A variety of Beauty Contest games

Rosemarie Nagel

UPF-ICREA-GSE

2012

RulesChoose a number between 0 and 100. The winner is the person whose number is closest to 2/3 times the average of all chosen numbers

I. Basic Beauty Contest Game

• The rules of the basic beauty-contest game: • N participants are asked to guess a number from the

interval 0 to 100. N=2 is very different from N>2 (dominant strategy

equilibrium vs iterated elimination of dominated strategies) • The winner is the person whose guess is closest to 2/3

times the mean of the choices of all players. • The winner gets a fixed prize of $20. In case of a tie the

prize is split amongst those who tie.• The same game may be repeated several periods• History: subjects are informed of the mean, 2/3 mean

and all choices in each period. • Time to think: up to two weeks• Participants: students, theorists, “newspaper readers” etc

• text in bold italics indicates the variations in the different experiments

Beauty Contest GameOr, to change the metaphor slightly, professional investment

may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a

hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average

preferences of the competitors as a whole; so that each competitor has to pick not those faces which he himself finds

prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those

which, to the best of one’s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest.

We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the

average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.

Keynes (1936, p. 156)

What happened?

Hypotheses? • Are players rational? • What does “rationality” imply in this game? • How should a rational player behave in a

population in which not everyone is perfectly rational?

=> More general: What expectations do we have about others?

What other questions can you answer with this game

What other experiments can you do with this game?

=> New designs

Why is a study of human behavior with this game interesting?

• clear distinction between bounded rationality and game theoretic solution

• game with unique game theoretic solution• separation of strategic factors from motivational factors

(as e.g. fairness, cooperation)• pure strategic game (constant some game)• behavior can be interpreted and visualized as “pure

bounded rationality” “detection” of different levels of reasoning via – iterated best reply– iterated elimination of dominated strategies

• each single aspect can be found in other games but the combination of all five are not easily met at once in other games

2/3-mean lab-students

0.00

0.05

0.10

0.15

0.20

67

chosen numbers

rela

tive

freq

uenc

ies

22 50 10033

mean: 36.732/3-mean: 23.49

14

6. Newspaper experiments (15-17)

0,00

0,02

0,04

0,06

0,08

0,10

10022 50

mean: 23.082/3mean: 15.39

33

2/3-mean, gametheorists and experimenters

0,00

0,05

0,10

0,15

0,20

chosen numbers

rela

tive

fre

qu

ence

s

22 50 10033

mean: 18.982/3-mean: 12.65

0 14

First period results with different populations (Nagel 1995, Bosch et al. 2002)

Rules, theories, and data for the basic game

RulesChoose a number between 0 and 100. The winner is the person whose number is closest to 2/3 times the average of all chosen numbers

3 Newspaper experiments (Spektrum, Financial Times, Expansion)

0,00

0,02

0,04

0,06

0,08

0,10

10022 50

average: 23.08

33

1. iterated elimination of dominated strategies Equilibrium ITERATION

... ... E(4) E(3) E(2) E(1) E(0)

0 13.17 19.75 29.63 44.44 66.66 100

2. iterated best response ... ... E(3) E(2) E(0)

E(1)

0 14.89 22.22 33.33 50 100 Main problem: starting point=level 0

Iterated best reply model characteristics

• Not equilibrium model=strategies of players don’t have to be best reply to each other

• No common knowlegde of rationality requirement

• Limited reasoning• Best reply to own belief (no consistent beliefs)• Purely strategic• Random behavior is also a strategy • Theoretical value plus noise (e.g. 50*pk+/-є,where p

is parameter of game and k is level of reasoning)

• Problem: what is level zero

Mean behavior over time

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10

time

mea

n

4/3-mean

0.7-mean, 3 players

2/3-mean, 15-18players

1/2-median

some variations

Nagel 1995, Camerer, Ho AER 1998)

More Variations

Slonim, Experimental Economics 200?

How to design an experiment to separate two hypotheses?

1.(Many) people don’t play equilibrium because they are confused. 2.(Many) people don’t play equilibrium because doing so (choosing 0) doesn‟t win; rather they are cleverly anticipating the behavior of others, with noise.

2 person guessing gamesby Brit Grosskopf & Nagel (GEB 2008)

• Many experiments have shown that participants do not necessarily behave according to equilibrium predictions.

• Lots of explanations, here are two:– No clue about equilibrium behavior.– A fully rational player might realize what equilibrium behavior looks like,

however doubts that all choose it.• Doubt about other players' rationality.• Belief about other players' doubts about rationality of Co-players

• Hard to separate observationally, since equilibrium strategies are not in general best replies to non-equilibrium choices of other players.

• We focus now on 2-person Beauty-Contest Games– Rational player chooses weakly dominant strategy 0.– Boundedly rational player

Survey• Using guesses to measure expectations and maybe to figure out what

people consider as “right”.

• Incentivizing actions

• To induce policy change

• Let’s look at ESA EXECUTIVE COMMITTEE COMPOSITION GUESSING GAME (we show some questions from survey

Motivation

• Create awareness about the situation of women in the executive committee of ESA

• Create cognitive dissonance about own guess /guess of guesses and reality

• To make that people give suggestions how to change the situation

• To show that survey methods can be helpful

=> result: there will be changes.

Guessing games in the Lab, Field, Brain, and Surveys:

Preparation for a general lecture Lab experiment: Level of Reasoning (descriptive

theory vs. full rationality=fix point)

Field experiment: going public/informing the public, loosing control => consequences?

Brain: biological data to connect with/understand behavior

Survey: Policy Advise with survey usage, creating awareness, cognitive dissonance (your guess, other peoples guesses=> activate protest.

Coauthors on the guessing game (in order of appearance): R Selten J Duffy A Bosch J Montalvo A Satorra B Grosskopf G Coricelli C Plott E Chou M McConnell V Crawfort M CostaGomes C Bühren B Frank H Llavador M Nagel A Perdomo

Conclusion: Guesses are everywhere and it is all the same

Guessing Games in the Lab, Field, fMRI Machine, and Survey: Level of Reasoning-Unravelling-GFR, Parallelism-Mixture Models, mPFC-ACC Activity, and Policy Change

By Rosemarie Nagel, ICREA, BGSE & Universitat Pompeu Fabra, SABE ’12, Granada Abstract

2. iterated best response ... ... E(3) E(2) E(0)

E(1)

0 14.89 22.22 33.33 50 100

Lab: Level of reasoning-Unravelling- NoGameFormRecognition (GFR) for N=2

Lab, classroom, conference, internet, newspaperFIELD: Parallelism-mixture models

fMRI Machine mPFC-ACC activity Survey: Policy Change

Before survey

After survey

To be known in ESA NY June 2012

Enlarge to 400% to read rules of the Beauty Contest Game (BCG) & theoretic approach

Keynes Ledoux

N>2

??????

p=2/3

A=level 1 (33)B=level 2 (22)C=level 3 (15)

Level ≥2 level1

Lit: AER’02, ExpEc’10, Expansion’97, FT’97, Spektrum’98, ‘08

p>1p=2/3

The fathers of BCG

Lit: AER’95, EJ’97, GEB’08, ExpEc09

Lit: PNAS’09, Hist. of Econ. Ideas’09, bookEconomía Experimental y del Comportamiento’11

Surveys: Nagel’98,’04,HBExRe’08,Festschrift Selten’10, Camerer 2003, Crawford et al JEL forthcoming

Lit:

6000 chess players participate in beauty contest