1 Teck-Hua Ho CH Model March – June, 2003 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

2 Teck-Hua Ho CH Model March – June, 2003 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.

3 Teck-Hua Ho CH Model March – June, 2003 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)

4 Teck-Hua Ho CH Model March – June, 2003 Modeling Principles PrincipleNash Thinking Strategic Thinking   Best Response   Mutual Consistency 

5 Teck-Hua Ho CH Model March – June, 2003 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)

6 Teck-Hua Ho CH Model March – June, 2003 Example 1: “zero-sum game” Messick(1965), Behavioral Science

7 Teck-Hua Ho CH Model March – June, 2003 Nash Prediction: “zero-sum game”

8 Teck-Hua Ho CH Model March – June, 2003 CH Prediction: “zero-sum game”

9 Teck-Hua Ho CH Model March – June, 2003 Empirical Frequency: “zero-sum game”

10 Teck-Hua Ho CH Model March – June, 2003 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)

11 Teck-Hua Ho CH Model March – June, 2003 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

12 Teck-Hua Ho CH Model March – June, 2003 Example 1: “zero-sum game”

13 Teck-Hua Ho CH Model March – June, 2003Implications  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

14 Teck-Hua Ho CH Model March – June, 2003 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

15 Teck-Hua Ho CH Model March – June, 2003 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)…

16 Teck-Hua Ho CH Model March – June, 2003 Implications  A1  Poisson distribution with mean and variance =   A1,A2  Poisson distribution, 1<   A1,A3  Poisson,    a  b)  Poisson,  golden ratio Φ)

17 Teck-Hua Ho CH Model March – June, 2003 Poisson Distribution  f(k) with mean step of thinking  :

18 Teck-Hua Ho CH Model March – June, 2003 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

19 Teck-Hua Ho CH Model March – June, 2003 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

20 Teck-Hua Ho CH Model March – June, 2003 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

21 Teck-Hua Ho CH Model March – June, 2003 Example 2: Entry games (data)

22 Teck-Hua Ho CH Model March – June, 2003 Behaviors of Level 0 and 1 Players (  =1.25) Level 0 Level 1 % of Entry Demand (as % of # of players)

23 Teck-Hua Ho CH Model March – June, 2003 Behaviors of Level 0 and 1 Players(  =1.25) Level 0 + Level 1 % of Entry Demand (as % of # of players)

24 Teck-Hua Ho CH Model March – June, 2003 Behaviors of Level 2 Players (  =1.25) Level 2 Level 0 + Level 1 % of Entry Demand (as % of # of players)

25 Teck-Hua Ho CH Model March – June, 2003 Behaviors of Level 0, 1, and 2 Players(  =1.25) Level 2 Level 0 + Level 1 Level 0 + Level 1 + Level 2 % of Entry Demand (as % of # of players)

26 Teck-Hua Ho CH Model March – June, 2003 Entry Games (Imposing Monotonicity on CH Model)

27 Teck-Hua Ho CH Model March – June, 2003 Estimates of Mean Thinking Step 

28 Teck-Hua Ho CH Model March – June, 2003 CH Model: CI of Parameter Estimates

29 Teck-Hua Ho CH Model March – June, 2003 Nash versus CH Model: LL and MSD

30 Teck-Hua Ho CH Model March – June, 2003 CH Model: Theory vs. Data (Mixed Games)

31 Teck-Hua Ho CH Model March – June, 2003 Nash: Theory vs. Data (Mixed Games)

32 Teck-Hua Ho CH Model March – June, 2003 CH Model: Theory vs. Data (Entry and Mixed Games)

33 Teck-Hua Ho CH Model March – June, 2003 Nash: Theory vs. Data (Entry and Mixed Games)

34 Teck-Hua Ho CH Model March – June, 2003 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?

35 Teck-Hua Ho CH Model March – June, 2003 Nash versus CH Model: Economic Value

36 Teck-Hua Ho CH Model March – June, 2003 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

37 Teck-Hua Ho CH Model March – June, 2003 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 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

38 Teck-Hua Ho CH Model March – June, 2003 Results from Caltech Board of Trustees

39 Teck-Hua Ho CH Model March – June, 2003 Results from Two Other Smart Subject Pools

40 Teck-Hua Ho CH Model March – June, 2003 Results from College Students

41 Teck-Hua Ho CH Model March – June, 2003 CH Model: Parameter Estimates

42 Teck-Hua Ho CH Model March – June, 2003 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