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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)

Outline

- Motivation
- Distributive versus Peer-induced Fairness
- The Model
- Equilibrium Analysis and Hypotheses
- Experiments and Results

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)

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

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 manyone-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.

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)

Outline

- Motivation
- Distributive versus Peer-induced Fairness
- The Model
- Equilibrium Analysis and Hypotheses
- Experiments and Results

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

Ultimatum Experimental Sites

Henrich et. al (2001; 2005)

Ultimatum Offers Across 16 Small Societies (Mean Shaded, Mode is Largest Circle…)

Mean offers

Range 26%-58%

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)

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.

Distributional and Peer-Induced Fairness

distributional

fairness

distributional

fairness

peer-induced fairness

A Market Interpretation

SELLER

posted price

posted price

distributional

fairness

distributional

fairness

take it or

leave it?

BUYER

BUYER

peer-induced fairness

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)

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

Outline

- Motivation
- Distributive versus Peer-induced Fairness
- The Model
- Equilibrium Analysis and Hypotheses
- Experiments and Results

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

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.

Revised Utility Function: Follower 1

- The leader divides the pie:
- Follower 1’s utility is:
- Follower 1 does not like to be behind the leader (dB > 0)

Revised Utility Function: Follower 2

- 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 (d > 0) and does not like to receive a worse offer than Follower 1 (r > 0)

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

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,

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.

29

Sequence of Events

Ultimatum Game 2

Leader : Follower 2

Ultimatum Game 1

Leader : Follower 1

Noise Generation

Uniform Noise

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

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)

Tests of Hypothesis 1: Logistic Regression

- Follower 2’s utility is:
- Probability of accepting is:

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

On Par

Being Behind

Being Ahead

Tests of Hypothesis 2: Simple Regression

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

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)

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

Is Self-Interested Assumption a Reasonable Approximation? No

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

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

Model Applications

- Price discrimination
- Executive compensation
- Union negotiation

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

Standard Assumptions in Equilibrium Analysis

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