Behavioral Game Theory:A Brief Introduction

Networked LifeCSE 112

Spring 2005Prof. Michael Kearns

Supplementary slides courtesy of Colin Camerer, CalTech

Behavioral Game Theoryand Game Practice

• Game theory: how rational individuals should behave

• Who are these rational individuals?• BGT: looks at how people actually behave

– experiment by setting up real economic situations– account for people’s economic decisions– don’t break game theory when it works

• Fit a model to observations, not “rationality”

Feeling in ultimatum games: How Feeling in ultimatum games: How much do you offer out of $10? much do you offer out of $10?

Proposer has $10Proposer has $10 Offers x to Responder (keeps $10-x)Offers x to Responder (keeps $10-x) What should the Responder do? What should the Responder do?

Self-interest: Take any x>0Self-interest: Take any x>0 Empirical: Empirical: Reject x=$2 half the Reject x=$2 half the

timetime

How People Ultimatum-Bargain

Thousands of games have been played in experiments…

• In different cultures around the world• With different stakes• With different mixes of men and women• By students of different majors• Etc. etc. etc.

Pretty much always, two things prove true:

1. Player 1 offers close to, but less than, half (40% or so)

2. Player 2 rejects low offers (20% or less)

Ultimatum offer experimental sites

Ultimatum Bargaining across Cultures

Sharing norms differ in the industrialized worldJapan, Israel lowest (Roth et al. 1991)

Machiguenga farmers in Peru (Henrich 2000)Offered 26% on average, accepted all but 1 offerVery socially disconnected

Ache in Paraguay, Lamelara in IndonesiaMade hyperfair (more than 50%) offers Headhunters (potlatch culture), whalers

slash & burngathered foods

fishinghunting

The Machiguengaindependent families

cash cropping

Fair offers correlate with market integration (top), Fair offers correlate with market integration (top), cooperativeness in everyday life (bottom)cooperativeness in everyday life (bottom)

Ultimatum offers across societies Ultimatum offers across societies (mean shaded, mode is largest circle…)(mean shaded, mode is largest circle…)

Ultimatum Bargaining across Majors

Economics majors offer 7% less, accept 7% less(Carter and Irons 1991)

They must have learned game theory!

… but this behavior is consistent across years of study (freshman to seniors) … maybe their game-theoretic nature made them want to study economics?

Other studies show no correlation, or that econ/business students offer more.

Ultimatum Bargaining and Looks

70 University of Miami students, photographed and rated for attractiveness (Schweitzer and Solnick 1999)

Man as player 1, attractive woman as player 2…Doesn’t make much difference

Woman as player 1, attractive man as player 2…Average offer is 50.7% (hyperfair!)Small percentage (1 or 2?) offer almost everything

Stakes, Entitlement, Framing

Indonesia: from a day’s wages to a month’s wagesNo difference…

Florida: answer questions to get $400 pie instead of $20More low offers at $400 … but subjects earned it

Framing it as a buyer/seller exchange lowers offers 10%

Framing it as a resource competition raises them slightly(Hoffman et al. 1994)

Ultimatum offers of children who Ultimatum offers of children who failed/passed false belief testfailed/passed false belief test

Subject (autistic?) complaining post-Subject (autistic?) complaining post-experiment (Zamir, 2000)experiment (Zamir, 2000)

Feeling: This is your brain on unfairnessFeeling: This is your brain on unfairness(Sanfey et al, Sci 13 March ’03)(Sanfey et al, Sci 13 March ’03)

1.1. Limited equilibrationLimited equilibrationBeauty contest gameBeauty contest game

N players choose numbers xN players choose numbers xii in in [0,100][0,100]

Compute target (2/3)*(Compute target (2/3)*( x xii /N) /N)

Closest to target wins $20Closest to target wins $20

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 9

17

25

33

41

49

57

65

73

81

89

97

number choices

pre

dic

ted

fre

qu

en

cy

Beauty contest results (Expansion, Financial Times, Spektrum)

0.00

0.05

0.10

0.15

0.20

numbers

rela

tive

fr

eq

ue

nci

es

22 50 10033

average 23.07

0

Beauty Contest

Some number of players try to guess a number that is 2/3 of the average guess.

The answer can’t be between 68 and 100 - no use guessing in that interval. It is dominated.

But if no one guesses in that interval, the answer won’t be greater than 44.

But if no one guesses more than 44, the answer won’t be greater than 29…

Everyone should guess 0! And good game theorists would…

But they’d lose…

Iterated Dominance

People don’t instantly compute all the way to 0

The median subject uses 1 or 2 rounds of iteration (25, 35)

Guessing 0 on the first round (game theorist) is poor

Guessing 30 (behavioral game theory) is much better

But 30 isn’t a good guess the seventh time you play…

A New Theory…

We could create new per-game theories…But this would be useless.

We could consider these as repeated games of some sort…

But that complicates a lot of things.

Maybe we can make a small change to something underlying…

What if people don’t only care about their own payoffs?

A New Theory of Utility

Consider that people still like their payoffsThey also dislike others having more money, with some coefficient

.And they dislike having more money than others, with coefficient

.

U_1 is player 1’s utility; P_1 & P_2 are the players’ payoffs.

U_1 = P_1 - (max[P_2 - P_1, 0]) - (max[P_1 - P_2,0])

is “envy” is “guilt”

0 <= < 1 < Different players can have different and

Inequality Aversion

U_1 = P_1 - _1(max[P_2 - P_1, 0]) - _1(max[P_1 - P_2,0])

(Fehr and Schmidt 1999)

Now, we can do classical game theory, but with U, not P

Player 2 should reject any offer < _2/(1 + 2_2)If = 1/3, player 2 should reject any offer less than 20%

Player 1 offers will depend onEstimates of player 2 envy (_2) distribution

and Player 1 guilt (_1)

Inequality Aversion: Advantages

• Model generalizes easily to more than 2 players• = 1/3, = 0 can explain a lot!• Ultimatum bargaining• Multi-player ultimatum bargaining (“Market

game”)• Even dictator games

• Parameters can be tuned for cultures or individuals• Does not break most of the existing, correct

predictions of non-IA game theory

Inequality Aversion on Graphs

For games where IA game theory works, we could put these games on graphs.

Do players care about global inequalities or neighborhood inequalities?

Our guesses may agree, but it’s an open question: no experiment has been done!