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# 1 / 2 bases + coordination
Sacha Bourgeois-Gironde
1. Coordination under cognitive hierarchies
outline
• Session one :
• Quick outlook of basic GT analysis
• Introducing CHT / => Applications to behavioral analyses of different games
Coordination problems #1: selecting single equilibria among several available (pure coordination games, BoS).
Coordination problems #2: heterogeneously bounded cognition among players in presence of one single players (beauty contests, Hotelling games).
Laquelle pensez-vous qu’on pense en général être la plus belle?
1 2 3
4 5 6
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)
Instructions
• Choisissez de manière privée un nombre entre 0 et 100. Le vainqueur sera celui ou celle dont le nombre est le plus proche de la moitié de la moyenne de tous les nombres choisis.
The Beauty Contest
A rational player does not simply choose a random number or her favourite number, nor does she choose a number above 67 (100 x 2/3), since it is dominated by 67.
Moreover, if she believes that the other participants are rational as well, she will not pick a number above 100 x 2/3 x 2/3; and if she believes that the others are rational and that they also believe that all are rational, she will not pick a number above 100 x 2/3 x 2/3 x 2/3 and so on, until all numbers are eliminated but zero.
The Beauty-contest game is an ideal tool to study whether individuals reason in steps and how many iterated levels subjects actually apply.
1. A quick outlook of basic GT analysis
What is a game?
• : a set of players, descriptions of their information, and a fix order of the sequence of choices by different players / a function mapping players’ choices and information to outcomes.
• The specification of a game is completed by a payoff function that attaches a numerical value (or utility) to each outcome.
Standard analysis
• => compute an equilibrium point
• = a set of strategies for each player which are simultaneously best responses to one another (Nash 1950)
• = = solving simultaneous equations in which each player’s strategy is an input to the other player’s calculation of expected payoff.
Stringent epistemic requirements
• Common prior beliefs about chance events
• Belief by players that all players are rational
• Know that their beliefs are common knowledge (KKKKKK….)
Think otherwise than purely standard
• (anticipated by Nash)
How equilibrium might arise (behaviorally)
=> asymptotic ideas:
• # of players (mass action): populations learn about what others do and adjust their strategies toward optimization.
* time, repetition.
2. A Cognitive Hierarchy (CH) Model of Games
Camerer, Ho, and Chong (2004)
The Quarterly Journal of Economics
Motivation
Nash equilibrium and its refinements: Dominant theories in economics for predicting behaviors in competitive situations.
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 games). Modeling heterogeneity really matters in games.
Main Goals
Provide a behavioral theory to explain and predict behaviors in any one-shot gameNormal-form games (e.g., zero-sum game, p-beauty
contest)Extensive-form games (e.g., centipede)
Provide an empirical alternative to Nash equilibrium (Camerer, Ho, and Chong, QJE, 2004) and backward induction principle (Ho, Camerer, and Chong, 2005)
Modeling Principles
Principle Nash CH
Strategic Thinking
Best Response
Mutual Consistency
5 elements to any CH model
1. Distribution of level k types• In Camerer, Ho & Chong (2004) the distribution
of level k types is assumed to follow a Poisson distribution with a mean value tau.
• Once the value of tau is chosen, the complete distribution is known.
• A nice property of a Poisson distribution is that the frequency of a very high level k drops off quickly for higher values of k.
2. A specification of the action of level 0 players
• Level 0 types are usually assumed to choose strategy equally often (random).
• What else can we think of?
3.Beliefs of level k players about other players
• In CH level k players know the correct proportion of lower-level players (but « overconfidence).
• Alternative models: level k modeling (all other players are level k-1)
• ECH : no overconfidence.
4. Assessing the expected payoffs based on the beliefs in (3)
• Each player in a hierarchy can compute the expected payoffs to different strategies
• Level 1s compute their expeccted payoffs, knowing what levels 0 will do.
• Level 2s compute their expected payoffs given their guess about what levels 1s and 0s do, and how frequent these level types are, etc.
5. A choice response function based on the expected payoffs
in (4).
• In the simplest case players choose the strategy with the highest expected payoff : the best response.
• (what else can we imagine?).
Example 1: “zero-sum game”
COLUMNL C R
T 0,0 10,-10 -5,5
ROW M -15,15 15,-15 25,-25
B 5,-5 -10,10 0,0
Messick(1965), Behavioral Science
Nash Prediction: “zero-sum game”
Nash COLUMN Equilibrium
L C RT 0,0 10,-10 -5,5 0.40
ROW M -15,15 15,-15 25,-25 0.11
B 5,-5 -10,10 0,0 0.49Nash
Equilibrium 0.56 0.20 0.24
CH Prediction: “zero-sum game”
Nash CH ModelCOLUMN Equilibrium ( = 1.55)
L C RT 0,0 10,-10 -5,5 0.40 0.07
ROW M -15,15 15,-15 25,-25 0.11 0.40
B 5,-5 -10,10 0,0 0.49 0.53Nash
Equilibrium 0.56 0.20 0.24CH Model( = 1.55) 0.86 0.07 0.07
Empirical Frequency: “zero-sum game”
Nash CH Model EmpiricalCOLUMN Equilibrium ( = 1.55) Frequency
L C RT 0,0 10,-10 -5,5 0.40 0.07 0.13
ROW M -15,15 15,-15 25,-25 0.11 0.40 0.33
B 5,-5 -10,10 0,0 0.49 0.53 0.54Nash
Equilibrium 0.56 0.20 0.24CH Model( = 1.55) 0.86 0.07 0.07Empirical
Frequency 0.88 0.08 0.04
The Cognitive Hierarchy (CH) Model
People are different and have different decision rules
Modeling heterogeneity (i.e., distribution of types of players). Types of players are denoted by levels 0, 1, 2, 3,…,
Modeling decision rule of each type
Modeling Decision Rule
Proportion of k-step is f(k)
Step 0 choose randomly
k-step thinkers know proportions f(0),...f(k-1)
Form beliefs and best-respond based on beliefs
Iterative and no need to solve a fixed point
gk (h) f (h)
f (h ' )h ' 1
K 1
Implications
Poisson distribution with mean and variance =
in 24 beauty contests).
!)(
kekf
k
Theoretical Implications
Exhibits “increasingly rational expectations”
Normalized gK(h) approximates f(h) more closely as k ∞∞ (i.e., highest level types are
“sophisticated” (or "worldly") and earn the most.
Highest level type actions converge as k ∞∞
marginal benefit of thinking harder 00
Poisson Distribution
f(k) with mean step of thinking :!
)(k
ekfk
Poisson distributions for various
00.05
0.10.15
0.20.25
0.30.35
0.4
0 1 2 3 4 5 6
number of steps
fre
qu
en
cy
COLUMNL C R
T 0,0 10,-10 -5,5
ROW M -15,15 15,-15 25,-25
B 5,-5 -10,10 0,0
K's K+1's ROW COLLevel (K) Proportion Belief T M B L C R
0 0.212 1.00 0.33 0.33 0.33 0.33 0.33 0.33Aggregate 1.00 0.33 0.33 0.33 0.33 0.33 0.33
0 0.212 0.39 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.61 0 1 0 1 0 0
Aggregate 1.00 0.13 0.74 0.13 0.74 0.13 0.130 0.212 0.27 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.41 0 1 0 1 0 02 0.255 0.32 0 0 1 1 0 0
Aggregate 1.00 0.09 0.50 0.41 0.82 0.09 0.09
K Proportion, f(k)0 0.2121 0.3292 0.2553 0.132
>3 0.072
COLUMNL C R
T 0,0 10,-10 -5,5
ROW M -15,15 15,-15 25,-25
B 5,-5 -10,10 0,0
K's K+1's ROW COLLevel(K) Proportion Belief T M B L C R
0 0.212 1.00 0.33 0.33 0.33 0.33 0.33 0.33Aggregate 1.00 0.33 0.33 0.33 0.33 0.33 0.33
0 0.212 0.39 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.61 0 1 0 1 0 0
Aggregate 1.00 0.13 0.74 0.13 0.74 0.13 0.130 0.212 0.27 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.41 0 1 0 1 0 02 0.255 0.32 0 0 1 1 0 0
Aggregate 1.00 0.09 0.50 0.41 0.82 0.09 0.090 0.212 0.23 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.35 0 1 0 1 0 02 0.255 0.28 0 0 1 1 0 03 0.132 0.14 0 0 1 1 0 0
Aggregate 1.00 0.08 0.43 0.50 0.85 0.08 0.08
Theoretical Properties of CH Model
Advantages over Nash equilibrium
Can “solve” multiplicity problem (picks one statistical distribution)
Sensible interpretation of mixed strategies (de facto purification)
Theory: τ∞ converges to Nash equilibrium in (weakly)
dominance solvable games
Figure 3b: Predicted Frequencies of Nash Equilibrium for Entry and Mixed Games
y = 0.707x + 0.1011
R2 = 0.4873
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Empirical Frequency
Pre
dic
ted
Fre
qu
en
cy
Nash: Theory vs. Data
Figure 2b: Predicted Frequencies of Cognitive Hierarchy Models for Entry and Mixed Games (common )
y = 0.8785x + 0.0419
R2 = 0.8027
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Empirical Frequency
Pre
dic
ted
Fre
qu
ency
CH Model: Theory vs. Data
A variety of Beauty Contest games
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.
limits
• 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.
The Colonel Blotto Game
Imagine you are a colonel in command of an army during wartime. You and thecolonel of the enemy’s army each command 120 troops. Your troops will engage the enemy in 6 battles on 6 separate battlefields.It is the night before the battles and each of you must decide how to deploy your forces across the 6 battlefields. In the morning, you will win a battle if the number of troops you have assigned to a particular battlefield is higher than that assigned by your opponent. In the case that you have both allocated the same number of troops to a particular battlefield, the outcome of the battle will be a loss for both of you.Your deployment of troops will face that of each of the other participants in thetournament. Your total score will be the number of battles you win against all the other participants.How will you deploy your 120 troops?
Emile Borel 1921 / Ariel Rubinstein 2009
