A variety of Beauty Contest games
Rosemarie Nagel
UPF-ICREA-GSE
2012
Rules
Choose 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 Game
Or, 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, gametheorists and
experimenters
2/3-m e a n la b -stu d e n ts
0.20
0,20
relative frequences
re la t iv e f re q u e n c ie s
0.15
m ean: 36.73
2/3-m ean: 23.49
0.10
0.05
0,15
mean: 18.98
2/3-mean: 12.65
0,10
0,05
0.00
0,00
14
22
33
50
100
67
0
14
22
33
50
chosen num bers
ch o s e n n u m b e r s
6. Newspaper experiments (15-17)
0,10
First period results with different
populations (Nagel 1995, Bosch et al.
2002)
0,08
mean: 23.08
2/3mean: 15.39
0,06
0,04
0,02
0,00
22
33
50
100
100
Rules, theories, and data for the basic game
3 Newspaper experiments (Spektrum, Financial
Times, Expansion)
Rules
Choose 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
0,10
0,08
average: 23.08
0,06
0,04
0,02
0,00
22
33
100
50
1 . ite ra te d e lim in a tio n o f d o m in a te d stra te g ie s
     IT E R A T IO N
E q u ilib riu m
...
0
... E (4 ) E (3 )
E (2 )
1 3 .1 7 1 9 .7 5 2 9 .6 3
2. iterated best response
... ... E (3) E (2)
E (1)
E (1 )
4 4 .4 4
E (0 )
6 6 .6 6
100
E (0)
Main problem: starting point=level 0
0
14.89 22.22
33.33
50
100
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
M ean behavior over tim e
som e variations
100
mean
80
4/3-m ean
60
0.7-m ean, 3 players
40
2/3-m ean, 15-18
players
20
0
1/2-m edian
1
2
3
4
5
6
7
8
9
10
tim e
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 games
by 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.
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
Enlarge to 400% to read rules of the Beauty
Abstract
Contest Game (BCG) & theoretic approach
The fathers of BCG
A=level 1 (33)
B=level 2 (22)
C=level 3 (15)
6000 chess players participate in beauty contest
p>1
p=2/3
Keynes
Ledoux
Lab: Level of reasoning-Unravelling- NoGameFormRecognition
(GFR) for N=2
p=2/3
2. iterated best response
... ... E (3) E (2)
E (1)
0
14.89 22.22
33.33
N>2
E (0)
Lab, classroom, conference, internet, newspaper
50
100
Lit: AER’95, EJ’97, GEB’08, ExpEc09
FIELD: Parallelism-mixture models
Lit: AER’02, ExpEc’10, Expansion’97, FT’97, Spektrum’98, ‘08
Lit: PNAS’09, Hist. of Econ. Ideas’09, bookEconomía Experimental y del Comportamiento’11
Before survey
Lit:
Level ≥2
level1
After survey
??????
fMRI Machine
mPFC-ACC activity
Surveys: Nagel’98,’04,HBExRe’08,Festschrift Selten’10, Camerer 2003, Crawford et al JEL forthcoming
Survey: Policy Change
To be known in ESA
NY June 2012
Conclusion: Guesses are everywhere and it is all the same
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