1 university of michigan, ann arbor a cognitive hierarchy (ch) model of games
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1University of Michigan, Ann Arbor
A Cognitive Hierarchy (CH) A Cognitive Hierarchy (CH) Model of GamesModel of Games
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2University of Michigan, Ann Arbor
MotivationMotivationNash 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.
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3University of Michigan, Ann Arbor
Main GoalsMain 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)
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4University of Michigan, Ann Arbor
Modeling PrinciplesModeling Principles
Principle Nash CH
Strategic Thinking
Best Response
Mutual Consistency
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5University of Michigan, Ann Arbor
Modeling PhilosophyModeling Philosophy
Simple (Economics)General (Economics)Precise (Economics)Empirically disciplined (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)
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6University of Michigan, Ann Arbor
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7University of Michigan, Ann Arbor
Example 1: “zero-sum game”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
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8University of Michigan, Ann Arbor
Nash Prediction: Nash Prediction: “zero-sum game”“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
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9University of Michigan, Ann Arbor
CH Prediction: CH Prediction: “zero-sum game”“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
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10University of Michigan, Ann Arbor
Empirical Frequency: Empirical Frequency: “zero-sum game”“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
http://groups.haas.berkeley.edu/simulations/CH/
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11University of Michigan, Ann Arbor
The Cognitive Hierarchy (CH) The Cognitive Hierarchy (CH) ModelModelPeople 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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12University of Michigan, Ann Arbor
Modeling Decision RuleModeling Decision RuleProportion 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
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13University of Michigan, Ann Arbor
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
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14University of Michigan, Ann Arbor
Theoretical ImplicationsTheoretical 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
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15University of Michigan, Ann Arbor
Modeling Heterogeneity, Modeling Heterogeneity, f(k)f(k)
A1:
sharp drop-off due to increasing difficulty in simulating others’ behaviors
A2: f(0) + f(1) = 2f(2)
kkf
kf
)1(
)(
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16University of Michigan, Ann Arbor
ImplicationsImplications
!)(
kekf
k A1 Poisson distribution with mean and variance =
A1,A2 Poisson, golden ratio Φ)
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17University of Michigan, Ann Arbor
La loi de Poisson a été introduite en 1838 par Siméon Denis Poisson (1781–1840), dans son ouvrage Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Le sujet principal de cet ouvrage consiste en certaines variables aléatoires N qui dénombrent, entre autres choses, le nombre d'occurrences (parfois appelées « arrivées ») qui prennent place pendant un laps de temps donné.Si le nombre moyen d'occurrences dans cet intervalle est λ, alors la probabilité qu'il existe exactement k occurrences (k étant un entier naturel, k = 0, 1, 2, ...) est:
Où:e est la base de l'exponentielle (2,718...)k! est la factorielle de kλ est un nombre réel strictement positif.On dit alors que X suit la loi de Poisson de paramètre λ.Par exemple, si un certain type d'évènements se produit en moyenne 4 fois par minute, pour étudier le nombre d'évènements se produisant dans un laps de temps de 10 minutes, on choisit comme modèle une loi de Poisson de paramètre λ = 10×4 = 40.
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18University of Michigan, Ann Arbor
Poisson DistributionPoisson Distribution f(k) with mean step of thinking :
!)(
kekf
k
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
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19University of Michigan, Ann Arbor
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
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20University of Michigan, Ann Arbor
Theoretical Properties of CH Theoretical Properties of CH ModelModelAdvantages 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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21University of Michigan, Ann Arbor
Estimates of Mean Thinking Estimates of Mean Thinking Step Step
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22University of Michigan, Ann Arbor
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
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23University of Michigan, Ann Arbor
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
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24University of Michigan, Ann Arbor
Economic ValueEconomic 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?
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25University of Michigan, Ann Arbor
Nash versus CH Model: Economic Value
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26University of Michigan, Ann Arbor
Application: Strategic IQhttp://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 poolsSoldiers
Writers
Chess players
Patients with brain damages
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27University of Michigan, Ann Arbor
Example 2Example 2: P: P-Beauty Contest-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 closest to
the Target Number
The prize to the winner is US$20Ho, Camerer, and Weigelt (AER, 1998)
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28University of Michigan, Ann Arbor
A Sample of CEOsA Sample of CEOs
David Baltimore President California Institute of Technology
Donald L. Bren
Chairman of the BoardThe Irvine Company
• Eli BroadChairmanSunAmerica Inc.
• Lounette M. Dyer Chairman Silk Route Technology
• 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
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29University of Michigan, Ann Arbor
Results in various p-BC gamesResults in various p-BC games
Subject Pool Group Size Sample Size Mean Error (Nash) Error (CH)
CEOs 20 20 37.9 -37.9 -0.1 1.0
80 year olds 33 33 37.0 -37.0 -0.1 1.1
Economics PhDs 16 16 27.4 -27.4 0.0 2.3
Portfolio Managers 26 26 24.3 -24.3 0.1 2.8
Game Theorists 27-54 136 19.1 -19.1 0.0 3.7
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30University of Michigan, Ann Arbor
SummarySummary
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
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31University of Michigan, Ann Arbor
Research AgendaResearch AgendaBounded Rationality in Markets
Revised Utility Functions
Empirical Alternatives to Nash Equilibrium
A New Taxonomy of Games
Neural Foundation of Game Theory
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32University of Michigan, Ann Arbor
Bounded Rationality in Markets: Revised Utility Function
Behavioral Regularities Standard Assumption New Model Specification Marketing ApplicationReference Example Example
1. Revised Utility Function
- Reference point and - Expected Utility Theory - Prospect Theory - Two-part tariff - double loss aversion Kahneman and Tversky (1979) marginalization problem
- Fairness - Self-interested - Inequality aversion - Price discrimination Fehr and Schmidt (1999)
- Impatience - Exponential discounting - Hyperbolic Discounting - Price promotion and Ainslie (1975) packaging size design
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33University of Michigan, Ann Arbor
Bounded Rationality in Markets: Alternative Solution Concepts
Behavioral Regularities Standard Assumption New Model Specification Marketing ApplicationExample Example
2. Bounded Computation Ability
- Nosiy Best Response - Best Response - Quantal Best Response - NEIO McKelvey and Palfrey (1995)
- Limited Thinking Steps - Rational expectation - Cognitive hierarchy - Market entry competition Camerer, Ho, Chong (2004)
- Myopic and learn - Instant equilibration - Experience weighted attraction - Lowest price guarantee Camerer and Ho (1999) competition
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34University of Michigan, Ann Arbor
Neural Foundations of Game Theory
Neural foundation of game theory
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35University of Michigan, Ann Arbor
Strategic IQ: A New Taxonomy of Games
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36University of Michigan, Ann Arbor
Nash versus CH Model: LL and MSD (in-sample)
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37University of Michigan, Ann Arbor
Economic Value:Economic Value:Definition and MotivationDefinition 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)
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38University of Michigan, Ann Arbor
Alternative SpecificationsAlternative 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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39University of Michigan, Ann Arbor
Example 3: Centipede GameExample 3: Centipede Game
1 2 2 21 1
0.400.10
0.200.80
1.600.40
0.803.20
6.401.60
3.2012.80
25.606.40
Figure 1: Six-move Centipede Game
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40University of Michigan, Ann Arbor
CH vs. Backward Induction CH vs. Backward Induction Principle (BIP)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)?
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41University of Michigan, Ann Arbor
BIP consists of three premisesBIP 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.
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42University of Michigan, Ann Arbor
Truncation ConsistencyTruncation Consistency
VS.
1 2 2 21 1
0.400.10
0.200.80
1.600.40
0.803.20
6.401.60
3.2012.80
25.606.40
Figure 1: Six-move Centipede game
1 2 21
0.400.10
0.200.80
1.600.40
0.803.20
6.401.60
Figure 2: Four-move Centipede game (Low-Stake)
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43University of Michigan, Ann Arbor
Subgame ConsistencySubgame Consistency
1 2 2 21 1
0.400.10
0.200.80
1.600.40
0.803.20
6.401.60
3.2012.80
25.606.40
VS.
2 21 1
1.600.40
0.803.20
6.401.60
3.2012.80
25.606.40
Figure 1: Six-move Centipede game
Figure 3: Four-move Centipede game (High-Stake (x4))
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44University of Michigan, Ann Arbor
Implied Take ProbabilityImplied Take ProbabilityImplied 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.
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45University of Michigan, Ann Arbor
Prediction on Implied Take Prediction on Implied Take ProbabilityProbability
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.
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46University of Michigan, Ann Arbor
Data: Truncation & Subgame Data: Truncation & Subgame ConsistenciesConsistencies
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
4-move(High Stake) 0.15 0.44 0.67 0.69
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47University of Michigan, Ann Arbor
KK-Step Look-ahead (Planning)-Step Look-ahead (Planning)
1 2 2 21 1
0.400.10
0.200.80
1.600.40
0.803.20
6.401.60
3.2012.80
25.606.40
1 2
0.400.10
0.200.80
V1
V2
Example: 1-step look-ahead
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48University of Michigan, Ann Arbor
Limited thinking and PlanningLimited thinking and PlanningXk (k), 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(1, 2, 3)
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49University of Michigan, Ann Arbor
Estimation ResultsEstimation Results6 stages All sessions
Low-Stake High-StakeSample Size 281 100 281 662
CalibrationSample Size 197 70 197 464
Agent Quantal Response Eqlbm (AQRE) -287.0 -106.8 -409.8 -848.2
Extensive Cognitive Hierarchy (xCH) -276.1 -105.9 -341.2 -753.0xCH (1=2=0) -276.1 -105.9 -341.2 -753.0
ValidationSample Size 84 30 84 198
Agent Quantal Response Eqlbm (AQRE) 281.0 100.0 281.0 662.0
Extensive Cognitive Hierarchy (xCH) -132.8 -41.5 -120.7 -293.9xCH (1=2=0) -132.8 -41.5 -121.1 -293.9
4 stages
Thinking steps and steps of planning are perfectly correlated
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50University of Michigan, Ann Arbor
Data and xCH Prediction: Data and xCH Prediction: Truncation & Subgame ConsistenciesTruncation & Subgame Consistencies
CH Prediction
6-move 0.06 0.16 0.15 0.48 0.90 0.99
4-move(Low-Stake) 0.15 0.31 0.76 0.97
4-move(High-Stake) 0.21 0.34 0.71 0.95
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
4-move(High Stake) 0.15 0.44 0.67 0.69