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Teck H. Ho 1
Peer-induced Fairness in Games
Teck H. HoUniversity of California, Berkeley
(Joint Work with Xuanming Su)
October, 2009
Teck H. Ho 2
Outline
Motivation
Distributive versus Peer-induced Fairness
The Model
Equilibrium Analysis and Hypotheses
Experiments and ResultsMarch, 2009
Teck H. Ho 3
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)
March, 2009
Teck H. Ho 4
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
March, 2009
Teck H. Ho 5
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
Teck H. Ho 6
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
Teck H. Ho 7
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)
March, 2009
Teck H. Ho 8
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)
March, 2009
Teck H. Ho 9
Outline
Motivation
Distributive versus Peer-induced Fairness
The Model
Equilibrium Analysis and Hypotheses
Experiments and ResultsMarch, 2009
Teck H. Ho 12
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
March, 2009
Teck H. Ho 14
Ultimatum Offers Across 16 Small Societies (Mean Shaded, Mode is Largest Circle…)
Mean offersRange 26%-58%
March, 2009
Teck H. Ho 15
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)
March, 2009
Teck H. Ho 16
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
March, 2009
Teck H. Ho
distr
ibut
iona
l
fairn
ess
distributional
fairness
Distributional and Peer-Induced Fairness
peer-induced fairness
March, 2009 18
Teck H. Ho
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?
March, 2009
Teck H. Ho 20
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)
March, 2009
Teck H. Ho 21
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
March, 2009
Teck H. Ho 22
Outline
Motivation
Distributive versus Peer-induced Fairness
The Model
Equilibrium Analysis and Hypotheses
Experiments and ResultsMarch, 2009
Teck H. Ho 23
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
March, 2009
Teck H. Ho 24
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.
March, 2009
Teck H. Ho 25
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
March, 2009
Teck H. Ho 26
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
March, 2009
Teck H. Ho 27
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
March, 2009
Teck H. Ho 28
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
March, 2009
Teck H. Ho 2929
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.
March, 2009
Teck H. Ho 30
Sequence of Events
Ultimatum Game 1Leader : Follower 1
Ultimatum Game 2Leader : Follower 2
Noise GenerationUniform Noise
March, 2009
Teck H. Ho 31
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
March, 2009
Teck H. Ho 32
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
March, 2009
Teck H. Ho 33
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 ?
March, 2009
Teck H. Ho 34
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%)
March, 2009
Teck H. Ho 35
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
March, 2009
Teck H. Ho 36
Test of Hypothesis 2: Second Offer vis-à-vis the Expectation of the First Offer
Being AheadBeing Behind
On Par
March, 2009
Teck H. Ho 37
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
March, 2009
Teck H. Ho 38
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)
March, 2009
Teck H. Ho 39
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
March, 2009
Teck H. Ho 42
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
March, 2009
Teck H. Ho 44
Model Applications
Price discrimination
Executive compensation
Union negotiation
March, 2009
Teck H. Ho46
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