A Cognitive Hierarchy Theory 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
March – June, 2003
CH Model
Teck-Hua Ho
1
Motivation
 Nash equilibrium and its refinements: Dominant theories
in economics for predicting behaviors in games.
 Subjects in experiments hardly play Nash in the first
round but do often converge to it eventually.
 Multiplicity problem (e.g., coordination games)
 Modeling heterogeneity really matters in games.
March – June, 2003
CH Model
Teck-Hua Ho
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Research Goals
How to model bounded rationality (first-period
behavior)?
Cognitive Hierarchy (CH) model
How to model equilibration?
EWA learning model (Camerer and Ho,
Econometrica, 1999; Ho, Camerer, and Chong,
2003)
How to model repeated game behavior?
Teaching model (Camerer, Ho, and Chong,
Journal of Economic Theory, 2002)
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CH Model
Teck-Hua Ho
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Modeling Principles
Principle
Nash
Thinking
Strategic Thinking


Best Response


Mutual Consistency

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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)
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CH Model
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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
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CH Model
Teck-Hua Ho
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Nash Prediction:
“zero-sum game”
ROW
Nash
Equilibrium
March – June, 2003
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
7
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/
March – June, 2003
CH Model
Teck-Hua Ho
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Empirical Frequency:
“zero-sum game”
ROW
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
Nash
Equilibrium
CH Model
(t = 1.55)
Empirical
Frequency
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CH Model
Teck-Hua Ho
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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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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
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CH Model
Teck-Hua Ho
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Example 1: “zero-sum game”
ROW
March – June, 2003
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
CH Model
Teck-Hua Ho
12
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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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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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)
 A4a: f(0)+f(1)=f(2)+f(3)+f(4)…
 A4b: f(2)=f(3)+f(4)+f(5)…
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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,A4a,A4b)  Poisson, t1.618..(golden ratio Φ)
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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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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
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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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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
March – June, 2003
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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Behaviors of Level 0 and 1
Players (t =1.25)
Level 1
Level 0
Demand (as % of # of players)
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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
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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
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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)
March – June, 2003
CH Model
Teck-Hua Ho
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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)
March – June, 2003
CH Model
Teck-Hua Ho
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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
March – June, 2003
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
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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
March – June, 2003
Stahl &
Wilson (1995)
Lower
Upper
Cooper &
Van Huyck
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
Costa-Gomes
et al.
Lower
Upper
Mixed
Lower
Upper
Entry
Lower
Upper
Teck-Hua Ho
28
Nash versus CH Model:
LL and MSD
Table 2: Model Fit (Log Likelihood LL and Mean-squared Deviation MSD)
Stahl &
Wilson (1995)
Data set
Cooper &
Van Huyck
Costa-Gomes
et al.
Within-dataset Forecasting
Cognitive Hierarchy (Game-specific t ) 1
LL
-721
-1690
-540
MSD
0.0074
0.0079
0.0034
Cognitive Hierarchy (Common t )
LL
-918
-1743
-560
MSD
0.0327
0.0136
0.0100
Cross-dataset Forecasting
Cognitive Hierarchy (Common t )
LL
-941
-1929
-599
MSD
0.0425
0.0328
0.0257
Nash Equilibrium
LL
MSD
Mixed
Entry
-824
0.0097
-150
0.0004
-872
0.0179
-150
0.0005
-884
0.0216
-153
0.0034
-1641
0.0521
-154
0.0049
2
-3657
0.0882
-10921
0.2040
-3684
0.1367
Note 1: The scale sensitivity parameter l for the Cognitive Hierarchy models is set to infinity. The results reported
in Camerer, Ho & Chong(2001) presented at the Nobel Symposium 2001 are for models where l is estimated.
Note 2: The Nash Equilibrium result is derived by allowing a non-zero mass of 0.0001 on non-equilibrium strategies.
March – June, 2003
CH Model
Teck-Hua Ho
29
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
March – June, 2003
CH Model
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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
March – June, 2003
CH Model
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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
March – June, 2003
CH Model
Teck-Hua Ho
32
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
March – June, 2003
CH Model
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33
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, would they have made a higher payoff?
March – June, 2003
CH Model
Teck-Hua Ho
34
Nash versus CH Model:
Economic Value
Table 3: Economic Value for Cognitive Hierarchy and Nash Equilibrium
Data set
Total Payoff (% Improvement)
Actual Subject Choices
Ex-post Maximum
Within-dataset Estimation
Cognitive Hierarchy (Game-specific t )
Cognitive Hierarchy (Common t )
Cross-dataset Estimation
Cognitive Hierarchy (Common t )
Nash Equilibrium
Stahl &
Wilson (1995)
Cooper &
Van Huyck
Costa-Gomes
et al.
Mixed
Entry
384
685
79%
1169
1322
13%
530
615
16%
328
708
116%
118
176
49%
401
4%
418
9%
1277
9%
1277
9%
573
8%
573
8%
471
43%
471
43%
128
8%
128
8%
418
9%
398
4%
1277
9%
1230
5%
573
8%
556
5%
460
40%
274
-16%
128
8%
112
-5%
Note 1: The economic value is the total value (in USD) of all rounds that a "hypothetical" subject will earn using the respective model
to predict other's behavior and best responds with the strategy that yields the highest expected payoff in each round.
March – June, 2003
CH Model
Teck-Hua Ho
35
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$20
March – June, 2003
CH Model
Teck-Hua Ho
36
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
March – June, 2003
• 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
37
Results from Caltech Board
of Trustees
Caltech Board of Trustees
ALL
CEOs only
Mean
Target
Standard Deviation
Sample Size
March – June, 2003
42.6
29.8
23.4
70
CH Model
37.8
26.5
18.9
20
Teck-Hua Ho
38
Results from Two Other Smart
Subject Pools
Mean
Target
Standard Deviation
Sample Size
March – June, 2003
Portfolio
Managers
Economics
PhDs
24.3
17.0
16.2
26
27.4
19.2
18.7
16
CH Model
Teck-Hua Ho
39
Results from College
Students
Mean
Target
Standard Deviation
Sample Size
March – June, 2003
Caltech
UCLA
Wharton
Germany
Singapore
21.9
42.3
37.9
15.3
29.6
26.5
10.4
18.0
18.8
27
28
35
36.7
25.7
20.2
67
46.1
32.2
28.0
98
CH Model
Teck-Hua Ho
40
CH Model: Parameter
Estimates
Table 1: Data and estimates of t in pbc games
(equilibrium = 0)
Mean
Steps of
subjects/game
Data
CH Model
Thinking
game theorists
19.1
19.1
3.7
Caltech
23.0
23.0
3.0
newspaper
23.0
23.0
3.0
portfolio mgrs
24.3
24.4
2.8
econ PhD class
27.4
27.5
2.3
Caltech g=3
21.5
21.5
1.8
high school
32.5
32.7
1.6
1/2 mean
26.7
26.5
1.5
70 yr olds
37.0
36.9
1.1
Germany
37.2
36.9
1.1
CEOs
37.9
37.7
1.0
game p=0.7
38.9
38.8
1.0
Caltech g=2
21.7
22.2
0.8
PCC g=3
47.5
47.5
0.1
game p=0.9
49.4
49.5
0.1
PCC g=2
54.2
49.5
0.0
mean
1.56
median
1.30
March – June, 2003
CH Model
Teck-Hua Ho
41
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
March – June, 2003
CH Model
Teck-Hua Ho
42