Teck H. Ho 1

Peer-induced Fairness in Games

Teck H. HoUniversity of California, Berkeley

(Joint Work with Xuanming Su)

October, 2009

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Outline

Motivation

Distributive versus Peer-induced Fairness

The Model

Equilibrium Analysis and Hypotheses

Experiments and ResultsMarch, 2009

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Dual Pillars of Economic Analysis

Specification of Utility Only final allocation matters Self-interest Exponential discounting

Solution Method Nash equilibrium and its refinements (instant

equilibration)

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Motivation: Utility Specification

Reference point matters: People care both about the final allocation as well as the changes with respect to a target level

Fairness: John cares about Mary’s payoff. In addition, the marginal utility of John with respect to an increase in Mary’s income increases when Mary is kind to John and decreases when Mary is unkind

Hyperbolic discounting: People are impatient and prefer instant gratification

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Motivation: Solution Method

Nash equilibrium and its refinements: Dominant theories in marketing for predicting behaviors in non-cooperative games.

Subjects do not play Nash in many one-shot games. Behaviors do not converge to Nash with repeated

interactions in some games. Multiplicity problem (e.g., coordination and

infinitely repeated games). Modeling subject heterogeneity really matters in

games.March, 2009

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Bounded Rationality in Markets: Revised Utility Function

Behavioral Regularities Standard Assumption New Model Specification Marketing Application ExampleReference Example

1. Revised Utility Function

- Reference point and - Expected Utility Theory - Prospect Theory - Ho and Zhang (2008) loss aversion Kahneman and Tversky (1979)

- Fairness - Self-interested - Inequality aversion - Cui, Raju, and Zhang (2007) Fehr and Schmidt (1999)

- Impatience - Exponential discounting - Hyperbolic Discounting - Della Vigna and Malmendier (2004) Ainslie (1975)

Ho, Lim, and Camerer (JMR, 2006)March, 2009

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Bounded Rationality in Markets: Alternative Solution Methods

Behavioral Regularities Standard Assumption New Model Specification Marketing Application ExampleExample

2. Bounded Computation Ability

- Nosiy Best Response - Best Response - Quantal Best Response - Lim and Ho (2008) McKelvey and Palfrey (1995)

- Limited Thinking Steps - Rational expectation - Cognitive hierarchy - Goldfrad and Yang (2007) Camerer, Ho, Chong (2004)

- Myopic and learn - Instant equilibration - Experience weighted attraction - Amaldoss and Jain (2005) Camerer and Ho (1999)

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Modeling Philosophy

Simple (Economics)General (Economics)Precise (Economics)Empirically disciplined (Psychology)

“the empirical background of economic science is definitely inadequate...it would have been absurd in physics to expect Kepler and Newton without Tycho Brahe” (von Neumann & Morgenstern ‘44)

“Without having a broad set of facts on which to theorize, there is a certain danger of spending too much time on models that are mathematically elegant, yet have little connection to actual behavior. At present our empirical knowledge is inadequate...” (Eric Van Damme ‘95)

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Outline

Motivation

Distributive versus Peer-induced Fairness

The Model

Equilibrium Analysis and Hypotheses

Experiments and ResultsMarch, 2009

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Distributive Fairness

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Ultimatum Game

Yes? No?

Split pie accordingly

Both getnothing

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Empirical Regularities in Ultimatum Game

Proposer offers division of $10; responder accepts or rejects

Empirical Regularities:

There are very few offers above $5

Between 60-80% of the offers are between $4 and $5

There are almost no offers below $2

Low offers are frequently rejected and the probability of rejection decreases with the offer

Self-interest predicts that the proposer would offer 10 cents to the respondent and that the latter would accept

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Ultimatum Experimental Sites

Henrich et. al (2001; 2005)

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Ultimatum Offers Across 16 Small Societies (Mean Shaded, Mode is Largest Circle…)

Mean offersRange 26%-58%

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Modeling Challenges & Classes of Theories

The challenge is to have a general, precise, psychologically plausible model of social preferences

Three major theories that capture distributive fairness Fehr-Schmidt (1999) Bolton-Ockenfels (2000) Charness-Rabin (2002)

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A Model of Social Preference(Charness and Rabin, 2002)

Blow is a general model that captures both classes of theories. Player B’s utility is given as:

B’s utility is a weighted sum of her own monetary payoff and A’s payoff, where the weight places on A’s payoff depend on whether A is getting a higher or lower payoff than B.

otherwise. 0 and , if 1

otherwise; 0 and , if 1 where

)1()(),(

ss

rr

srsrU

AB

AB

BABAB

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Peer-induced Fairness

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distr

ibut

iona

l

fairn

ess

distributional

fairness

Distributional and Peer-Induced Fairness

peer-induced fairness

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distr

ibut

iona

l

fairn

ess

distributional

fairness

19

A Market Interpretation

peer-induced fairness

SELLER

BUYER BUYER

posted priceposted price

take it or leave it?

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Examples of Peer-Induced Fairness

Price discrimination (e.g., iPhone)

Employee compensation (e.g., your peers’ pay)

Parents and children (favoritism)

CEO compensation (O’Reily, Main, and Crystal, 1988)

Labor union negotiation (Babcock, Wang, and Loewenstein, 1996)

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Social Comparison

Theory of social comparison: Festinger (1954)

One of the earliest subfields within social psychology

Handbook of Social Comparison (Suls and Wheeler, 2000)

WIKIPEDIA: http://en.wikipedia.org/wiki/Social_comparison_theory

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Outline

Motivation

Distributive versus Peer-induced Fairness

The Model

Equilibrium Analysis and Hypotheses

Experiments and ResultsMarch, 2009

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Modeling Differences between Distributional and Peer-induced Fairness

2-person versus 3-person

Reference point in peer-induced fairness is derived from how a peer is treated in a similar situation

1-kink versus 2-kink in utility function specification

People have a drive to look to their peers to evaluate their endowments

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The Model Setup

3 Players, 1 leader and 2 followers

Two independent ultimatum games played in sequence

The leader and the first follower play the ultimatum game first.

The second follower receives a noisy signal about what the first follower receives. The leader and the second follower then play the second ultimatum game.

Leader receives payoff from both games. Each follower receives only payoff in their respective game.

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Revised Utility Function: Follower 1

.0 if ,0 1. if })(,0max{

),(1

1111111 a

asssasU F

The leader divides the pie:

Follower 1’s utility is:

Follower 1 does not like to be behind the leader (B > 0)

) ,( 11 ss

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Revised Utility Function: Follower 2

.0 if ,0

1. if }s-(z)ˆ max{0,)(ˆ - })(,0max{)|,(2

221222222 a

aszpssszasU F

Follower 2 believes that Follower 1 receives

The leader divides the pie:

Follower 2’s utility is:

Follower 2 does not like to be behind the leader ( > 0) and does not like to receive a worse offer than Follower 1 ( > 0)

1̂s

) ,( 22 ss

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Revised Utility Function: The Leader

.0 if ,0

1. if )}(,0max{)|,(2

222222, a

assszasU IIL

The leader receives utilities from both games

In the second ultimatum game:

In the first ultimatum game:

Leader does not like to be behind both followers

.0 if ,0

1. if )}(,0max{),(1

111111, a

asssasU IL

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Hypotheses

Hypothesis 1: Follower 2 exhibits peer-induced fairness. That is, > 0.

Hypothesis 2: If > 0, The leader’s offer to the second follower depends on Follower 2’s expectation of what the first offer is. That is,

)0|ˆ( 1*2 sfs

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Economic Experiments

Standard experimental economics methodology: Subjects’ decisions are consequential

75 undergraduates, 4 experimental sessions.

Subjects were told the following: Subjects were told their cash earnings depend on their and others’

decisions

15-21 subjects per session; divided into groups of 3

Subjects were randomly assigned either as Leader or Follower 1, or Follower 2

The game was repeated 24 times

The game lasted for 1.5 hours and the average earning per subject was $19.

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Sequence of Events

Ultimatum Game 1Leader : Follower 1

Ultimatum Game 2Leader : Follower 2

Noise GenerationUniform Noise

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Subjects’ Decisions

Leader to Follower 1

to Follower 2 after observing the random draw (-20, - 10, 0, 10, 20)

Follower 1 Accept or reject

Follower 2 (i.e., a guess of what is after observing )

Accept or reject

Respective payoff outcomes are revealed at the end of both games

1s

2s

1̂s 1s Xs 1

X

1a

2a

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Hypotheses

Hypothesis 1: Follower 2 exhibits peer-induced fairness. That is, > 0.

Hypothesis 2: If > 0, The leader’s offer to the second follower depends on Follower 2’s expectation of what the first offer is. That is, (Proposition 1)

)0|ˆ( 1*2 sfs

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Tests of Hypothesis 1: Follower 2’s Decision

Being Ahead On Par Being Behind

N Number of Rejection

N Number of Rejection

N Number of Rejection

165 ? 110 ? 179 ?

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Tests of Hypothesis 1: Follower 2’s Decision

Being Ahead On Par Being Behind

N Number of Rejection

N Number of Rejection

N Number of Rejection

165 6 (3.6%) 110 5 (4.5%) 179 42 (23.5%)

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Tests of Hypothesis 1: Logistic Regression

Follower 2’s utility is:

Probability of accepting is:

)05.0( 024.0ˆ2 p

.0 if ,0

1. if }s-(z)ˆ max{0,)(ˆ - })(,0max{)|,(2

221222222 a

aszpssszasU F

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Test of Hypothesis 2: Second Offer vis-à-vis the Expectation of the First Offer

Being AheadBeing Behind

On Par

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Tests of Hypothesis 2: Simple Regression

The theory predicts that is piecewise linear in

That is, we have

2s 1̂s

01

)01.0( 09.0ˆ1 p

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Implication of Proposition 1: S2* > S1*

Method 1: Each game outcome involving a triplet in a round as an

independent observation

Wilcoxon signed-rank test (p-value = 0.03)

Method 2: Each subject’s average offer across rounds as an

independent observation

Compare the average first and second offers

Wilcoxon signed-rank test (p-value = 0.04)

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Structural Estimation

The target outlets are economics journals

We want to estimate how large is compared to (important for field applications)

Is self-interested assumption a reasonable approximation?

Understand the degree of heterogeneity

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Is Self-Interested Assumption a Reasonable Approximation? No

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Is Peer-Induced Fairness Important? YES

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Latent-Class Model

The population consists of 2 groups of players: Self-interested and fairness-minded players

The proportion of fairness-minded

See paper for Propositions 5 and 6: depends on

*2s

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Is Subject Pool Heterogeneous? 50% of Subjects are Fairness-minded

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Model Applications

Price discrimination

Executive compensation

Union negotiation

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Price Discrimination

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Summary

Peer-induced fairness exists in games

Leader is strategic enough to exploit the phenomenon

Peer-induced fairness parameter is 2 to 3 times larger than distributional fairness parameter

50% of the subjects are fairness-minded

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Standard Assumptions in Equilibrium Analysis

Assumptions Nash Cognitive QRE EWAEquilbirum Hierarchy Learning

Solution Method

Strategic Thinking X X X X

Best Response X X

Mutual Consistency X X

Instant Convergence X X X

March, 2009