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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. Motivation.

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A Cognitive Hierarchy Theory of One-Shot Games

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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


  • 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.

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)

Modeling Principles


Strategic Thinking 

Best Response 

Mutual Consistency 

Modeling Philosophy

General(Game Theory)

Precise(Game Theory)

Empirically disciplined(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)

Example 1: “zero-sum game”

Messick(1965), Behavioral Science

Nash Prediction: “zero-sum game”

CH Prediction: “zero-sum game”


Empirical Frequency: “zero-sum game”

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)

Modeling Decision Rule

  • f(0) step 0 choose randomly

  • f(k) k-step thinkers know proportions f(0),...f(k-1)

  • Normalize and best-respond

Example 1: “zero-sum game”


  • 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

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

Modeling Heterogeneity, f(k)

  • A1:

    • 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)…


  • A1 Poisson distribution with mean and variance = t

  • A1,A2 Poisson distribution, 1< t < 2

  • A1,A3  Poisson, t=2=1.414..

  • (A1,A4a,A4b)  Poisson, t=1.618..(golden ratio Φ)

Poisson Distribution

  • f(k) with mean step of thinking t:

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

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

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

Example 2: Entry games (data)

Behaviors of Level 0 and 1

Players (t =1.25)

Level 1

% of Entry

Level 0

Demand (as % of # of players)

Behaviors of Level 0 and 1

Players(t =1.25)

Level 0 + Level 1

% of Entry

Demand (as % of # of players)

Behaviors of Level 2 Players

(t =1.25)

Level 2

Level 0 + Level 1

% of Entry

Demand (as % of # of players)

Behaviors of Level 0, 1, and

2 Players(t =1.25)

Level 2

Level 0 + Level 1 +

Level 2

% of Entry

Level 0 +

Level 1

Demand (as % of # of players)

Entry Games (Imposing Monotonicity on CH Model)

Estimates of Mean Thinking Step t

CH Model: CI of Parameter Estimates

Nash versus CH Model:

LL and MSD

CH Model: Theory vs. Data

(Mixed Games)

Nash: Theory vs. Data

(Mixed Games)

CH Model: Theory vs. Data

(Entry and Mixed Games)

Nash: Theory vs. Data

(Entry and Mixed Games)

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?

Nash versus CH Model:

Economic Value

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

A Sample of Caltech Board of Trustees

  • 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

Results from Caltech Board of Trustees

Results from Two Other Smart Subject Pools

Results from College Students

CH Model: Parameter Estimates


  • 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

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