Peer-induced Fairness in Games
Teck H. Ho
University 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 Results
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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.
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Bounded Rationality in Markets:
Revised Utility Function
Behavioral Regularities
Standard Assumption
New Model Specification
Reference Example
Marketing Application Example
1. Revised Utility Function
- Reference point and
loss aversion
- Expected Utility Theory
- Prospect Theory
- Ho and Zhang (2008)
Kahneman and Tversky (1979)
- Fairness
- Self-interested
- Inequality aversion
Fehr and Schmidt (1999)
- Cui, Raju, and Zhang (2007)
- Impatience
- Exponential discounting
- Hyperbolic Discounting
Ainslie (1975)
- Della Vigna and Malmendier (2004)
Ho, Lim, and Camerer (JMR, 2006)
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Bounded Rationality in Markets:
Alternative Solution Methods
Behavioral Regularities
New Model Specification
Example
Standard Assumption
Marketing Application Example
2. Bounded Computation Ability
- Nosiy Best Response
- Best Response
- Quantal Best Response
McKelvey and Palfrey (1995)
- Lim and Ho (2008)
- Limited Thinking Steps
- Rational expectation
- Cognitive hierarchy
Camerer, Ho, Chong (2004)
- Goldfrad and Yang (2007)
- Myopic and learn
- Instant equilibration
- Experience weighted attraction
Camerer and Ho (1999)
- Amaldoss and Jain (2005)
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Modeling Philosophy
Simple
General
Precise
Empirically disciplined
(Economics)
(Economics)
(Economics)
(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 Results
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Distributive Fairness
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Ultimatum Game
Yes? No?
Split pie
accordingly
Both get
nothing
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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 offers
Range 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:
U B ( A ,  B )  (   r    s )   A  (1    r    s)   B
where
r  1 if  B   A , and r  0 otherwise;
s  1 if  B   A , and s  0 otherwise.
 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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Peer-induced Fairness
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Distributional and Peer-Induced Fairness
peer-induced fairness
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A Market Interpretation
posted price
SELLER
posted price
take it or
leave it?
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BUYER
peer-induced fairness
BUYER
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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 Results
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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
 The leader divides the pie: (  s1 , s1 )
 Follower 1’s utility is:
s1    max{ 0, (  s1 )  s1} if a1  1.

U F1 ( s1 , a1 )  
if a1  0.
0,
 Follower 1 does not like to be behind the leader (B > 0)
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Revised Utility Function: Follower 2
 Follower 2 believes that Follower 1 receives ŝ1
 The leader divides the pie: (  s2 , s2 )
 Follower 2’s utility is:
ˆ
ˆ
U F 2 (s2 , a2 | z )  s2    max{ 0, (  s2 )  s2 } -   p( z )  max{0, s1 (z) - s 2 } if a2  1.
if a2  0.
0,
 Follower 2 does not like to be behind the leader ( > 0) and does
not like to receive a worse offer than Follower 1 ( > 0)
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Revised Utility Function: The Leader
 The leader receives utilities from both games
 In the second ultimatum game:
  s2    max{ 0, s2  (  s2 )}
U L, II ( s2 , a2 | z )  
0,

if a2  1.
if a2  0.
 In the first ultimatum game:
  s1    max{ 0, s1  (  s1 )}
U L, I ( s1 , a1 )  
0,

if a1  1.
if a1  0.
 Leader does not like to be behind both followers
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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, s2*  f (sˆ1 |   0)
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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 2
Leader : Follower 2
Ultimatum Game 1
Leader : Follower 1
Noise Generation
Uniform Noise
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Subjects’ Decisions
 Leader

s1 to Follower 1
 s2 to Follower 2 after observing the random draw X (-20, - 10, 0, 10,
20)
 Follower 1
 Accept or reject
a1
 Follower 2


ŝ1 (i.e., a guess of what s1 is after observing s1  X)
Accept or reject a 2
 Respective payoff outcomes are revealed at the end of both games
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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, s2*  f (sˆ1 |   0)
(Proposition 1)
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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:
ˆ
ˆ
U F 2 (s2 , a2 | z )  s2    max{ 0, (  s2 )  s2 } -   p( z )  max{0, s1 (z) - s 2 } if a2  1.
if a2  0.
0,
 Probability of accepting is:

ˆ2  0.024 ( p  0.05)
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Test of Hypothesis 2: Second Offer vis-à-vis
the Expectation of the First Offer
On Par
Being Behind
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Being Ahead
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Tests of Hypothesis 2: Simple Regression
 The theory predicts that
 That is, we have

s2 is piecewise linear in ŝ1
1  0
ˆ1  0.09 ( p  0.01)
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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 
*
s
 See paper for Propositions 5 and 6: 2 depends on 
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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
Di ( pi )  Ai  pi
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i  L, H
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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
Equilbirum
Cognitive
Hierarchy
QRE
EWA
Learning
Strategic Thinking
X
X
X
X
Best Response
X
X
Mutual Consistency
X
Solution Method
Instant Convergence
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X
X
X
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X
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