peer-induced fairness in games - berkeley haas

Peer-induced Fairness in Games

Teck H. HoUniversity of California, Berkeley

(Joint Work with Xuanming Su)

Teck H. Ho 1March, 2009

Outline

Motivation

Distributive versus Peer-induced Fairness

The ModelThe Model

Equilibrium Analysis and HypothesesEquilibrium Analysis and Hypotheses

Experiments and ResultsTeck H. Ho 2

Experiments and ResultsMarch, 2009

Dual Pillars of Economic Analysis

S ifi ti f UtilitSpecification of UtilityOnly final allocation mattersSelf interestSelf-interestExponential discounting

Solution MethodNash equilibrium and its refinements (instantNash equilibrium and its refinements (instant equilibration)


Motivation: Utility Specificationy p

Reference point matters: People care both about the finalReference 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 unkindwhen Mary is unkind

Hyperbolic discounting: People are impatient and prefer yp g p p pinstant gratification


Motivation: Solution Method

Nash equilibrium and its refinements: DominantNash 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 i i iinteractions in some games.Multiplicity problem (e.g., coordination and infinitel repeated games)infinitely repeated games).Modeling subject heterogeneity really matters in games

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games.March, 2009

Bounded Rationality in Markets: R i d Utilit F tiRevised Utility Function

Behavioral Regularities Standard Assumption New Model Specification Marketing Application ExampleBehavioral 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)

Teck H. Ho 6Ho, Lim, and Camerer (JMR, 2006)

March, 2009

Bounded Rationality in Markets: Alternative Solution MethodsAlternative Solution Methods

Behavioral Regularities Standard Assumption New Model Specification Marketing Application ExampleBehavioral 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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March, 2009

Standard Assumptions in Equilibrium AnalysisEquilibrium Analysis

A ti N h C iti QRE EWAAssumptions 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 XTeck H. Ho 8

Instant Convergence X X XMarch, 2009

Modeling Philosophyg p y

Simple (Economics)p ( )General (Economics)Precise (Economics)Empirically disciplined (Psychology)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 Tychowould have been absurd in physics to expect Kepler and Newton without TychoBrahe” (von Neumann & Morgenstern ‘44)

“With t h i b d t f f t hi h t th i th i t i“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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knowledge is inadequate... (Eric Van Damme 95)

March, 2009

Outline

Motivation


The ModelThe Model




Distributive Fairness


Ultimatum GameUltimatum Game

Yes? No?Yes? No?

Split pie accordingly

Both getnothing

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Empirical Regularities in Ultimatum GameUltimatum Game

Proposer offers division of $10; responder accepts or rejectsProposer offers division of $10; responder accepts or rejects

Empirical Regularities:

There are very few offers above $5There are very few offers above $5

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

There are almost no offers below $2There are almost no offers below $2

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

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


Ultimatum Experimental SitesUltimatum Experimental Sites

Henrich et al (2001; 2005)

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Henrich et. al (2001; 2005)March, 2009

Ultimatum Offers Across 16 Small Societies (M Sh d d M d i L t Ci l )(Mean Shaded, Mode is Largest Circle…)

Mean offersRange 26%-58%Range 26% 58%


Modeling Challenges & Classes of TheoriesModeling Challenges & Classes of Theories

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

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


A Model of Social Preference(Ch d R bi 2002)(Charness and Rabin, 2002)

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

)1()(),( ⋅⋅−⋅−+⋅⋅+⋅= srsrU BABAB πσρπσρππ

otherwise;0 and,if1 where

)1()(),(

=>=

++

rr

srsrU

AB

BABAB

ππ

πσρπσρππ

otherwise. 0 and , if 1 =<= ss AB

AB

ππ

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.

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g g g p y

March, 2009

Peer-induced Fairness


Distributional and Peer-Induced FairnessDistributional and Peer Induced Fairness

peer-induced fairness

Teck H. Ho

p

March, 2009 19

A Marketing Interpretationg p

SELLERSELLERposted price

posted price

take it or

peer-induced fairnessBUYER BUYERleave it?

Teck H. Ho20

March, 2009

Examples of Peer-Induced Fairness

Price discrimination (e.g., iPhone)

Employee compensation (e g your peers’ pay)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)


Social Comparison

Theory of social comparison: Festinger (1954)

One of the earliest subfields within social psychology

H db k f S i l C i (S l d Wh l 2000)Handbook of Social Comparison (Suls and Wheeler, 2000)

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


Outline

Motivation


The ModelThe Model




Modeling Differences between Distributional and Peer induced FairnessDistributional and Peer-induced Fairness

2-person versus 3-person2-person versus 3-person

Reference point in peer-induced fairness is derived from how a i d i i il i ipeer is treated in a similar situation

1-kink versus 2-kink in utility function specification1 kink versus 2 kink in utility function specification

People have a drive to look to their peers to evaluate their d tendowments


The Model Setup

3 Players, 1 leader and 2 followers3 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 gamesecond ultimatum game.

Leader receives payoff from both games. Each follower receives

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only payoff in their respective game.March, 2009

Revised Utility Function: Follower 1

The leader divides the pie: )( ssπThe leader divides the pie:

Follower 1’s utility is:

) ,( 11 ss−π

⎩⎨⎧ =−−⋅−= 0if0

1. if })(,0max{),( 1111111 a

asssasU Fπδ

⎩⎨ = .0if ,0),(

1111 aF

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


Revised Utility Function: Follower 2

Follower 2 believes that Follower 1 receives sFollower 2 believes that Follower 1 receives

The leader divides the pie:

1s

) ,( 22 ss−π

Follower 2’s utility is:

⎩⎨⎧

==⋅⋅−−⋅−= .0 if ,0

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

221222222 a

aszpssszasUFρπδ

Follower 2 does not like to be behind the leader (δ > 0) and does

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not like to receive a worse offer than Follower 1 (ρ > 0) March, 2009

Revised Utility Function: The Leader

The leader receives utilities from both gamesThe leader receives utilities from both games

In the second ultimatum game:

⎩⎨⎧

==−−⋅−−= .0 if ,0

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

222222, a

assszasU IILπδπ

In the first ultimatum game:

⎨⎧=−−⋅−−= 1.if )}(,0max{)( 1111 asssasU πδπ

Leader does not like to be behind both followers

⎩⎨ == .0if ,0)}({),(

1

111111, aasU IL


Hypotheses

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

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

*2 >= ρsfs


Economic Experiments

Standard experimental economics methodology: Subjects’ p gy jdecisions are consequential75 undergraduates, 4 experimental sessions.Subjects were told the following:

Subjects were told their cash earnings depend on their and others’ decisions15-21 subjects per session; divided into groups of 3Subjects were randomly assigned either as Leader or Follower 1, or Follower 2Follower 2The game was repeated 24 timesThe game lasted for 1.5 hours and the average earning per subject was $19

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$19.March, 2009

Sequence of EventsSequence of Events

Ultimatum Game 1Leader : Follower 1

Ultimatum Game 2Leader : Follower 2

Noise GenerationUniform Noise


Subjects’ DecisionsSubjects Decisions

Leader to Follower 1

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

1s2s X

20)

Follower 1Accept or reject aAccept or reject

Follower 2(i e a guess of what is after observing )1s 1s Xs +1

1a

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

Accept or reject

Respective payoff outcomes are revealed at the end of both games

1s 1s Xs +1

2a


Hypotheses

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

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

)0|ˆ( 1*2 >= ρsfs

( p )


Tests of Hypothesis 1: Follower 2’s Decision

Being Ahead On Par Being Behind

N Number of

R j ti

N Number of

R j ti

N Number of Rejection

Rejection Rejection

165 ? 110 ? 179 ?


Tests of Hypothesis 1: Follower 2’s Decision

Being Ahead On Par Being Behind

N Number of

R j ti

N Number of

R j ti

N Number of Rejection

Rejection Rejection

165 6 (3.6%) 110 5 (4.5%) 179 42 (23 5%)(23.5%)


Tests of Hypothesis 1: Logistic Regression

Follower 2’s utility is:y

⎩⎨⎧

==⋅⋅−−⋅−= .0 if ,0

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

221222222 a

aszpssszasUFρπδ

Probability of accepting is:

)05.0( 024.0ˆ 2 <−= pγ


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

On Par

Being AheadBeing Behind


Tests of Hypothesis 2: Simple Regression

The theory predicts that is piecewise linear in 2s 1sy p p2 1

That is, we have 01 >α

)01.0( 09.0ˆ1 <= pα


Implication of Proposition 1: S2* > S1*

Method 1:Each game outcome involving a triplet in a round as an independent observationWilcoxon signed-rank test (p-value = 0.03)

Method 2:Each subject’s average offer across rounds as an independent observationobservationCompare the average first and second offersWilcoxon signed-rank test (p-value = 0 04)

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Wilcoxon signed-rank test (p-value 0.04)March, 2009

Structural EstimationStructural Estimation

Th t t tl t i j lThe target outlets are economics journals

W i h l i dWe want to estimate how large is compared to (important for field applications)

ρ δ

Is self-interested assumption a reasonable approximation?

Understand the degree of heterogeneity


Is Self-Interested Assumption a Reasonable Approximation? Noa Reasonable Approximation? No


Is Peer-Induced Fairness Important? YES


Latent-Class ModelLatent Class Model

Th l ti i t f 2 f l S lf i t t dThe population consists of 2 groups of players: Self-interested and fairness-minded players

The proportion of fairness-minded θ

See paper for Propositions 5 and 6: depends on θ*2s


Is Subject Pool Heterogeneous? 50% of Subjects are Fairness minded50% of Subjects are Fairness-minded

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Model ApplicationsModel Applications

Price discrimination

Executive compensation

Union negotiationUnion negotiation


SummarySummary

Peer-induced fairness exists in gamesPeer induced fairness exists in games

Leader is strategic enough to exploit the phenomenonLeader is strategic enough to exploit the phenomenon

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

50% of the subjects are fairness-minded

Teck H. Ho46

peer-induced fairness in games - berkeley haas

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