A Cognitive Hierarchy Model of
One-Shot Games
Teck H. Ho
Haas School of Business
University of California, Berkeley
Joint work with Colin Camerer, Caltech
Juin-Kuan Chong, NUS
April 15, 2004
CH Model
Teck-Hua Ho
1
Motivation
Nash equilibrium and its refinements: Dominant
theories in economics for predicting behaviors in
games.
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.
April 15, 2004
CH Model
Teck-Hua Ho
2
Behavioral Game Theory
How to model bounded rationality in one-shot
games?
Cognitive Hierarchy (CH) model (Camerer, Ho, and
Chong, QJE, 2004)
How to model equilibration?
EWA learning model (Camerer and Ho, Econometrica,
1999; Ho, Camerer, and Chong, 2004)
How to model repeated game behavior?
Teaching model (Camerer, Ho, and Chong, JET, 2002)
April 15, 2004
CH Model
Teck-Hua Ho
3
First-Shot Games
The FCC license auctions, elections, military
campaigns, legal disputes
Many marketing models and simple game
experiments
Initial conditions for learning
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CH Model
Teck-Hua Ho
4
Modeling Principles
Principle
Nash
CH
Strategic Thinking


Best Response


Mutual Consistency

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CH Model
Teck-Hua Ho
5
Modeling Philosophy
General
Precise
Empirically disciplined
(Game Theory)
(Game Theory)
(Experimental Econ)
“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)
April 15, 2004
CH Model
Teck-Hua Ho
6
Example 1: “zero-sum game”
ROW
T
L
0,0
COLUMN
C
10,-10
R
-5,5
M
-15,15
15,-15
25,-25
B
5,-5
-10,10
0,0
Messick(1965), Behavioral Science
April 15, 2004
CH Model
Teck-Hua Ho
7
Nash Prediction:
“zero-sum game”
ROW
Nash
Equilibrium
April 15, 2004
Nash
Equilibrium
T
L
0,0
COLUMN
C
10,-10
R
-5,5
0.40
M
-15,15
15,-15
25,-25
0.11
B
5,-5
-10,10
0,0
0.49
0.56
0.20
0.24
CH Model
Teck-Hua Ho
8
CH Prediction:
“zero-sum game”
ROW
Nash
Equilibrium
CH Model
(t = 1.55)
Nash
CH Model
Equilibrium (t = 1.55)
T
L
0,0
COLUMN
C
10,-10
R
-5,5
0.40
0.07
M
-15,15
15,-15
25,-25
0.11
0.40
B
5,-5
-10,10
0,0
0.49
0.53
0.56
0.20
0.24
0.86
0.07
0.07
http://groups.haas.berkeley.edu/simulations/CH/
April 15, 2004
CH Model
Teck-Hua Ho
9
Empirical Frequency:
“zero-sum game”
ROW
Nash
Equilibrium
CH Model
(t = 1.55)
Empirical
Frequency
April 15, 2004
Nash
CH Model Empirical
Equilibrium (t = 1.55) Frequency
T
L
0,0
COLUMN
C
10,-10
R
-5,5
0.40
0.07
0.13
M
-15,15
15,-15
25,-25
0.11
0.40
0.33
B
5,-5
-10,10
0,0
0.49
0.53
0.54
0.56
0.20
0.24
0.86
0.07
0.07
0.88
0.08
0.04
CH Model
Teck-Hua Ho
10
The Cognitive Hierarchy (CH)
Model
People are different and have different decision rules
Modeling heterogeneity (i.e., distribution of types of
players)
Modeling decision rule of each type
Guided by modeling philosophy (general, precise, and
empirically disciplined)
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CH Model
Teck-Hua Ho
11
Modeling Decision Rule
 f(0) step 0 choose randomly
 f(k) k-step thinkers know proportions f(0),...f(k-1)
 Normalize g (h) 
f ( h)
K 1
 f (h )
and best-respond
'
h ' 1
April 15, 2004
CH Model
Teck-Hua Ho
12
ROW
t1 .5 5
Level
0
Proportion
0.212
0
1
0.212
0.329
0
1
2
0.212
0.329
0.255
0
1
2
3
0.212
0.329
0.255
0.132
0
1
2
3
4
0.212
0.329
0.255
0.132
0.051
Aggregate
Aggregate
Aggregate
Aggregate
Aggregate
April 15, 2004
T
L
0,0
COLUMN
C
10,-10
R
-5,5
M
-15,15
15,-15
25,-25
B
5,-5
-10,10
0,0
Belief
1.00
0.39
0.61
1.00
0.27
0.41
0.32
1.00
0.23
0.35
0.28
0.14
1.00
0.22
0.34
0.26
0.13
0.05
1.00
T
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
0.33
0
0
0
0
0.07
ROW
M
0.33
0.33
0.33
1
0.74
0.33
1
0
0.50
0.33
1
0
0
0.43
0.33
1
0
0
0
0.41
CH Model
B
0.33
0.33
0.33
0
0.13
0.33
0
1
0.41
0.33
0
1
1
0.50
0.33
0
1
1
1
0.51
L
0.33
0.33
0.33
1
0.74
0.33
1
1
0.82
0.33
1
1
1
0.85
0.33
1
1
1
1
0.85
COL
C
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
0.33
0
0
0
0
0.07
R
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
0.33
0
0
0
0
0.07
Teck-Hua Ho
13
Implications
Exhibits “increasingly rational expectations”
 Normalized g(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 0
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CH Model
Teck-Hua Ho
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Alternative Specifications
Overconfidence:
k-steps think others are all one step lower (k-1) (Stahl, GEB,
1995; Nagel, AER, 1995; Ho, Camerer and Weigelt, AER, 1998)
“Increasingly irrational expectations” as K ∞
Has some odd properties (e.g., cycles in entry games)
Self-conscious:
k-steps think there are other k-step thinkers
Similar to Quantal Response Equilibrium/Nash
Fits worse
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CH Model
Teck-Hua Ho
15
Modeling Heterogeneity, f(k)
 A1:
f (k )
1
f (k )
t
 

f (k  1)
k
f (k  1)
k
sharp drop-off due to increasing working memory
constraint
 A2: f(1) is the mode
 A3: f(0)=f(2) (partial symmetry)
 A4: f(0) + f(1) = 2f(2)
April 15, 2004
CH Model
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Implications
 A1 Poisson distribution
and variance = t
f (k )  e
t

tk
k!
with mean
A1,A2 Poisson distribution, 1< t < 2
A1,A3  Poisson, t21.414..
A1,A4  Poisson, t1.618..(golden ratio Φ)
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CH Model
Teck-Hua Ho
17
Poisson Distribution
 f(k) with mean step of thinking t:
f (k )  e
t

tk
k!
frequency
Poisson distributions for
various t
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
t1
t1.5
t2
0
1
2
3
4
5
6
number of steps
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CH Model
Teck-Hua Ho
18
ROW
t1 .5 5
Level
0
Proportion
0.212
0
1
0.212
0.329
0
1
2
0.212
0.329
0.255
0
1
2
3
0.212
0.329
0.255
0.132
0
1
2
3
4
0.212
0.329
0.255
0.132
0.051
Aggregate
Aggregate
Aggregate
Aggregate
Aggregate
April 15, 2004
T
L
0,0
COLUMN
C
10,-10
R
-5,5
M
-15,15
15,-15
25,-25
B
5,-5
-10,10
0,0
Belief
1.00
0.39
0.61
1.00
0.27
0.41
0.32
1.00
0.23
0.35
0.28
0.14
1.00
0.22
0.34
0.26
0.13
0.05
1.00
T
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
0.33
0
0
0
0
0.07
ROW
M
0.33
0.33
0.33
1
0.74
0.33
1
0
0.50
0.33
1
0
0
0.43
0.33
1
0
0
0
0.41
CH Model
B
0.33
0.33
0.33
0
0.13
0.33
0
1
0.41
0.33
0
1
1
0.50
0.33
0
1
1
1
0.51
L
0.33
0.33
0.33
1
0.74
0.33
1
1
0.82
0.33
1
1
1
0.85
0.33
1
1
1
1
0.85
COL
C
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
0.33
0
0
0
0
0.07
R
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
0.33
0
0
0
0
0.07
Teck-Hua Ho
19
Historical Roots
 “Fictitious play” as an algorithm for computing Nash
equilibrium (Brown, 1951; Robinson, 1951)
 In our terminology, the fictitious play model is equivalent to
one in which f(k) = 1/N for N steps of thinking and N  ∞
 Instead of a single player iterating repeatedly until a fixed
point is reached and taking the player’s earlier tentative
decisions as pseudo-data, we posit a population of players
in which a fraction f(k) stop after k-steps of thinking
April 15, 2004
CH Model
Teck-Hua Ho
20
Theoretical Properties of
CH Model
Advantages over Nash equilibrium
Can “solve” multiplicity problem (picks one statistical
distribution)
Solves refinement problems (all moves occur in
equilibrium)
Sensible interpretation of mixed strategies (de facto
purification)
Theory:
τ∞ converges to Nash equilibrium in (weakly)
dominance solvable games
Equal splits in Nash demand games
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CH Model
Teck-Hua Ho
21
Example 2: Entry games
 Market entry with many entrants:
Industry demand D (as % of # of players) is announced
Prefer to enter if expected %(entrants) < D;
Stay out if expected %(entrants) > D
All choose simultaneously
 Experimental regularity in the 1st period:
 Consistent with Nash prediction, %(entrants) increases with D
 “To a psychologist, it looks like magic”-- D. Kahneman ‘88
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CH Model
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Example 2: Entry games
(data)
How entry varies with industry demand
D, (Sundali, Seale & Rapoport, 2000)
1
0.9
0.8
% entry
0.7
entry=demand
experimental data
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Demand (as % of number of
players )
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CH Model
Teck-Hua Ho
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Behaviors of Level 0 and 1
Players (t =1.25)
Level 1
Level 0
Demand (as % of # of players)
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CH Model
Teck-Hua Ho
24
Behaviors of Level 0 and 1
Players(t =1.25)
Level 0 + Level 1
Demand (as % of # of players)
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CH Model
Teck-Hua Ho
25
Behaviors of Level 2 Players
(t =1.25)
Level 2
Level 0 + Level 1
Demand (as % of # of players)
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CH Model
Teck-Hua Ho
26
Behaviors of Level 0, 1, and
2 Players(t =1.25)
Level 2
Level 0 + Level 1 +
Level 2
Level 0 +
Level 1
Demand (as % of # of players)
April 15, 2004
CH Model
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27
Entry Games (Imposing
Monotonicity on CH Model)
How entry varies with demand (D),
experimental data and thinking model
1
0.9
0.8
% entry
0.7
entry=demand
0.6
experimental data
0.5
t1.25
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Demand (as % of # of players)
April 15, 2004
CH Model
Teck-Hua Ho
28
Estimates of Mean Thinking
Step t
Table 1: Parameter Estimate t for Cognitive Hierarchy Models
Data set
Game-specific t
Game 1
Game 2
Game 3
Game 4
Game 5
Game 6
Game 7
Game 8
Game 9
Game 10
Game 11
Game 12
Game 13
Game 14
Game 15
Game 16
Game 17
Game 18
Game 19
Game 20
Game 21
Game 22
Median t
Common t
April 15, 2004
Stahl &
Wilson (1995)
Cooper &
Van Huyck
Costa-Gomes
et al.
2.93
0.00
1.35
2.34
2.01
0.00
5.37
0.00
1.35
11.33
6.48
1.71
16.02
1.04
0.18
1.22
0.50
0.78
0.98
1.42
2.16
2.05
2.29
1.31
1.71
1.52
0.85
1.99
1.91
2.30
1.23
0.98
2.40
1.86
1.01
1.54
0.80
CH Model
Mixed
Entry
0.69
0.83
0.73
0.69
1.91
0.98
1.71
0.86
3.85
1.08
1.13
3.29
1.84
1.06
2.26
0.87
2.06
1.88
9.07
3.49
2.07
1.14
1.14
1.55
1.95
1.68
3.06
1.77
1.69
1.48
0.73
0.71
Teck-Hua Ho
29
CH Model: CI of Parameter
Estimates
Table A1: 95% Confidence Interval for the Parameter Estimate t of Cognitive Hierarchy Models
Data set
Game-specific t
Game 1
Game 2
Game 3
Game 4
Game 5
Game 6
Game 7
Game 8
Game 9
Game 10
Game 11
Game 12
Game 13
Game 14
Game 15
Game 16
Game 17
Game 18
Game 19
Game 20
Game 21
Game 22
Common t
April 15, 2004
Stahl &
Wilson (1995)
Lower
Upper
Cooper &
Van Huyck
Lower
Upper
Costa-Gomes
et al.
Lower
Upper
2.40
0.00
0.75
2.34
1.61
0.00
5.20
0.00
1.06
11.29
5.81
1.49
3.65
0.00
1.73
2.45
2.45
0.00
5.62
0.00
1.69
11.37
7.56
2.02
15.40
0.83
0.11
1.01
0.36
0.64
0.75
1.16
16.71
1.27
0.30
1.48
0.67
0.94
1.23
1.72
1.58
1.44
1.66
0.91
1.22
0.89
0.40
1.48
1.28
1.67
0.75
0.55
1.75
3.04
2.80
3.18
1.84
2.30
2.26
1.41
2.67
2.68
3.06
1.85
1.46
3.16
0.67
0.98
0.57
2.65
0.70
0.87
2.45
1.21
0.62
1.34
0.64
1.40
1.64
6.61
2.46
1.45
0.82
0.78
1.00
1.28
0.95
1.70
1.22
2.37
1.37
4.26
1.62
1.77
3.85
2.09
1.64
3.58
1.23
2.35
2.15
10.84
5.25
2.64
1.52
1.60
2.15
2.59
2.21
3.63
0.21
0.73
0.56
0.26
1.43
0.88
1.09
1.58
1.39
1.67
0.74
0.87
1.53
2.13
1.30
1.78
0.42
1.07
CH Model
Mixed
Lower
Upper
Entry
Lower
Upper
Teck-Hua Ho
30
Nash versus CH Model:
LL and MSD
Table 2: Model Fit (Log Likelihood LL and Mean-squared Deviation MSD)
Stahl &
Wilson (1995)
Cooper &
Van Huyck
Costa-Gomes
et al.
Mixed
Entry
-824
0.0097
-150
0.0004
-872
0.0179
-150
0.0005
Cross-dataset Forecasting
Cognitive Hierarchy (Common t )
-599
-1929
-941
LL
0.0257
0.0328
0.0425
MSD
-884
0.0216
-153
0.0034
Nash Equilibrium 1
LL
MSD
-1641
0.0521
-154
0.0049
Data set
Within-dataset Forecasting
Cognitive Hierarchy (Game-specific t )
-540
-1690
-721
LL
0.0034
0.0079
0.0074
MSD
Cognitive Hierarchy (Common t )
-560
-1743
-918
LL
0.0100
0.0136
0.0327
MSD
-3657
0.0882
-10921
0.2040
-3684
0.1367
Note 1: The Nash Equilibrium result is derived by allowing a non-zero mass of 0.0001 on non-equilibrium strategies.
April 15, 2004
CH Model
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31
CH Model: Theory vs. Data
(Mixed Games)
Figure 2a: Predicted Frequencies of Cognitive Hierarchy Models
for Matrix Games (common t )
1
0.9
y = 0.868x + 0.0499
R2 = 0.8203
Predicted Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Empirical Frequency
April 15, 2004
CH Model
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32
Nash: Theory vs. Data
(Mixed Games)
Figure 3a: Predicted Frequencies of Nash Equilibrium for Matrix
Games
1
0.9
y = 0.8273x + 0.0652
Predicted Frequency
0.8
2
R = 0.3187
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Empirical Frequency
April 15, 2004
CH Model
Teck-Hua Ho
33
Nash vs CH (Mixed Games)
April 15, 2004
CH Model
Teck-Hua Ho
34
CH Model: Theory vs. Data
(Entry and Mixed Games)
Figure 2b: Predicted Frequencies of Cognitive Hierarchy Models
for Entry and Mixed Games (common t )
1
0.9
y = 0.8785x + 0.0419
R2 = 0.8027
Predicted Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Empirical Frequency
April 15, 2004
CH Model
Teck-Hua Ho
35
Nash: Theory vs. Data
(Entry and Mixed Games)
Figure 3b: Predicted Frequencies of Nash Equilibrium for Entry
and Mixed Games
1
0.9
y = 0.707x + 0.1011
0.8
2
Predicted Frequency
R = 0.4873
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Empirical Frequency
April 15, 2004
CH Model
Teck-Hua Ho
36
CH vs. Nash (Entry and Mixed
Games)
April 15, 2004
CH Model
Teck-Hua Ho
37
Economic Value
 Evaluate models based on their value-added rather than statistical
fit (Camerer and Ho, 2000)
 Treat models like consultants
 If players were to hire Mr. Nash and Ms. CH as consultants and
listen to their advice (i.e., use the model to forecast what others will
do and best-respond), would they have made a higher payoff?
 A measure of disequilibrium
April 15, 2004
CH Model
Teck-Hua Ho
38
Nash versus CH Model:
Economic Value
April 15, 2004
CH Model
Teck-Hua Ho
39
Example 3: P-Beauty
Contest
 n players
 Every player simultaneously chooses a number from 0
to 100
 Compute the group average
 Define Target Number to be 0.7 times the group
average
 The winner is the player whose number is the closet to
the Target Number
 The prize to the winner is US$10
April 15, 2004
CH Model
Teck-Hua Ho
40
April 15, 2004
CH Model
Teck-Hua Ho
41
A Sample of Caltech Board
of Trustees
 David Baltimore
President
California Institute of
Technology
 Donald L. Bren
Chairman of the Board
The Irvine Company
• Eli Broad
Chairman
SunAmerica Inc.
• Lounette M. Dyer
Chairman
Silk Route Technology
April 15, 2004
• David D. Ho
Director
The Aaron Diamond AIDS Research Center
• Gordon E. Moore
Chairman Emeritus
Intel Corporation
• Stephen A. Ross
Co-Chairman, Roll and Ross Asset Mgt Corp
• Sally K. Ride
President Imaginary Lines, Inc., and
Hibben Professor of Physics
CH Model
Teck-Hua Ho
42
Summary
 CH Model:
Discrete thinking steps
Frequency Poisson distributed
 One-shot games
Fits better than Nash and adds more economic value
Explains “magic” of entry games
Sensible interpretation of mixed strategies
Can “solve” multiplicity problem
 Initial conditions for learning
April 15, 2004
CH Model
Teck-Hua Ho
43