peer-induced fairness in games - berkeley haas
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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)
Teck H. Ho 3March, 2009
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
Teck H. Ho 4March, 2009
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
Distributive versus Peer-induced Fairness
The ModelThe Model
Equilibrium Analysis and HypothesesEquilibrium Analysis and Hypotheses
Experiments and ResultsTeck H. Ho 10
Experiments and ResultsMarch, 2009
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
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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%
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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)
Teck H. Ho 16March, 2009
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
Distributional and Peer-Induced FairnessDistributional and Peer Induced Fairness
peer-induced fairness
Teck H. Ho
p
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A Marketing Interpretationg p
SELLERSELLERposted price
posted price
take it or
peer-induced fairnessBUYER BUYERleave it?
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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)
Teck H. Ho 21March, 2009
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
Teck H. Ho 22March, 2009
Outline
Motivation
Distributive versus Peer-induced Fairness
The ModelThe Model
Equilibrium Analysis and HypothesesEquilibrium Analysis and Hypotheses
Experiments and ResultsTeck H. Ho 23
Experiments and ResultsMarch, 2009
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
Teck H. Ho 24March, 2009
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)
Teck H. Ho 26March, 2009
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
Teck H. Ho 28March, 2009
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
Teck H. Ho 29March, 2009
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
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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
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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 )
Teck H. Ho 33March, 2009
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 ?
Teck H. Ho 34March, 2009
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%)
Teck H. Ho 35March, 2009
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
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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
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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
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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
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Is Self-Interested Assumption a Reasonable Approximation? Noa Reasonable Approximation? No
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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
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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
Teck H. Ho 45March, 2009
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
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