Network Effects - Columbia Universityac3827/CodesSlides/NetworkEffects.pdfNetwork effects captured...
Transcript of Network Effects - Columbia Universityac3827/CodesSlides/NetworkEffects.pdfNetwork effects captured...
Network Effects
Motivation01
CONTENTS Network Effect Recap02
Program Demo03
MOTIVATION
01PART
MOTIVATION
Ideal Model for ImplementationThe majority of the analysis is based on the generic form of the utility function: π’ π π,π‘ , π₯π,π‘ , ππ‘ , ππ‘β1 =
π₯π πβ ππ‘β1 + 1 β π β ππ‘ π π,π‘ β ππ π,π‘ and the distribution πΊ(π₯)
Key Characteristic of NetworksNetwork effects captured one widely observed phenomenon of the behavior of agents in a network,
especially in social network settings
Intuitive Study ToolA graphical representation of the result we got in class will be a very intuitive study tool for
students to better understand the theories on network effect
NETWORK EFFECTRECAP
02PART
NETWORK EFFECT RECAP β SET UP
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π,, ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Type DistributionEach agent has a type π₯π, which describes his/her utility from taking action π = 1. π₯π comes from the
distribution πΊ(π₯) (CDF)
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Type DistributionEach agent has a type π₯π, which describes his/her utility from taking action π = 1. π₯π comes from the
distribution πΊ(π₯) (CDF)
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
Network Effect Functionβ π is an increasing function that captures positive network effect on agents
NETWORK EFFECT RECAP β SET UP
Homogeneous Utility FunctionEach agent has the utility function: π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘, where π π,π‘ β {0,1}
denotes the strategy of agent π at time π‘, and ππ‘ denotes the fraction of population that choose π = 1
Type DistributionEach agent has a type π₯π, which describes his/her utility from taking action π = 1. π₯π comes from the
distribution πΊ(π₯) (CDF)
Infinite AgentsModelling a community with a large number of agents. For instance, a continuum πΌ β [0,1] of agents
Network Effect Functionβ π is an increasing function that captures positive network effects on agents
Other Constant Parametersπ: cost of taking action π = 1
π: weight of the network effect from the previous period
NETWORK EFFECT RECAP β MAIN RESULT
Fixed Point Equation
Only agents that that will have positive utilities from taking action π = 1 will take this action. Recall that the utility
function has the form:
π’ π π,π‘ , π₯π , ππ‘ , ππ‘β1 = π₯π πβ ππ‘β1 + 1 β π β ππ‘ β π π π,π‘
Thus, only the agents with the following type will take action π = 1:
π₯π >π
πβ ππ‘β1 + 1 β π β ππ‘
Thus, we have the fixed point equation for the fraction of population that take action π = 1:
ππ‘ = 1 β πΊ(π
πβ ππ‘β1 + 1 β π β ππ‘)
PROGRAM DEMO
03PART
PROGRAM DEMO β STRUCTURE
Part 1:
Solve for equilibrium
Part 2-1:
Finite Agents Simulation
Part 2-2:
Analytical Form Analysis
User input 1: type distribution, network effect function, parametersβ¦
Output of equilibrium from an analysis using a finite number of agents
User input 2: input an initial state from which an equilibrium will be derived
Output of equilibrium from the closed form analysis based on the fixed point equation
PROGRAM DEMO β ASSUMPTIONS
Type Distribution
πΊ π₯ =πΎ + π½
1 + π½
1 + π½ πβπΌβ²
π₯
πΎ + π½πβπΌβ²
π₯
Network Effect Function
πΊβ1 π =βπΌβ²
πππ(πΎπ
πΎ + π½ β π½π)
β π = π
PROGRAM DEMO β ASSUMPTIONS
PROGRAM DEMO
Demo in Spyder
THANKS