A Cognitive Hierarchy (CH)
Model of Games
Teck H. Ho
Haas School of Business
University of California, Berkeley
Joint work with Colin Camerer, Caltech
Juin-Kuan Chong, NUS
Teck H. Ho
University of Michigan, Ann Arbor
1
Motivation
Nash equilibrium and its refinements: Dominant
theories in economics and marketing 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.
Teck H. Ho
University of Michigan, Ann Arbor
2
Main Goals
Provide a behavioral theory to explain and
predict behaviors in any one-shot game
Normal-form games (e.g., zero-sum game, pbeauty 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)
Teck H. Ho
University of Michigan, Ann Arbor
3
Modeling Principles
Principle
Nash
CH
Strategic Thinking


Best Response


Mutual Consistency

Teck H. Ho
University of Michigan, Ann Arbor
4
Modeling Philosophy
Simple
General
Precise
Empirically disciplined
(Economics)
(Economics)
(Economics)
(Psychology)
“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)
Teck H. Ho
University of Michigan, Ann Arbor
5
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
Teck H. Ho
University of Michigan, Ann Arbor
6
Nash Prediction:
“zero-sum game”
ROW
Nash
Equilibrium
Teck H. Ho
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
University of Michigan, Ann Arbor
7
CH Prediction:
“zero-sum game”
ROW
Nash
Equilibrium
CH Model
(t = 1.55)
Teck H. Ho
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
University of Michigan, Ann Arbor
8
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/
Teck H. Ho
University of Michigan, Ann Arbor
9
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
Teck H. Ho
University of Michigan, Ann Arbor
10
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
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

Teck H. Ho
University of Michigan, Ann Arbor
11
ROW
K
0
1
2
3
>3
Teck H. Ho
Proportion, f(k)
0.212
0.329
0.255
0.132
0.072
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
T
University of Michigan, Ann Arbor
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
R
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
12
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
Teck H. Ho
University of Michigan, Ann Arbor
13
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)
Teck H. Ho
University of Michigan, Ann Arbor
14
Implications
 A1 Poisson distribution
and variance = t
f (k )  e
t

tk
k!
with mean
A1,A2  Poisson, t1.618..(golden ratio Φ)
Teck H. Ho
University of Michigan, Ann Arbor
15
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
Teck H. Ho
University of Michigan, Ann Arbor
16
ROW
t1.55
Level(K)
K's
Proportion
0
0.212
Aggregate
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
Aggregate
Aggregate
Aggregate
Teck H. Ho
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+1's
Belief
1.00
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
T
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
ROW
M
0.33
0.33
0.33
1
0.74
0.33
1
0
0.50
0.33
1
0
0
0.43
University of Michigan, Ann Arbor
B
0.33
0.33
0.33
0
0.13
0.33
0
1
0.41
0.33
0
1
1
0.50
L
0.33
0.33
0.33
1
0.74
0.33
1
1
0.82
0.33
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
R
0.33
0.33
0.33
0
0.13
0.33
0
0
0.09
0.33
0
0
0
0.08
17
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
Teck H. Ho
University of Michigan, Ann Arbor
18
Estimates of Mean Thinking
Step t
Teck H. Ho
University of Michigan, Ann Arbor
19
Nash: Theory vs. Data
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
Teck H. Ho
University of Michigan, Ann Arbor
20
CH Model: Theory vs.
Data
Figure 2b: Predicted Frequencies of Cognitive Hierarchy
Models for Entry and Mixed Games (common t)
1
0.9
y = 0.8785x + 0.0419
Predicted Frequency
0.8
R2 = 0.8027
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
Teck H. Ho
University of Michigan, Ann Arbor
21
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?
Teck H. Ho
University of Michigan, Ann Arbor
22
Nash versus CH Model:
Economic Value
Teck H. Ho
University of Michigan, Ann Arbor
23
Application: Strategic IQ
http://128.32.67.154/siq13/default1.asp
A battery of 30 "well-known" games
Measure a subject's strategic IQ by how much money she
makes (matched against a defined pool of subjects)
Factor analysis + fMRI to figure out whether certain brain
region accounts for superior performance in "similar" games
Specialized subject pools
Soliders
Writers
Chess players
Patients with brain damages
Teck H. Ho
University of Michigan, Ann Arbor
24
Example 2: 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
Ho, Camerer, and Weigelt (AER, 1998)
Teck H. Ho
University of Michigan, Ann Arbor
25
A Sample of CEOs
 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
Teck H. Ho
• 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
University of Michigan, Ann Arbor
26
Results in various p-BC games
Subject Pool
CEOs
80 year olds
Economics PhDs
Portfolio Managers
Game Theorists
Teck H. Ho
Group Size
20
33
16
26
27-54
Sample Size
20
33
16
26
136
Mean
37.9
37.0
27.4
24.3
19.1
University of Michigan, Ann Arbor
Error (Nash) Error (CH)
t
-37.9
-0.1
1.0
-37.0
-0.1
1.1
-27.4
0.0
2.3
-24.3
0.1
2.8
-19.1
0.0
3.7
27
Summary
 CH Model:
Discrete thinking steps
Frequency Poisson distributed
 One-shot games
Fits better than Nash and adds more economic value
Sensible interpretation of mixed strategies
Can “solve” multiplicity problem
 Application: Measurement of Strategic IQ
Teck H. Ho
University of Michigan, Ann Arbor
28
Research Agenda
 Bounded Rationality in Markets
Revised Utility Functions
Empirical Alternatives to Nash Equilibrium
(Ho, Lim, and Camerer, JMR, forthcoming)
 A New Taxonomy of Games
Neural Foundation of Game Theory
Teck H. Ho
University of Michigan, Ann Arbor
29
Bounded Rationality in Markets: Revised
Utility Function
Behavioral Regularities
Standard Assumption
New Model Specification
Reference Example
Marketing Application
Example
1. Revised Utility Function
- Reference point and
loss aversion
- Expected Utility Theory - Prospect Theory
Kahneman and Tversky (1979)
- Two-part tariff - double
marginalization problem
- Fairness
- Self-interested
- Price discrimination
- Impatience
- Exponential discounting - Hyperbolic Discounting
Ainslie (1975)
Teck H. Ho
- Inequality aversion
Fehr and Schmidt (1999)
University of Michigan, Ann Arbor
- Price promotion and
packaging size design
30
Bounded Rationality in Markets: Alternative
Solution Concepts
Behavioral Regularities
Standard Assumption
New Model Specification
Example
Marketing Application
Example
2. Bounded Computation Ability
- Nosiy Best Response
- Best Response
- Quantal Best Response
McKelvey and Palfrey (1995)
- NEIO
- Limited Thinking Steps
- Rational expectation
- Cognitive hierarchy
Camerer, Ho, Chong (2004)
- Market entry competition
- Myopic and learn
- Instant equilibration
- Experience weighted attraction
Camerer and Ho (1999)
- Lowest price guarantee
competition
Teck H. Ho
University of Michigan, Ann Arbor
31
Neural Foundations of Game Theory
Neural foundation of game theory
Teck H. Ho
University of Michigan, Ann Arbor
32
Strategic IQ: A New Taxonomy of Games
Teck H. Ho
University of Michigan, Ann Arbor
33
Teck H. Ho
University of Michigan, Ann Arbor
34
Nash versus CH Model:
LL and MSD (in-sample)
Teck H. Ho
University of Michigan, Ann Arbor
35
Economic Value:
Definition and Motivation
“A normative model must produce strategies
that are at least as good as what people can do
without them.” (Schelling, 1960)
A measure of degree of disequilibrium, in
dollars.
If players are in equilibrium, then an equilibrium theory will
advise them to make the same choices they would make anyway,
and hence will have zero economic value
If players are not in equilibrium, then players are mis-forecasting
what others will do. A theory with more accurate beliefs will have
positive economic value (and an equilibrium theory can have
negative economic value if it misleads players)
Teck H. Ho
University of Michigan, Ann Arbor
36
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
Teck H. Ho
University of Michigan, Ann Arbor
37
Teck H. Ho
University of Michigan, Ann Arbor
38
Example 3: Centipede Game
1
2
1
2
0.40
0.10
0.20
0.80
1.60
0.40
0.80
3.20
1
2
25.60
6.40
6.40 3.20
1.60 12.80
Figure 1: Six-move Centipede Game
Teck H. Ho
University of Michigan, Ann Arbor
39
CH vs. Backward Induction
Principle (BIP)
Is extensive CH (xCH) a sensible empirical
alternative to BIP in predicting behavior in an
extensive-form game like the Centipede?
Is there a difference between steps of thinking
and look-ahead (planning)?
Teck H. Ho
University of Michigan, Ann Arbor
40
BIP consists of three premises
 Rationality: Given a choice between two alternatives, a
player chooses the most preferred.
 Truncation consistency: Replacing a subgame with its
equilibrium payoffs does not affect play elsewhere in the
game.
 Subgame consistency: Play in a subgame is
independent of the subgame’s position in a larger game.
Binmore, McCarthy, Ponti, and Samuelson (JET, 2002) show violations of
both truncation and subgame consistencies.
Teck H. Ho
University of Michigan, Ann Arbor
41
Truncation Consistency
1
2
0.40 0.20
0.10 0.80
1
1.60
0.40
2
1
2
25.60
6.40
0.80 6.40 3.20
3.20 1.60 12.80
Figure 1: Six-move Centipede game
1
2
1
2
1.60
0.40
0.80
3.20
VS.
0.40 0.20
0.10 0.80
6.40
1.60
Figure 2: Four-move Centipede game (Low-Stake)
Teck H. Ho
University of Michigan, Ann Arbor
42
Subgame Consistency
1
2
0.40 0.20
0.10 0.80
2
1
1.60
0.40
1
2
25.60
6.40
0.80 6.40 3.20
3.20 1.60 12.80
Figure 1: Six-move Centipede game
1
2
VS.
1
2
25.60
6.40
1.60
0.40
0.80 6.40 3.20
3.20 1.60 12.80
Figure 3: Four-move Centipede game (High-Stake (x4))
Teck H. Ho
University of Michigan, Ann Arbor
43
Implied Take Probability
Implied take probability at each stage, pj
Truncation consistency: For a given j, pj is identical
in both 4-move (low-stake) and 6-move games.
Subgame consistency: For a given j, pn-j (n=4 or 6)
is identical in both 4-move (high-stake) and 6-move
games.
Teck H. Ho
University of Michigan, Ann Arbor
44
Prediction on Implied Take
Probability
Implied take probability at each stage, pj
Truncation consistency: For a given j, pj is identical
in both 4-move (low-stake) and 6-move games.
Subgame consistency: For a given j, pn-j (n=4 or 6)
is identical in both 4-move (high-stake) and 6-move
games.
Teck H. Ho
University of Michigan, Ann Arbor
45
Data: Truncation & Subgame
Consistencies
Data
p1
p2
p3
p4
p5
p6
6-move
0.01
0.06
0.21
0.53
0.73
0.85
4-move(Low Stake)
0.07
0.38
0.65
0.75
0.15
0.44
0.67
0.69
4-move(High Stake)
Teck H. Ho
University of Michigan, Ann Arbor
46
K-Step Look-ahead (Planning)
Example: 1-step look-ahead
1
2
0.40 0.20
0.10 0.80
1
2
2
1
1.60
0.40
1
2
25.60
6.40
0.80 6.40 3.20
3.20 1.60 12.80
V1
V2
0.40 0.20
0.10 0.80
Teck H. Ho
University of Michigan, Ann Arbor
47
Limited thinking and
Planning
Xk (lk), k=1,2,3 follow independent Poisson
distributions
X3=common thinking/planning; X1=extra thinking,
X2=extra planning
X (thinking) =X1+X3 ; Y (planning) =X2 +X3
follow jointly a bivariate Poisson distribution BP(l1,
l 2 , l3 )
Teck H. Ho
University of Michigan, Ann Arbor
48
Estimation Results
4 stages
Low-Stake High-Stake
281
100
Sample Size
Calibration
Sample Size
6 stages
All sessions
281
662
197
70
197
464
Agent Quantal Response Eqlbm (AQRE)
-287.0
-106.8
-409.8
-848.2
Extensive Cognitive Hierarchy (xCH)
xCH (l1=l2=0)
-276.1
-276.1
-105.9
-105.9
-341.2
-341.2
-753.0
-753.0
84
30
84
198
281.0
100.0
281.0
662.0
-132.8
-132.8
-41.5
-41.5
-120.7
-121.1
-293.9
-293.9
Validation
Sample Size
Agent Quantal Response Eqlbm (AQRE)
Extensive Cognitive Hierarchy (xCH)
xCH (l1=l2=0)
 Thinking steps and steps of planning are perfectly correlated
Teck H. Ho
University of Michigan, Ann Arbor
49
Data and xCH Prediction:
Truncation & Subgame Consistencies
Data
p1
p2
p3
p4
p5
p6
6-move
0.01
0.06
0.21
0.53
0.73
0.85
4-move(Low Stake)
0.07
0.38
0.65
0.75
0.15
0.44
0.67
0.69
4-move(High Stake)
CH Prediction
6-move
0.06
0.16
0.15
0.48
4-move(Low-Stake)
0.15
0.31
0.76
0.97
0.21
0.34
4-move(High-Stake)
Teck H. Ho
University of Michigan, Ann Arbor
0.90
0.99
0.71
0.95
50