], negative social contact also exists and change people’s attitudes toward outgroupmembers in the other direction. Enos studied the impact of intergroup contact in a transportationcontext by asking Spanish-speaking confederates to show up on commuter rail stations, and sur-veyed the attitudes of riders on immigration policies before and after this indirect contact. Theexperiment showed a mixed results on both sides. After a 3-day period of interaction, the attitudestoward immigrants became more exclusionary, while after 10 days, the exclusionary attitudes re-duced. Therefore, in the long run the preference change may be in either direction before furtherinvestigation is conducted. In this chapter, I will assume three different scenarios: a dominantpositive force toward integration, a dominant negative force toward segregation, and a scenariowith competing two forces. The detailed investigation of how the three scenarios lead to differentequilibria of integration will be discussed in Section 5.4.

5.2.3 Mobility sharing for social integration

Trip matching in mobility sharing have been widely studied by scholars, and different matchingalgorithms have been proposed [1–3] to maximize the system efficiency.

Nonetheless, mobility sharing with social dimensions as the optimization objectives are lessstudied. Thaithatkul et al., and Zhang and Zhao studied the usage of preference for fellow passengersto match trips, and analyzed the trade-off of efficiency [23, 34].

Preference is related to social integration but the implication could be twofold. If there isexisting discrimination in people’s preference for fellow passengers, the preference-based matchingmay exacerbate segregation, while if people’s preferences are in line with integration, the matchingalgorithm can also amplify the effect.

Librino et al. specifically target intergroup contact as the objective for mobility sharing [50].In the matching models in this paper, maximum number of cross-group pairs and maximum socialmingling time were used as the matching objectives. However, this paper only discussed the resultof different matching in one round, but not how the matching results lead to further change ofintegration and the feedbacks between the two. I will use the methods proposed in this paper for

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the matching component in the reverse Schelling model, and further analyze how different matchingalgorithms can change the equilibria of integration.

5.2.4 Mathematical formulation of dynamic systems

Vinković and Kirman analyzed the Schelling model as a dynamic system and its similarity withphysical systems [92]. There are many other dynamic systems which have analogues in both naturalenvironments and social environments. For example, the Lotka–Volterra equations were used bothin the predator–prey model, and in the growth models in economics [97].

While dynamic models sometimes can be represented using one or multiple differential equations,and then the analysis of the model can be conducted through analyzing the properties of thedifferential equations, the differential equation formulation is not always easy to derive. Brandt et al.provided a rigorous formulation of one-dimensional Schelling process in 2012 [98], and manifesteddifficulties of the formulation. A rigorous mathematical formulation of the Schelling model in twodimensions was not found in literature.

In addition of the difficulty of formulating the Schelling model with stochastic processes anddifferential equations, the matching models in existing literature are mostly formulated using graphtheory and discrete optimization. The combination of differential equations, which consider thepopulation of agents as aggregated numbers, and discrete models, which needs to consider thematching on specific individuals, can be challenging. Therefore, in this chapter I will still use agent-based simulation to analyze the reverse Schelling model, and leave a comprehensive mathematicalmodeling combining the components for future research.

5.3 Model Formulation

In this section, first, I will introduce the structure of the reverse Schelling model. Then, I willdiscuss the detailed formulation, assumptions, and algorithms for key components of the model.

5.3.1 Overall structure

The flow chart of the model is shown in Figure 5-2, with notations explained in Table 5.1. Similarto in the Schelling model, the “world” of the reverse Schelling model is a grid of cells with each cellrepresenting a possible location for agents, and each cell can only be occupied by at most one agentat a time. The agents are divided into two groups: Group 1 and Group 2. After fixing the systemparameters—density d, and population share of Group 2, s—the locations of agents are randomlygenerated such that the ratio of occupied cells by agents in all cells is d, and among all agents, theproportion of agents in Group 2 is s. The left graph of Figure 5-1 illustrates a random initializationof agent locations with d = 50% and s = 50%.

Opposite to in the Schelling model, I assume that at beginning the two groups are completelysegregated. Group 1 agents occupy the sector with angle (1 − s)360◦ of the space, and Group 2agents occupy the rest sector with angle s · 360◦. If s = 50%, both groups will take half of the plane.

Each agent k has a preference threshold p(k)i at tick i to limit its minimum acceptable share of

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Figure 5-1: A random generation of initial locations of all agents (left) and a sample of locations ofagents after 200 ticks (right) with d = 50% and s = 50%

ingroup members in its eight surrounding cells, or its preference threshold. For example, if agentk has p(k)i = 0.3 and has 6 of the 8 neighboring cells occupied by other agents, the agent needs 2(the smallest integer greater than 0.3 × 6) out of the 6 agents to be its ingroup members in orderto accept the cell as an acceptable destination. In accordance with Schelling’s assumption in hismodel, in the initialization we assume each agent to have p(k)0 = 0.5, i.e., everybody just does notwant to be outnumbered in its 3 × 3 “local.”

After initialization, the model starts from asking each agent to identify the next location for its“destination.” The next destination needs to meet the preference threshold p(k)i of the agent makingthe choice, and each cell can only be identified by one agent as its destination. Formally, each agentwill randomly pick a cell, identify the acceptance by checking the surrounding cells and the currentoccupancy of the cell, and keep searching until an accept destination is confirmed.

Once each agent has a destination identified, the matching system will collect the origins anddestinations (OD) of all trips. Then, for each trip pair, based on whether sharing them will incurdetours longer than the system detour cap ∆, a shareability network is constructed. A shareabilitynetwork is a graph in which each node represents a trip, and an edge connecting two nodes repre-senting whether the two are shareable. Santi et al. proposed the shareability network model andimplemented it with a road network in New York City [2]. The model can be generalized to anyspatial context, and in this chapter I used grid cells and Manhattan distance as the base layer forthe construction of the network.

With the shareability network, the trips are then matched with a matching algorithm µ. Inthis chapter three matching algorithms were implemented and compared—maximum cardinalitymatching (MC) to maximize the number of cross-group pairs, maximum weight matching (MW) tomaximize the total mingling time of the two groups, and random greedy matching (RD). The firsttwo matching algorithms were introduced by Librino et al. in [50], and random greedy matchingis used as a baseline for comparison. The details of the algorithms will be described later in thissection.

After matching, all agents will move to destinations according to the matching, and the contacttime will be recorded. Generally, after trip sharing, as contact time increases the preference thresholdwill decrease, and people can accept cells with fewer ingroup neighbors to be destinations. Therefore,as time goes on while repeating this matching process, the space will become more integrated, asshown in the right graph of Figure 5-1. For a more precise measure of the evolution of the level

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of integration, after each tick the index of dissimilarity and Gini coefficient are calculated. Theformal definitions of the two metrics will also be given later in this section.

Different from Schelling’s interpretation of considering the occupied cells of agents to be theirhome locations, in this model I generalize the destinations of agents to include a broader settingof location choices, including all types of activities. Thence, a series of choice of destinations canbe interpreted as a chain of trips, instead of just home relocation. At the beginning of the model,when the preference threshold p is large and the initial segregation level is high, agents can onlychoose destinations in their own sectors or at the border between the two groups. Nonetheless, asthey are paired with outgroup members and gain more contact time with them, their preferencethresholds decrease, and as time goes on, they can accept trips in more diverse locations.

This destination choice, intergroup contact, preference update process keeps iterating after eachtick. I will first describe more details about the key components in the model, and then in the nextsection discuss even with similar assumptions, how the system will evolve to diverse equilibria.

Randomly gener-ate agent locations

with parameters d, s

Initialize preferencep(k)0 = 0.5 for all

agents; Set tick i = 0

For each agent k, finda non-overlapping des-tination meeting its

preference threshold p(k)i

Collect OD of all agents;Construct a shareability

network with parameter ∆

Match trips withmatching algorithm µ

All agents move todestinations; Update

contact time h(k)i

Update preference p(k)i+1

based on h(k)i for all k

Update systemmetrics li, gi

Tic

ki+1

Figure 5-2: Flow chart of the reverse Schelling model

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5.3.2 Relationship between contact time and preference threshold

According to the intergroup contact theory [84] and the recent experiment by Enos [89], the ac-ceptance to outgroup members will increase with more contact with them, and thus the minimumpreference threshold for ingroup members can potentially decrease accordingly. However, there isno empirical evidence specifically measuring the relationship between the length of contact time andthe change of preference.

In this section, a negative exponential function is used to to approximate this relationship, asexplicitly given in Eq. 5.1. In this equation h(k)i is the contact time of agent k with outgroup membersat tick i, p(k)i is the preference threshold at tick i, and α is the changing rate between contact timeand preference threshold. This function is selected based on three reason: 1) Its agreeing propertieswith the intergroup contact theory: it is a monotonic decreasing function with contact time; 2) Ourrequirement to the preference threshold to be between 0 and 1: it has a natural lower bound zeroand upper bound p(k)i ; 3) The interpretability of the formulation: it has a constant parameter αrepresenting the changing rate.

p(k)i+1 = p(k)i e−αh(k)i (5.1)

While the negative exponential function is assumed here, the empirical relationship can befurther estimated using real world data in different preference cultivation and expression settings.With a better empirical understanding of this relationship, the formulation used in this model canalways be substituted.

In Section 5.5, we will explore another possible case when preference threshold does not onlydecrease with contact time, but can also increase with it by including adversarial agents in the model[84, p. 263]. For adversarial agents, the relationship between preference threshold and contact timeis given in Eq. 5.2. With this formulation, the more adversarial agents interact with outgroupmembers, the more they would like to find destinations with fewer neighboring cells with outgroupmembers. The formulation of Eq. 5.2 is a flipped negative exponential function which is symmetric tothat of the non-adversarial agents. Similarly, this function is selected because 1) it is monotonicallyincreasing; 3) it has an upper bound one and lower bound p(k)i ; and 3) it has the same constantchanging rate α.

p(k)i+1 = 1 −[1 − p(k)i

]e−αh

(k)i (5.2)

The detailed investigation on how adversarial agents will change the convergence of integrationbe given in Section 5.5, and the formulation of the adversarials is introduced here just to completethe discussion. In this section, only Eq. 5.1 is used.

5.3.3 Manhattan distance and maximum detour

In the calculation of trip distance and detour, Manhattan distance is used. Between two grid cellsa1 = (x1, y1) and a2 = (x2, y2), the Manhattan distance between the two cells is defined as

π(a1, a2) = |x1 − x2 | + |y1 − y2 |. (5.3)

The usage of Manhattan distance provides simplicity for modeling but without losing generality.

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First, Manhattan distance preserves the order of measures with Euclidean distance, i.e., if thedistance between two points a1 and a2 is shorter than the distance between a3 and a4, or formallyd(a1, a2) < d(a3, a4), we must have π(a1, a2) < π(a3, a4). Second, the calculation of Manhattandistances includes only addition and subtraction, which is faster than calculating squares and squareroots for massive computation. Third, with Manhattan distance there exists more than one shortestpaths between two grid cells in most cases. Therefore, the model does not need to consider theimpact of route choice and congestion. Fourth, in reality, the street networks of many cities aregrid-based.

After formulating distance, the detour of sharing two trips can be explicitly expressed. For twoagents who are traveling from origins o1 and o2 to destinations d1 and d2, respectively, there areonly four possible orders of pick-ups and drop-offs for them to share the trip.

• o1, o2, d1, d2: in this case, the detour for agent 1 will be π(o1, o2) + π(o2, d1) − π(o1, d1), andthe detour for agent 2 will be π(o2, d1) + π(d1, d2) − π(o2, d2);

• o2, o1, d2, d1: this case is symmetric to the first case, but only needs to switch the order of 1and 2;

• o1, o2, d2, d1: in this case, agent 2 does not need to make a detour, so the only detour will beπ(o1, o2) + π(o2, d1) + π(d1, d2) − π(o1, o2) for agent 1;

• o2, o1, d1, d2: symmetric to the third case.

Similar to in [2], a system parameter ∆ is used to control the maximum detour allowed for all agents.Overall, two trips are considered shareable if at least one of the the above four pick-up and drop-offorders will not incur a detour longer than ∆ for either of the two agents. If with all four routings atleast one of the two agents has to take a detour longer than ∆, the two agents cannot share at thistick.

5.3.4 Shareability network and matching algorithms

If the maximum detour requirement can be satisfied for two trips, they are considered shareable.Then, based on the shareability of all trip pairs, a shareability network can be constructed. Formally,a shareability network is a graph G(V, E) in which the node set V represents all trips, and if twotrips are shareable, there is an edge connecting the two trips in the edge set E .

In this chapter, upon the shareability network at each tick i, three matching algorithms areused for comparison: random greedy matching (RD), maximum cardinality matching (MC), andmaximum weight matching (MW).

In the RD matching, the system iterates by randomly selecting an unmatched node, and thenrandomly selecting an edge connecting the node to another node which is currently unmatched andis from the other group. Then, both nodes will be marked as “matched”. The algorithm keepsiterating until no more match can be made to any node.

The MC and MW matchings follow the same formulation as in [50]. For the MC matching, thenumber of cross-group pairs is maximized in each time period, while for the MW matching, thetotal cross-group mingling time is maximized. I implemented both algorithms using the Pythonlibrary NetworkX [33].

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The shareability network is constructed at each tick right after the destinations are chosen byeach agent, and the respective matching algorithms will then be executed. In Section 5.4, I willdiscuss the impact of matching algorithms on the speed of convergence for integration.

Table 5.1: Notations used in this chapterSystem parametersd Density of agents: the ratio of agents to cellss Group share of the minoritiesµ Matching algorithm from set {MW, MC, RD}∆ Maximum detour contraint of matchingi Tick iM, N Number of agents from Group 1 and Group 2W Number of blocksms, nt Number of agents from Group 1 in block s, and number of agents

from Group 2 in block tα Rate between contact time and preference changeβ Rate of conversion between adversarial agents and non-adversarial

agentsκ Branching dissimilarity index of conversion between adversarial and

non-adversarial agentsa, b, c, e, d Coefficients of general logistic functionsMetricsli, gi Dissimilarity index and Gini coefficient at tick iπ(a1, a2) Manhattan distance from cell a1 to a2

p(k)i The preference threshold of agent k at tick ih(k)i Contact time of agent k with outgroup agents during tick ims, ns The number of agents from Group 1 and Group 2 in cell svi Proportion of adversarial agents at tick i

5.3.5 Index of dissimilarity and Gini coefficient

To measure the level of segregation and how it changes with time, I use the index of dissimilarity

1

2

W∑s=1

���ms

M−

nsN

��� , (5.4)

and Gini coefficient1

2

W∑s=1

W∑t=1

|msnt − mtns |MN

, (5.5)

as defined by Iceland and Weinberg [99]. In both definitions, the whole space is divided into a list ofnon-overlapping “blocks,” labeled from 1 to W , each containing a group of cells. ms and nt representthe numbers of agents from Group 1 and Group 2 in blocks s, t ∈ {1, . . . ,W}, and M and N denotethe total number of agents of the two groups in the whole space. Both indices range from 0 to 1,

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with 1 indicating fully segregated, and 0 fully integrated.

Intuitively, the index of dissimilarity measures the aggregated in-block difference between theproportions of two groups, while the Gini coefficient measures the total cross-block difference be-tween the proportions of the two groups for all block pairs in the space. As empirically shown inFigure 5-3 and later in the result section, the two metrics are highly correlated. However, to crossvalidate each other, throughout this chapter I will keep both indices in the discussion of results.

While using different block sizes, the exact values of index of dissimilarity and Gini coefficientdiffer, but the difference is a systematic rescaling. In other words, if a spatial distribution of agentshas a higher Gini coefficient than another distribution using one block size, it should also has ahigher Gini coefficient using another block size, as illustrated in Figure 5-3 with the result of onerun of the reverse Schelling model. In the graph, the trends of the two metrics with two block sizesare almost identical with only scaling differences. Moreover, the transition time points are all atthe same tick using both metrics and both blocks sizes. In this chapter, the block size is fixed tobe 4 × 4.

0 50 100 150 200

Ticks

0

0.2

0.4

0.6

0.8

1

Ind

ex

Dissim. index, block size 4 4

Dissim. index, block size 8 8

Gini coeff., block size 4 4

Gini coeff., block size 8 8

Figure 5-3: Indexes of dissimilarity and Gini coefficients with difference block sizes in one run ofthe reverse Schelling model

5.3.6 Unit system and equivalence of parameters

In the above discussion, all the metrics are “unitless.” Iterations of matching are in discrete “ticks,”distance are measured by the number of cells, and preference is measured as the minimum number ofingroup agents needed in neighbors. I would like to notate here that, although the measures of thisabstract model cannot be directly translated into the real-world unit system, e.g., the convergence ofintegration will happen in 10 years, the absolute value of the results are not the focus of the paper.Through the investigation between the direction of the change of parameters and the differentequilibria, the impact of each parameter can be shown.

Although the variables in the model are not measured with units, there are some variablesnaturally connected, and through these connections, some parameters are equivalent in their impacton the model, at a constant factor. Two equivalent parameter pairs are explained below.

• Contact time and distance of shared part of trips: In the model, the speed is assumed to beconstant for all trips, and no traffic congestion and routing choice is assumed. Therefore, the

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contact time in this model is only at a constant factor with the overlapping part of two tripsif shared.

• Conversion rate α and proportion of shareable trips: The conversion rate α determines howmuch contact time will change the preference threshold of agents. The greater α is, the fasterthe preference threshold decreases with contact time. Although at each tick, the contacttime of each agent is not uniform, through destination choices with randomness, overall theconversion rate should be uniformly apply to all agents. Due to the randomness, to have asmaller alpha would be equivalent to applying a constant smaller share of travelers who arewilling to share. Because of the equivalence of the two parameters, in this model I assume allagents are willing to share.

The model was implemented in NetLogo [100]. The initialization and relocation modules wererevised based on [101].

5.4 Results

In this section, I will summarize the results of the reverse Schelling model with different systemparameters, including matching algorithm, maximum detour allowed, density, group share, anddiscuss their impacts on the evolution of the level of integration.

5.4.1 Matching algorithms

Figure 5-4 shows the different convergence speeds of integration with the three matching algorithms,each with three different values of maximum detour allowed, from 10 to 30.

Controlling for the same maximum detour allowed, the maximum cross-group mingling time(MW) matching always leads to the fastest integration since it will generate the longest totalintergroup contact time. As the preference threshold is assumed to be monotonically decreasingwith contact time. to maximize the total intergroup contact time between the two groups leads tothe fastest convergence.

Maximum cross-group pair (MC) matching also leads to faster integration than the randomgreedy matching (RD) as a baseline. Although MC does not directly optimize on the total intergroupcontact time, it maximizes the number of cross-group pairs. At each tick, RD algorithm cannotmatch as many cross-group pairs as MC. Therefore, the total contact time at each tick with RD isshorter than that with MC.

The dash-dotted lines in Figure 5-4 show the corresponding indices when the agents are randomlydistributed, or the asymptotic converging equilibrium of each index. The theoretical lower boundsfor both indices are zero, only when the number of agents are completely equal in all blocks. However,if agents are randomly distributed, the probability of in each block the numbers of agents from twogroups being equal is very small. Therefore, the values with randomly distributed agents are moreappropriate benchmarks for the equilibrium level of integration. From the graphs, although thespeeds vary, all three algorithms converge to the same equilibrium level of integration after 250ticks.

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0 50 100 150 200 250 300

Ticks

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ind

ex o

f d

issim

ilarity

MW; = 10

MW; = 20

MW; = 30

MC; = 10

MC; = 20

MC; = 30

RD; = 10

RD; = 20

RD; = 30

0 50 100 150 200 250 300

Ticks

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gin

i co

eff

icie

nt

MW; = 10

MW; = 20

MW; = 30

MC; = 10

MC; = 20

MC; = 30

RD; = 10

RD; = 20

RD; = 30

Figure 5-4: Convergences with three algorithms (MW: maximum cross-group mingling time; MC:maximum number of cross-group pairs; RD: random greedy; ∆ indicates the maximum detourallowed; density = 60%; group share = 50%)

5.4.2 Maximum detour allowed, actual detour, and trip distance

The maximum detour parameter ∆ controls the maximum inefficiency that each traveler needs toendure [2]. When ∆ is large, or when longer detours are allowed, there are more trip pairs beingshareable, generating a denser shareability network. Then, it is more likely that there will be moretrips actually shared after matching, and thus longer total intergroup contact time.

Figure 5-4 shows the convergence of integration with different values of ∆, from 10 to 30. Asdiscussed, the longer the maximum detour is allowed, the longer the total intergroup contact timeis, and the faster the convergence of integration will be.

The trade-off of allowing longer maximum detour is that there will be trips actually takinglonger detours, and the average detour becomes longer. The simulation results show that theaverage detours with the MC algorithm are 3, 7, and 10 for ∆ being 10, 20, and 30, respectivelyafter convergence. The intrinsic contradiction with ∆ itself is that on one hand by allowing longerdetours, there is more intergroup contact time, which is in line with the objective of the model. Onthe other hand, detour per se is inefficient for transportation.

However, the detour also becomes shorter as integration level increases. As shown in Figure 5-5,under all three matching algorithms detours decrease with time. This is because at the beginningthe two groups of agents are segregated. In order to match trips between agents from the twogroups, they have to cross to the other half plane. As the space becomes more integrated, thedetour needed to reach an outgroup member becomes shorter. The average detour stabilizes whenthe integration level converges.

When the lengths of detour decrease as time goes on, the average distance from agents’ originsto destinations increases, shown by the blue curves in Figure 5-5. The explanation is that in theinitial segregated stage, agents can only accept locations in their own half-plane, which are closeto them. Nonetheless, as they are more acceptable to locations with more outgroup neighboringagents, their destination choices are more diverse spatially, but also further. The total trip distance

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100 200 3000

10

20

30

40

50

MC

Avg. detour

Avg. OD dist.

Avg. trip dist.

100 200 300

MW

100 200 300

RD

Ticks

Dis

tan

ce

Figure 5-5: Average detour, average trip distance without sharing, and average trip distance withsharing under different matching algorithms (∆ = 30)

is the sum of the distance from origins to destinations (OD), and the distance of detours. Withincreases in OD distance, and decrease in detour, the net effect is still a slight decline in the totaltrip distance, under all three matching algorithms. The implication is that although in the shortterm detour is inefficient, in the long run the inefficiency will reduce, and people go to spatiallymore diverse destinations.

5.4.3 Density

The density parameter d indicates how much percent of the cells are occupied by agents. If thereare more empty cells, i.e., when d is smaller, there is a higher probability that each agent is able tofind a cell that matches its preference threshold since cells surrounded by empty cells are consideredacceptable by either group. Intuitively, the empty cells act as the “transition areas” of integrationfor both groups.

Therefore, when density is lower, the convergence of integration is faster, as illustrated in Figure5-6. When 50% of cells are occupied, the highest speed of convergence happens at tick 80, whilewith 90% cells occupied, it is at tick 105.

Nonetheless, with higher density, the equilibrium dissimilarity indices and Gini coefficients canalso be lower, as shown by the dash-dotted lines in Figure 5-6. With more empty cells, in randomdistribution of agents it is less likely that each cell will be distributed with similar proportions ofagents from two groups. For example, if in a higher-density case on average each cell has 8 agents, tobe distributed (3, 5), (4, 4), or (5, 3) agents from the two groups will all result in similar proportionsof agents in this cell—37.5%, 50%, and 62.5%, respectively. However, if on average there are only4 agents in each cell, the numbers of agents have to be (2, 2) to get an even distribution, while (1,3) and (3, 1) will turn into 25% and 75%—thus less likely to reach a lower Gini coefficient.

To summarize, with higher density, it takes longer time to integrate, but an equilibrium of moreintegrated spatial distribution of agents is also more likely to happen.

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5.4.4 Group shares

In this chapter, the group share parameter s is defined as the proportion of Group 2, and the rangeof s is between 0 and 50% such that the population of Group 2 equals or less than the populationof Group 1. Considering that the square-shaped space is easier to be quadrisected or octosected, Ichose the values of s to be from 1/8 to 4/8—with Group 2 segregated in one to four of the octosectorsat the initial stage, respectively—and summarized results of convergence in Figure 5-7.

As shown in the results, the convergence of integration is the fastest when the shares of twogroups are even. The greater the difference is between the shares of the two groups, the longer timeit takes to reach the integration equilibrium. With 50% group share of each, the fastest convergencetime happens at around 100 ticks, but with 12.5% group share it happens at around 170 ticks. Theexplanation is that when the share of the smaller group is low, in the trip sharing although theagents from Group 2 can always be paired with agents from Group 1, the agents from Group 1only have small probabilities of being paired with agents from Group 2, due to the imbalance of thenumbers of agents. Therefore, the change of preference threshold of Group 1 becomes slower whenthe share difference is larger, as shown in the right graph of Figure 5-8.

Moreover, before the preference threshold of agents from Group 2 is sufficiently low, they stillneed to select destinations with some number of ingroup agents. Since the number of agents isalready small for this group, they form small clusters (left graph of Figure 5-8). These smallclusters before full convergence will also lead to higher dissimilarity index.

When the share of Group 2 is sufficiently low, e.g., 1/8, it cannot converge to the equilibrium ofa random distribution (the gap between the top curve and dash-dotted line in Figure 5-7). Thisis because in the assumption of the relationship between contact time and preference threshold,the negative exponential function is non-negative. Therefore, even if it is a small positive number,the agents still need to have at least an integer above that number of ingroup neighbors in thedestination choice, and the smallest integer that satisfies this criterion is one. Then, even if theagent can accept a destination as long as more than, e.g., 10−6 of the neighboring cells to be ingroup

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agents, it still needs at least one. To reduce this bias and allowing each agent to be able to acceptzero number of adjacent ingroup agents, a small turbulence was added to the preference thresholdfunction p(k)i+1 = p(k)i e−αh

(k)i −ε, where ε is a sufficiently small positive number. The effect of including

the small turbulence are shown in the insets of Figure 5-7. They show that for small group shares,if agents can accept zero adjacent ingroup agents in the destination choice, the integration can stillconverge to the equilibria of random distributions, after the average preference threshold of Group2 decreases below ε.

The implication of the impact of group shares on the convergence of integration is that when agroup only has a small population, even to actively match this group with outgroups still cannot letthe outgroup have sufficient contact to change acceptance due the limited population of this group.In this case, to have other formats of positive contact for the “majority” to this group, like througheducation or mass media, may be helpful to accelerate the convergence.

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5.4.5 Speed of convergence and the “tipping point”

To systematically understand the relationship between the speed of convergence and the systemparameters, I used the tick with the fastest convergence speed, or the “tipping point” to representthe time needed for convergence. Roughly, it is the “middle point” on the S-shaped curve, but tocompare the locations of the convergence results of different parameters, a systematic approach isneeded.

Noticing that the changes of Gini coefficient with time in all results are asymmetric sigmoidcurves—the curvatures of the two bends are different—I used the generalized logistic function

g(i) = a +a − d[

1 +

(ic

)b]e , (5.6)

to fit the curves, in which a, b, c, d, e are regression coefficients. In the coefficients, a and d deter-mine the maximum and minimum asymptotic values, c determines the horizontal location of thecurve, while b and d determine the curvatures. The left graph of Figure 5-9 shows one example ofgeneralized logistic regression with the result of d = 60% and s = 12.5%.

The tipping point is the time when the function declines the fastest, i.e., when d2g/di2 = 0. Tolet the second derivative of the function equal zero, we get the time of the tipping point

i |d2gdi2=0= c

(b − 1be + 1

) 1b

. (5.7)

The circle in the left graph of Figure 5-9 marks the tipping point in this example. After fitting thegeneralized function of the result with each parameter combination, the corresponding b, c, and ecan be estimated, and then using Eq. 5.7, the time of tipping point can be explicitly calculated.

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The right graph shows the relationship between the time of tipping points, and density andgroup share. All the tipping times were estimated using the above approach, with all R2 greater

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than or equal 0.99, and all root-mean-square deviations (RMSE) smaller than 0.013.

In addition to the findings discussed above that both higher density and lower group share lead toslower convergence of integration, the graph also shows the interaction effect of the two parameters.When density is high, the impact of lower group share is greater than that of when density is low.For example, with 50% density, changing group share from 50% to 12.5% will only put the tippingpoint 15 ticks later, while with 90% density, the difference is 55 ticks. Similarly, when group shareis smaller, the impact of density is greater than that of when group share is smaller (see the topcurve and the bottom curve of Figure 5-9 right graph). To summarize, the cross effect of densityand group share is stronger than with only one dimension of the two.

5.5 Counteracting Forces and Adversarial Agents

All the above discussion are based on the assumption that for all agents the preference thresholdkeeps decreasing with longer contact time. However, empirical evidence has also suggested thecounteracting force—for some adversarial people or during some adversarial time periods, the ac-ceptance of outgroup members could decrease with contact [84, 93]. In this case, the preferencethreshold will increase with longer contact time, i.e., the more they interact with outgroup members,the more ingroup members they need in the neighboring cells for their destination choices. Withlonger contact time, on the contrary to normal agents, the adversarial agents will tend to segregatemore, creating a segregative force counteracting to integration. Therefore, as introduced in 5.3.2,we assume the preference threshold to be an increasing function with contact time for adversarialagents

p(k)i+1 = 1 −[1 − p(k)i

]e−αh

(k)i .

Although the empirical study between the relationship between the integration force and thecounteracting force is lacking, there are at least three possible scenarios that can be assumed.

• The integration force is constantly stronger than the segregation force: then we can assumethe net force to be integrating, and thus the outcome of system will be as discussed above;

• The counteracting force is constantly stronger: the system will degenerate to the originalSchelling model or with even higher preference thresholds, turning into a converging equilib-rium of segregation in the end;

• The integration force and the counteracting force are competing with each other, and therelative strengths of the two depend on the current level of integration.

The first two cases have been discussed in the previous section and in [83]. In this section, we focuson the third scenario.

The relationship between the level of integration and the relative strengths of the two forces canbe very complicated, and due to many factors including the disparity in income, political views, etc.In this chapter I will only discuss one possible formulation of the relationship while acknowledgingthat with a better understanding of the relationship between the two forces and integration level,other formulations can be incorporated into the model.

The intuition of the following formulation is that the more integrated the space is, the counter-acting force becomes relatively stronger. On the other side, the more segregated the space is, the

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integration force is stronger. At a certain branching integration level, the sign of the net strengthof the two forces switches. This oscillation is a typical phenomenon frequently observed in bothnatural systems and human society, e.g., in the predator–prey model [102] and the business cycles[103].

Formally, let’s denote the share of adversarial agents at tick i as vi, dissimilarity index as li.Based on the assumption, at a certain integration level, labeled by a branching dissimilarity indexκ, the sign of the net force switches. When the space is more integrated than κ, the net force issegregative, and there will be agents converted from normal to adversarial. When the space is moresegregated than κ, the net force is integrative, and there will be agents converted from adversarialto normal. The further the current integration level is from κ, the stronger the net force becomesin either direction. The conversion rate from the difference between the current dissimilarity indexand κ, to the share of adversarial/normal agents converted is assumed to be a constant β. Bothβ and κ are system parameters depending on the nature of the system. The explicit form of theconversion rate is

vi+1 − vi =

{min {(β(κ − li), 1 − vi} , li 6 κ−min {β(li − κ), vi} , li > κ

, (5.8)

in which, the change in the share of adversarial agents depends on the difference between κ and thecurrent dissimilarity index. The min operations in the formula is used to control that the numberof agents to be converted does not exceed the number of agents in the opposite group.

With this additional formulation, another step is added to loop in the reverse Schelling model inFigure 5-2. After the last step of the iteration, the conversion rate between adversarial agents andnormal agents are calculated, and randomly the corresponding number of agents will be selected andconverted from/to adversarial to/from normal. Since being adversarial or not is independent withthe group affiliation, on average the proportions of adversarial and normal agents should be similarin both groups. Another change to the original reverse Schelling model is that, in each iteration thepreference update will depend on both contact time, and whether the agent is adversarial or not atthat tick.

To understand the impact of β and κ on the convergence of integration, different value combi-nations of the two parameters were tested on the model. Figure 5-10 shows the different systemoutcomes with κ = 0.2, 0.5, 0.8, and β = 0.01, . . . , 0.05, and 0.1. The combination of the two param-eters represent a rich set of possible equilibria of integration.

If the branching dissimilarity index κ is close to 1—the segregation end—even when the spaceis still relatively segregated, the segregation force has already become stronger than the integrationforce. Reversely, when κ is close to 0, the net force will only start to be segregative when the systemis highly integrated and will never become very strong due to the limited difference between κ andzero. The former case is shown by the top row of Figure 5-10—the integration level either convergesto or oscillate around κ = 0.8, resulting into a highly segregated equilibrium. In the bottom row,when κ = 0.2, which is even lower than the converging dissimilarity index 0.28 with d = 50%, thesystem will quickly convert all adversarial agents to normal agents, and then the system behavesthe same as without adversarial agents.

The different values of conversion rate β between adversarial and normal agents also lead to dif-ferent convergence patterns of the dissimilarity index, which is similar to “step sizes” in optimizationmodels. When β is small, the integration level converges in one or a few periods of oscillations—appearing as damped oscillations—as shown in the left four columns. However, when conversionrate β is sufficiently large, multiplying into big changes in the numbers of adversarial agents, the

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integration level cannot converge, and will keep oscillating, as shown in the last two columns ofFigure 5-10 and in Figure 5-11.

To analyze the change behind the scenes of the convergence or oscillation of integration level, thechange of the number of adversarial agents and average preference threshold by time of one dampedoscillation case (β = 0.04, κ = 0.5) were plotted in Figure 5-12 since it can illuminate on the reasonsof both oscillation and convergence. Comparing it with the respective change of dissimilarity index(the graph in the second row and the fifth column of Figure 5-10), we can find that with the changeof dissimilarity index, the number of adversarial agents change first, and the average preferencethreshold then changes with a time lag since the accumulation of contact time needs more ticks todrive the change. The change of preference threshold then conducts to the destination choices inlater ticks, and reflect back to the index of dissimilarity.

When the change of the number of adversarial agents is slow, the counteracting force will notswitch the current integration level too fast to the other side of κ, and then the amplitude ofoscillation is damped gradually. However, if the number of adversarial agents are converted at ahigh rate, the system will turn into a oscillation between two extreme segregation and integrationlevels without converging.

The interpretation of κ could be the integration level at which some people would start feelingthat the society needs to be changed, from both the normal side and the adversarial sides. Thiscould depend on the “ideal state” in people’s mind, which may be a result of ideology of the societyand education. The conversion rate β can be interpreted as the strengths of advocating of both

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groups of people who try to convince more people to join them.

In this section, in addition to the original reverse Schelling model, a model with both integra-tion force and counteracting force was investigated with one formulation of the feedback betweenintegration level and the relative strength of the two forces. The branching point of the feedback,and the rate of how feedbacks are conducted to preference changes were also further analyzed indifferent scenarios. With different values of the two parameters, the integration level can eitherdirectly converge without oscillating, turn into a damped oscillation, or keep oscillating betweenintegration and segregation.

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5.6 Discussion

In this chapter, built upon the Schelling segregation model, I proposed an abstract model which usesmobility sharing as a tool to create more opportunities for intergroup contact, and thus reverse thesegregation process in the Schelling model. I used this model to test the impact of a series of systemparameters, including density, group shares, maximum detour allowed, matching algorithms, on theequilibria of convergence of integration. Although the model is based on many assumptions, includ-ing the relationship between preference and contact, the choice of destinations, and the uniformityof agents, and thus shall not be considered as a replication of the past or forecast of the future, itcan be used to infer how the society may evolve to different equilibira under different assumptions,and what factors may need to be changed if a specific equilibrium is preferred by the society.

From the results of the model, if the positive relationship between intergroup contact and lowerpreference threshold can be assumed, once the maximum detour is fixed, different matching al-gorithms can all converge to the same level of integration. The maximum social mingling timematching can lead to the fastest convergence of integration, but the trade-off is that it will alsoincur the longest total detour to the system. If a ridesharing system is designed with social inte-gration as the objective, the balance between the speed of convergence and the average detour oftravelers will determine which matching algorithm or a combination of them will be a better option.

Also inferred from the model results, both uneven group shares and high densities will lead toslower convergence of integration. The more uneven the two group shares are and the denser thespace is, the slower will the convergence be, measured by the time of the tipping point. Further,the two factors have interaction effects—with a high density and uneven group shares at the sametime, the integration will converge even slower than with one factor of the two controlled. Thisresult could imply that in cities with higher densities or cities with very small minority groups itwould take longer to integrate using mobility sharing as the tool. However, as notated above, thisimplication might only be valid with all assumptions and with other factors to be controlled. In thereal world, big cities and small cities may differ in other aspects that could contradict the impactof density and group shares on the speed of convergence.

Based on the theories of positive and negative intergroup contact, three scenarios of the rela-tionship between the integrating force and the segregating force were explored. In the scenario ofthe two forces competing with each other, the impact of two parameters on the convergence of inte-gration was explored: the branching dissimilarity index, and the conversion rate from the differencebetween current integration level and the branching index to the proportion of agents convertedbetween adversarial and normal. Under different combinations of the two parameters, the systemcan evolve into three possible equilibria, direct convergence, damped oscillation, or lasting oscilla-tion. Implied from the model, if a converging integration without oscillation is the most desirableoutcome, the branching dissimilarity index needs to be lower than the equilibrium one. In the realworld, this could be possibly achieved through education or mass media.

Many research questions can extend and improve the model in this chapter: 1) Including asym-metric preference structure between the two groups. Empirical evidences include the asymmetricgender preference in Chapter 3, and the analysis of “dominant groups” and “subordinate groups” insocial psychology [104]; 2) In addition to limiting the cultivation and expression of preference both inthe transportation context, extending both to other fields, e.g., preference cultivation in education,residential contact, work place contact, online social networks, etc., and preference expression in theopinions to public policies, residential location choices, job location choices, etc. Besides, a compar-

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ison model can be added to understand the relative effectiveness of the preference cultivation andexpression in each field—e.g., whether transportation is a more effective tool for social integrationthan housing, or not; 3) Instead of complete segregation as the initial stage, investigate how “blurryboundary” at the beginning would change the dynamics of integration; 4) Implementing the modelwith real-world locations of city residents; 4) Understanding the difference and relationship betweenspatial integration and social integration, through the investigation of the difference between spatialproximity and social distance.

Moreover, many of the assumptions made in this chapter were based on the observations andabstractions of the situations in the U.S. or Europe, while in many countries in the global south, theintergroup relationships are different, and the forms of mobility sharing also vary [105]. A broaderinvestigation of how the model assumptions can be adjusted to be applied to the contexts of differentcountries can also be further studied in the future.

At the end of the chapter, I would like to again acknowledge that although the abstractness ofthe model may make it difficult to provide direct policy implications, the inferences from the modelcould still be useful for planners and policy makers. Further, even if it turns out that 100 ticksin this model would translate into 100 years in the real time unit, we should be confident aboutthe human wisdom of passing knowledge and compassion through generations, and still making theconvergence of integration happen in the future.

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6

Epilogue and Prospects

6.1 Summary of Findings

In this dissertation, I examined the role, structure, and potential use of the preference for fellowpassengers in mobility sharing. I investigated how stable preference between fellow passengerscan be used as an objective for matching shared trips; how gender, one key factor of the fellowpassenger preference, impacts the usage and satisfaction level in mobility sharing via the attitudestoward social interaction; and explored the potential of mobility sharing being used as a tool tochange intergroup attitudes and reduce segregation. Besides, in this dissertation I also developed anoptimization model combining mobility sharing with time flexibility to further reduce total vehicletravel time, and to improve the efficiency of mobility sharing.

Here, I would like to summarize the findings of this dissertation echoing the three overarchingquestions—how we understand, utilize, and shape preference in mobility sharing.

• On the understanding of the role of gender in the preference for fellow passengers, in Chapter3 I found that there was no significant gender difference in the usage and satisfaction levelsof mobility sharing. However, the indirect gender difference in usage and satisfaction wassignificant via the attitudes toward social interaction. The total effect combining the directand indirect effects was insignificant. Moreover, I found that user experiences would reducethis gender difference;

• On the utilization of preference for fellow passengers, in Chapter 2 I found that if stablepreference is used as the objective to match fellow passengers for mobility sharing trips, thereis only small efficiency trade-off at individual level, and moderate efficiency loss at aggregatelevel. However, the gain on the preference side is significant—passengers can be paired withtheir near-top preferred fellow passengers with preference-based matching, in comparison withbeing matched with averagely preferred fellow passengers under efficiency-based matchings.Besides, If there are people holding group-based preference, the preference gain will be reduced,especially when both groups prefer members of one of the two groups, as illustrated by thegender preference matching in Chapter 3;

• On the shaping of preference, in Chapter 5 with an abstract model I found that if people areactively paired to increase intergroup contact time, there is a potential that the preferencefor outgroup members can be more positive, and the society could evolve into equilibria with

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better integration. This integration process could lead to a reversion of the Schelling process,and several system parameters, including density, group shares, matching algorithms, andmaximum detours allowed will impact the speed and equilibrium of the convergence. Further,if there are competing integrating and segregating forces, I found that three equilibria ofsystems—direct convergence, damped oscillation, and lasting oscillation—could evolve basedon the branching state of integration, and the conversion rate from the two forces to thenumber of adversarial agents.

6.2 Areas for Future Research

Starting from this dissertation, many further research questions can be extended to complete thediscussion of preference in mobility sharing. Among the research questions, I would like to elaboratetwo of them, which are closely connected to this dissertation, and have potential significance boththeoretically and practically.

6.2.1 Boundary of respectable preference

Throughout this dissertation, it has been notated that the structure of preference for fellow passen-gers is complicated, and featured to be heterogeneous, dynamic, and more about compatibility—people have various factors in their preference for fellow passengers; the preference can change fromtime to time; and in a lot of cases, there is no universal criterion about which preference is betteras long as it is compatible between the fellow passengers. In a brainstorm with the students in theclass Behavior and Policy in the spring of 2019, over thirty factors were identified to be relevant tofellow passenger preference, ranging from sociodemographics, trip features, personality, behaviors,to personal hobbies and political views.

Behind this complexity of mobility sharing, many can be quickly identified as unrespectable andshould never be used in mobility sharing services, like race and religion, and some are acceptable tobe used by most people, like talkativeness and trip features. Nonetheless, there is also a grey area inthe middle—there are many preference factors which are acceptable to part of the population, whilenot acceptable to others. For example, gender to be used as factor for safety reasons is acceptable,but for other reasons is not. Moreover, many factors may be correlated, like trip destinations andsocial class, and thus bring more difficulty for drawing the boundary.

Based on different cultures, traditions, and regulation regimes, different countries and societiesmay have different delineations of the boundary. In this section, I would like to list a few generalprinciples that I recognized to be relevant to the respectfulness of preference. I acknowledge thatthey are far from a complete list, and may still be up for debate. While further studies are neededon the respectability of preference, this list could to be used as a starting point for discussion.

• Individual, socially existing, and socially ideal: Individual preference may be a reflection ofthe preference of social existence as historic residuals, and differ from the socially ideal. Theactual individual preference is difficult to regulate, but when a mobility sharing platform iscoded to incorporate preference into services, it should avoid exaggerating or even try to reflectthe existing unrespectable preference;

• Passivity: If a factor is only related to the passive characteristics of a fellow passenger, it

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is more likely that the preference factor should be cautioned, e.g., race, gender, and age.The active characteristics, like talkativeness, behaviors, and hobbies, are more likely to beacceptable ones;

• Inclusive vs. exclusive: Inclusive expression of preference is likely to be more acceptable thanexclusive ones, and is more related to the active characteristics of fellow passengers;

• Systematic vs. ad hoc: If a passenger uses a factor to systematically avoid trip matching witha certain social group, the factor is more likely to be unacceptable. If a factor is only used adhoc, it is more likely to be acceptable.

6.2.2 Duality of efficiency and preference

In Chapters 2 and 3, stable preference is used as the objective for fellow passenger matching, and thismatching is compared with matchings using efficiency criteria as the objectives. In the meantime,in the optimization of mobility sharing matching, both preference and efficiency criteria can also beused as constraints, e.g., the maximum detour, or certain preferences to be avoided. The possibilityof including efficiency and preference as both objectives and constraints form a natural duality. Asystematic mathematical investigation of the symmetry and asymmetry of the role of preference andefficiency in the optimizations can be another future research topic. The systematic analysis cangeneralize from specific efficiency measures, e.g., travel time, travel distance, etc., and from specificpreference measures, e.g., gender, talkative, etc., to more generic concepts of both, and the analysiscan be developed into a general framework for analyzing the symmetry and asymmetry betweentwo features of an optimization problem, and their relationships.

While a detailed investigation on the mathematical characteristics about the duality and whatmeaningful corollaries can be derived will be left for future research, I would like to lay out theinitial formulation of this analysis.

Similarly as assumed in the shareability network model, there is a node set of trips V . Some orall trip pairs are connected by edges in the edge set E . Then, two weights can be generally definedon the set, ws

e to measure the efficiency savings if two trips are shared, with any efficiency measure,and w

pe to measure the “compatibility” of two passengers. Both weights are assumed to be well

scaled and normalized across all passenger–pairs.

In an efficiency-based formulation of matching with preference as constraints, the objective canbe maximizing the total efficiency savings

s∗ = maxxe

fs(xe) = maxxe

∑e∈E

wsexe

s.t. ∀e ∈ E,wpe xe ≥ C, (6.1)

∀v ∈ V,∑e3v

xe = 1, (6.2)

where xe indicates whether an edge is included in the matching, and C is a system parameter forthe minimum compatibility of preference. The first constraint is to guarantee for each matched trippair, the preference constraint can be met, and the second constraint guarantees that nobody ismatched with more than one other passenger.

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In order to include both preference and efficiency into one formula, the Lagrange method canbe used. The Lagrange function of this optimization problem can be written as

L(xe, λe, µv) =∑e∈E

wsexe +

∑e∈E

λe(wpe xe − C) +

∑v∈V

µv

(∑e3v

xe − 1

)≥

∑e∈E

wsexe, λe ≥ 0, (6.3)

which provides an upper bound for the original problem in the feasible range of xe. The Lagrangefunction can also be used to express the primal problem

s∗ = maxxe

{min

λe≥0,µvL(xe, λe, µv)

}(6.4)

= maxxe

{min

λe≥0,µv

∑e∈E

wsexe +

∑e∈E

λe(wpe xe − C) +

∑v∈V

µv

(∑e3v

xe − 1

)}. (6.5)

Symmetrically, the Lagrange dual function can also be explicitly expressed with λe and µv

g(λe .µv) , maxxeL(xe, λe, µv) ≥

∑e∈E

wsexe (6.6)

for all xe in the feasible region and λe ≥ 0, and thus g(λe, µv) ≥ s∗. Then, the dual problem is

t∗ = minλe≥0,µv

g(λe .µv) (6.7)

= minλe≥0,µv

{maxxe

∑e∈E

wsexe +

∑e∈E

λe(wpe xe − C) +

∑v∈V

µv

(∑e3v

xe − 1

)}. (6.8)

Using the Lagrange function, we have written the efficiency weight wse and preference weight w

pe

into the same formulae in both the primal problem, and the dual problem.

Similarly, on the reverse side we can also formulate the matching to be preference-based, i.e., tomaximize the total compatibility of trips with efficiency constraints

p∗ = maxxe

fp(xe) = maxxe

∑e∈E

wpe xe

s.t. ∀e ∈ E,wsexe ≥ C̃, (6.9)

∀v ∈ V,∑e3v

xe = 1, (6.10)

where C̃ is another system parameter to constraint the efficiency savings for each matched trip pair.The two formulations are exactly symmetric to each other, and both problems consist the productswsexe and w

pe xe. Therefore, a further analysis can be conducted to bridge the relationship between

p∗ and s∗, and thus connect the problems of efficiency-based matching with preference constraints,and preference-based matching with efficiency constraints.

Beyond the boundary of respectable preference, and the duality between efficiency and prefer-ence, there are many other topics in mobility sharing that could be further explored, including thepricing model, integration of mobility sharing with public transit, and how the form of mobilitysharing will evolve with the advent of autonomous vehicles.

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6.3 Theoretical and Practical Implications

The findings in this dissertation could provide further implications for both theoretical investigation,and for planning, policy and system design practices.

From a behavior studies perspective, the findings on the interrelationships between group-basedpreference and matching outcomes, and between preference, social interaction, and usage and sat-isfaction of mobility sharing both suggested that the structure of preference does have an impacton efficiency and social outcomes. However, as revealed throughout the dissertation, the structureof the fellow passenger preference, featured with dynamism, heterogeneity and compatibility, stillneeds a comprehensive investigation. Moreover, how the role of driver changes this inter-passengerdynamics is also to be further explored.

From the mathematical perspective, this dissertation has revealed a lasting challenge of com-bining individual-based models and aggregate models, as shown in the time flexibility model whencombining the matching models and the traffic assignment models, and in the reverse Schellingmodel when analyzing the relationship between individual choices and the overall integration met-rics. This intrinsic difficulty is not only in this dissertation, but more broadly in the analysis of urbansystems, which invites further development in both mathematical tools targeting the combinationof the two types of systems, and more accurate representations of urban systems.

In addition to the theoretical investigation, the models, findings, and inferences in this disserta-tion could be used by policy makers, planners, and system designers for mobility sharing services.For policy makers and planners, this dissertation could partially answer or provide a framework ofanalysis to the following questions. 1) How could preference in mobility sharing services be used ormisused? What is the efficiency trade-off, and how to regulate the use of it? 2) What factors mayimpact the preference for fellow passengers? Are the preference factors respectable, and what fac-tors should be included/excluded in the mobility sharing services from a regulation perspective? 3)How can mobility sharing be actively used as a tool to encourage more social interaction, especiallyacross different social groups? What is the short-term cost, and the long-term benefit?

For the system designers of mobility sharing services, this dissertation can be used as a referencefor the development of a preference-based mobility sharing platform. The following questions havebeen traced, and the methods can be improved when more data are available to the system designers.1) If preference is to be used, what input data are needed, and how they need to be processed the forpreference-matching model? 2) What preference factors should be included in the system design,what factors should be used with caution, and what factors should be eliminated? 3) If timeflexibility of trips can be included in the system design, how much congestion can be reduced?What system design is needed in order to achieve this congestion reduction?

With the advancement of massive data and computation resources related to cities, the under-standing of algorithms can further enrich urban planners’ toolbox. Even if algorithms might be farfrom direct policy or regulation instruments, the understanding of them, especially how the differentstakeholders in the cities are using them for their own purposes, could benefit the planners’ abilityto further improve the life of urban residents.

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Appendix A

Universe of Preference

In the shareability network G(V, E), denote the degree of v ∈ V as deg(v) = |{u, {u, v} ∈ E}|. Then,the number of possible permutations as this passenger’s preference for shareable fellow passengersis deg(v)!. If the preferences of passengers are independent, the total possible number of preferenceassignments is

N =∏v∈V

[deg(v)!].

In the two-group scenarios, the preferences of passengers are not totally independent, but con-strained by the group affiliation of neighboring nodes. If both groups can be considered as evenlydistributed in space, like the case for gender, the expected number of same-group passengers andpassengers of the other group in the neighbors of a node are both approximately deg(v)/2. Then, ifthe preference for one group is always better than that for the other, the total number of preferenceassignments would be

N ′ =∏v∈V

{[deg(v)

2

]!

}2.

Because when x → +∞Γ(x/2)2

Γ(x)→ 0,

the number of preference possibilities is much smaller than without group-based preference.

However, the above discussion is under the assumption that the preferences of passengers areindependent, or are only constrained by group affiliation. In this case, the probability of each elementin this possible set of preferences is uniform. If the preferences of people are not independent, e.g.,some people are popular among all the others, or some are unwelcome by all the others, the universewould be much smaller. When interdependence of preference involves hard constraints, or some casesin the universe are extremely impossible, the probability of each preference assignment is stochasticwith a distribution.

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Appendix B

Results of Structural Equation Models

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Table B.1: SEM resultsEst. p-value

Factor Loadingspossoccircl 1.00+

netwk 1.03 .000atrct 0.79 .000frnds 1.06 .000smtlk 1.02 .000smevt 1.01 .000psginfrtngs 1.00+

pdemo 1.24 .000pictr 1.13 .000negsocprefi 1.00+

normi 0.92 .000unpls 0.90 .000prvcy 0.73 .000Regression Slopesntripsuser 1.50 .001gender (d1) 0.30 .594gender.user (f1) 0.27 .677age -0.11 .381income 0.25 .009edu 0.12 .530havchd 0.36 .342mr.single -0.43 .205mr.married -0.77 .067re.black 1.95 .019re.asian 0.83 .314re.hispanic 0.87 .310re.white 1.32 .064em.student -0.93 .033em.unemployed -1.32 .019possoc (b11) 0.36 .000psginf (b12) 0.12 .559negsoc (b13) -0.23 .270satisfuser -0.08 .621gender (d2) 0.18 .370gender.user (f2) 0.04 .866age -0.05 .276income 0.05 .107edu -0.00 .981havchd -0.28 .038mr.single -0.28 .018mr.married -0.20 .171re.black -0.08 .776re.asian -0.14 .638re.hispanic 0.24 .416re.white -0.02 .948em.student -0.09 .565em.unemployed -0.02 .913possoc (b21) 0.12 .001psginf (b22) 0.01 .847negsoc (b23) -0.18 .013

possocuser -0.34 .025gender (a1) -0.69 .000gender.user (e1) 0.45 .040age 0.05 .281income -0.02 .577edu -0.26 .000havchd -0.61 .000mr.single 0.32 .006mr.married -0.19 .187re.black -0.16 .572re.asian -0.09 .762re.hispanic -0.50 .091re.white -0.37 .127em.student -0.25 .094em.unemployed -0.34 .084psginfuser 0.18 .193gender (a2) 0.59 .001gender.user (e2) -0.37 .073age -0.01 .820income -0.01 .731edu 0.06 .308havchd -0.11 .350mr.single -0.08 .473mr.married -0.04 .790re.black 0.20 .451re.asian 0.40 .128re.hispanic 0.07 .797re.white 0.16 .489em.student 0.09 .522em.unemployed 0.07 .685negsocuser -0.10 .502gender (a3) 0.53 .008gender.user (e3) -0.17 .440age -0.04 .338income -0.02 .542edu 0.02 .768havchd -0.17 .195mr.single 0.02 .865mr.married -0.22 .138re.black 0.18 .532re.asian 0.41 .161re.hispanic -0.13 .658re.white 0.18 .474em.student 0.00 .997em.unemployed 0.16 .428Indirect and Total Effectsc1 -0.30 .023c2 -0.17 .002g1 0.16 .190g2 0.08 .133t1 0.01 .990t2 0.01 .973u1 0.43 .509u2 0.12 .597

Residual Variancescircl 0.64 .000netwk 0.60 .000atrct 1.40 .000frnds 0.51 .000smtlk 0.79 .000smevt 0.92 .000rtngs 1.16 .000pdemo 0.49 .000pictr 0.65 .000prefi 1.04 .000normi 1.07 .000unpls 1.72 .000prvcy 1.94 .000ntrips 18.41 .000satisf 2.24 .000user 0.19+

gender 0.25+

gender.user 0.21+

age 1.58+

income 2.40+

edu 0.64+

havchd 0.19+

mr.single 0.24+

mr.married 0.21+

re.black 0.08+

re.asian 0.08+

re.hispanic 0.07+

re.white 0.21+

em.student 0.12+

em.unemployed 0.06+

Latent Variancespossoc 2.06 .000psginf 1.72 .000negsoc 1.89 .000Latent Covariancespossoc∼psginf 0.29 .000psginf∼negsoc 1.39 .000

Fit Indicesχ2 844.33 .000

(df=233)CFI 0.94TLI 0.91RMSEA 0.05+Fixed parameter

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