Peer-induced Fairness in Games

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Peer-induced Fairness in Games
Teck H. Ho University of California,
Berkeley (Joint Work with Xuanming Su)
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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 Marys payoff. In
  addition, the marginal utility of John with
  respect to an increase in Marys 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
Ho, Lim, and Camerer (JMR, 2006)
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Bounded Rationality in Markets Alternative
Solution Methods
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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 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 Bs utility is given
  as
• Bs utility is a weighted sum of her own monetary
  payoff and As payoff, where the weight places on
  As 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
distributional fairness
distributional fairness
peer-induced fairness
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A Market Interpretation
SELLER
posted price
posted price
distributional fairness
distributional fairness
take it or leave it?
BUYER
BUYER
peer-induced fairness
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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 (OReily, 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_co
  mparison_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
• Follower 1s utility is
• Follower 1 does not like to be behind the leader
  (dB gt 0)

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Revised Utility Function Follower 2

• Follower 2 believes that Follower 1 receives
• The leader divides the pie
• Follower 2s utility is
• Follower 2 does not like to be behind the leader
  (d gt 0) and does not like to receive a worse
  offer than Follower 1 (r gt 0)

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Revised Utility Function The Leader

• 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

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Hypotheses

• Hypothesis 1 Follower 2 exhibits peer-induced
  fairness. That is,
• gt 0.
• Hypothesis 2 If gt 0, The leaders offer
  to the second follower depends on Follower 2s
  expectation of what the first offer is. That is,

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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
• 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

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Hypotheses

• Hypothesis 1 Follower 2 exhibits peer-induced
  fairness. That is,
• gt 0.
• Hypothesis 2 If gt 0, The leaders offer
  to the second follower depends on Follower 2s
  expectation of what the first offer is. That is,
  
• (Proposition 1)

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Tests of Hypothesis 1 Follower 2s Decision
Being Ahead Being Ahead On Par On Par Being Behind 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 2s Decision
Being Ahead Being Ahead On Par On Par Being Behind 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 2s utility is
• Probability of accepting is

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Test of Hypothesis 2 Second Offer vis-à-vis the
Expectation of the First Offer
On Par
Being Ahead
Being Behind
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Tests of Hypothesis 2 Simple Regression

• The theory predicts that is piecewise
  linear in
• That is, we have

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Implication of Proposition 1 S2 gt 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 subjects 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

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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