1 Teck H. Ho Spring 2006 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

2 Teck H. Ho Spring 2006 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.

3 Teck H. Ho Spring 2006 Main Goals  Provide a behavioral theory to explain and predict behaviors in any one-shot game.  Normal-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)

4 Teck H. Ho Spring 2006 Modeling Principles PrincipleNash CH Strategic Thinking   Best Response   Mutual Consistency 

5 Teck H. Ho Spring 2006 Modeling 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)

6 Teck H. Ho Spring 2006 Example 1: “zero-sum game” Messick(1965), Behavioral Science

7 Teck H. Ho Spring 2006 Nash Prediction: “zero-sum game”

8 Teck H. Ho Spring 2006 CH Prediction: “zero-sum game”

9 Teck H. Ho Spring 2006 Empirical Frequency: “zero-sum game”

10 Teck H. Ho Spring 2006 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.

11 Teck H. Ho Spring 2006 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 and best-respond based on beliefs  Iterative and no need to solve a fixed point

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13 Teck H. Ho Spring 2006 Theoretical Implications  Exhibits “increasingly rational expectations”   ∞  Normalized g K (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  ∞   0  marginal benefit of thinking harder  0

14 Teck H. Ho Spring 2006 Modeling Heterogeneity, f(k)  A1:  sharp drop-off due to increasing difficulty in simulating others’ behaviors  A2: f(0) + f(1) = 2f(2)

15 Teck H. Ho Spring 2006 Implications  A1  Poisson distribution with mean and variance =   A1,A2  Poisson,  golden ratio Φ)

16 Teck H. Ho Spring 2006 Poisson Distribution  f(k) with mean step of thinking  :

17 Teck H. Ho Spring 2006 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

18 Teck H. Ho Spring 2006 Estimates of Mean Thinking Step 

19 Teck H. Ho Spring 2006 Nash: Theory vs. Data

20 Teck H. Ho Spring 2006 CH Model: Theory vs. Data

21 Teck H. Ho Spring 2006 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?

22 Teck H. Ho Spring 2006 Nash versus CH Model: Economic Value

23 Teck H. Ho Spring 2006 Application: Strategic IQ  A battery of 30 "well-known" games  Measure CH player and a subject's strategic IQ by how much money they make (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  Soldiers  Writers  Chess players  Patients with brain damages

24 Teck H. Ho Spring 2006 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)

25 Teck H. Ho Spring 2006 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 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

26 Teck H. Ho Spring 2006 Results in various p-BC games

27 Teck H. Ho Spring 2006 Example 3: Centipede Game Figure 1: Six-move Centipede Game

28 Teck H. Ho Spring 2006 CH vs. Backward Induction Principle (BIP)  Can CH be extended (i.e., extensive CH) to provide an empirical alternative to BIP in predicting behavior in the Centipede?  Is there a difference between steps of thinking and look-ahead (planning)?

29 Teck H. Ho Spring 2006 The three underlying 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.

30 Teck H. Ho Spring 2006 Truncation Consistency VS Figure 1: Six-move Centipede game Figure 2: Four-move Centipede game (Low-Stake)

31 Teck H. Ho Spring 2006 The three underlying 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.

32 Teck H. Ho Spring 2006 Subgame Consistency VS Figure 1: Six-move Centipede game Figure 3: Four-move Centipede game (High-Stake (x4))

33 Teck H. Ho Spring 2006 Data: Truncation & Subgame Consistencies *Data from McKelvey and Palfrey (1992)

34 Teck H. Ho Spring 2006 Limited thinking and Planning Steps of Reasoning (People) Steps of Planning (Time) Bivariate Poisson

35 Teck H. Ho Spring 2006 Estimation Results (out-of- sample)  Thinking steps and steps of planning are perfectly correlated

36 Teck H. Ho Spring 2006 Data and xCH Prediction: Truncation & Subgame Consistencies

37 Teck H. Ho Spring 2006 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  xCH:  Provides an empirical alternative to BIP  Limited steps of thinking and planning are perfectly correlated

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39 Teck H. Ho Spring 2006 Bounded Rationality in Markets: Revised Utility Function

40 Teck H. Ho Spring 2006 Bounded Rationality in Markets: Alternative Solution Concepts

41 Teck H. Ho Spring 2006 Neural Foundations of Game Theory  Neural foundation of game theory

42 Teck H. Ho Spring 2006 Strategic IQ: A New Taxonomy of Games

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44 Teck H. Ho Spring 2006 First-Shot Games  The FCC license auctions, elections, military campaigns, legal disputes  Many marketing/IO models  Simple gane experiments in economics and marketing

45 Teck H. Ho Spring 2006 Nash versus CH Model: LL and MSD (in-sample)

46 Teck H. Ho Spring 2006 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)

47 Teck H. Ho Spring 2006 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

48 Teck H. Ho Spring 2006 Implied Take Probability  Implied take probability at each stage, p j  Truncation consistency: For a given j, p j is identical in both 4-move (low-stake) and 6-move games.  Subgame consistency: For a given j, p n-j (n=4 or 6) is identical in both 4-move (high-stake) and 6-move games.

49 Teck H. Ho Spring 2006 K-Step Look-ahead (Planning) V1V2V1V2 Example: 1-step look-ahead