Outline
 Motivation
 Mutual Consistency: CH Model
 Noisy Best-Response: QRE Model
 Instant Convergence: EWA Learning
August 2007
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1
Standard Assumptions in
Equilibrium Analysis
Assumptions
Nash
Equilbirum
Cognitive
Hierarchy
QRE
EWA
Learning
Strategic Thinking
X
X
X
X
Best Response
X
X
Solution Method
X
X
Mutual Consistency
Instant Convergence
August 2007
X
X
Duke PhD Summer Camp
X
X
2
Example A: Exercise
 Consider matching pennies games in which the row player
chooses between Top and Bottom and the column player
simultaneously chooses between Left and Right, as shown
below:
G1
G2
August 2007
Top
Bottom
Left
80,40
40,80
Right
40,80
80,40
Top
Bottom
Left
320,40
40,80
Right
40,80
80,40
Top
Bottom
Left
44,40
40,80
Right
40,80
80,40
Duke PhD Summer Camp
3
Example A: Exercise
 Consider matching pennies games in which the row player
chooses between Top and Bottom and the column player
simultaneously chooses between Left and Right, as shown
below:
G1
G2
August 2007
Top
Bottom
Left
80,40
40,80
Right
40,80
80,40
Top
Bottom
Left
320,40
40,80
Right
40,80
80,40
Top
Bottom
Left
44,40
40,80
Right
40,80
80,40
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Example A: Data
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Example B: Exercise
 The two players choose “effort” levels
simultaneously, and the payoff of each player is
given by pi = min (e1, e2) – c x ei
 Efforts are integer from 110 to 170.
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Example B: Exercise
 The two players choose “effort” levels
simultaneously, and the payoff of each player is
given by pi = min (e1, e2) – c x ei
 Efforts are integer from 110 to 170.
 C = 0.1 or 0.9.
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Example B: Data
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Motivation: CH
 Model heterogeneity explicitly (people are not equally
smart)
 Introduce the word surprise into the game theory’s
dictionary (e.g., Next movie)
 Generate new predictions (reconcile various treatment
effects in lab data not predicted by standard theory)
Camerer, Ho, and Chong (QJE, 2004)
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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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Nash Prediction: “zero-sum game”
ROW
Nash
Equilibrium
August 2007
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
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CH Prediction: “zero-sum game”
ROW
Nash
Equilibrium
CH Model
(t = 1.55)
August 2007
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
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Empirical Frequency: “zero-sum game”
ROW
Nash
Equilibrium
CH Model
(t = 1.55)
Empirical
Frequency
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
http://groups.haas.berkeley.edu/simulations/CH/
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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). Types of players are denoted by levels 0, 1, 2,
3,…,
Modeling decision rule of each type.
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Modeling Decision Rule
 Frequency of k-step is f(k)
 Step 0 choose randomly
 k-step thinkers know proportions f(0),...f(k-1)
 Form beliefs
beliefs
gk (h) 
f (h)
K 1
 f (h )
'
and best-respond based on
h ' 1
 Iterative
and no need to solve a fixed point

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ROW
K
0
1
2
3
>3
Proportion, f(k)
0.212
0.329
0.255
0.132
0.072
August 2007
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
K's
Level (K) Proportion
0
0.212
Aggregate
0
0.212
1
0.329
Aggregate
0
0.212
1
0.329
2
0.255
Aggregate
K+1's
Belief
1.00
1.00
0.39
0.61
1.00
0.27
0.41
0.32
1.00
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T
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
ROW
M
0.33
0.33
0.33
1
0.74
0.33
1
0
0.50
B
0.33
0.33
0.33
0
0.13
0.33
0
1
0.41
L
0.33
0.33
0.33
1
0.74
0.33
1
1
0.82
COL
C
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
16
R
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
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 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 )
t

f (k  1) k
sharp drop-off due to increasing difficulty in simulating
others’ behaviors
 A2: f(0) + f(1) = 2f(2)
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Implications
 A1 Poisson distribution
and variance = t
f (k )  e
t

tk
k!
with mean
A1,A2  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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Existence and Uniqueness:
CH Solution
 Existence: There is always a CH solution in any game
 Uniqueness: It is always unique
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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
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23
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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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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Behaviors of Level 2 Players
(t=1.25)
Level 2
Level 0 + Level 1
Demand (as % of # of players)
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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)
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CH Predictions in Entry Games
(t = 1.25)
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)
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Homework
 What value of tcan help to explain the data in Example
A?
 How does CH model explain the data in Example B?
August 2007
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Empirical Frequency: “zero-sum game”
ROW
Empirical
Frequency
August 2007
T
L
0,0
COLUMN
C
10,-10
R
-5,5
0.125
M
-15,15
15,-15
25,-25
0.333
B
5,-5
-10,10
0,0
0.542
0.875
0.083
0.042
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Frequency
32
MLE Estimation
T
M
B
L
C
R
Count
13
33
54
88
8
4
Label
N1
N2
N3
M1
M2
M3
LL(t )  p1 (t ) N1  p2 (t ) N2  (1  p1 (t )  p2 (t )) N3  q1 (t ) M1  q2 (t ) M 2  (1  q1 (t )  q2 (t )) M 3
August 2007
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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
August 2007
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
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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
34
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
August 2007
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
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Mixed
Lower
Upper
Entry
Lower
Upper
35
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.
August 2007
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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
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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
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Nash vs. CH (Mixed Games)
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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
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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
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CH vs. Nash
(Entry and Mixed Games)
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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
August 2007
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Nash versus CH Model:
Economic Value
August 2007
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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
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Results in various p-BC games
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Results in various p-BC games
Subject Pool
CEOs
80 year olds
Economics PhDs
Portfolio Managers
Game Theorists
August 2007
Group Size Sample Size
20
33
16
26
27-54
20
33
16
26
136
Duke PhD Summer Camp
Mean
37.9
37.0
27.4
24.3
19.1
Error
(Nash)
-37.9
-37.0
-27.4
-24.3
-19.1
Error
(CH)
-0.1
-0.1
0.0
0.1
0.0
t
1.0
1.1
2.3
2.8
3.7
47
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
August 2007
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Outline
 Motivation
 Mutual Consistency: CH Model
 Noisy Best-Response: QRE Model
 Instant Convergence: EWA Learning
August 2007
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