A Cognitive Hierarchy (CH) Model of Games

1. A Cognitive Hierarchy (CH) Model of Games

2. Motivation • Nash 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.

3. 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. Modeling Principles PrincipleNashCH Strategic Thinking Best Response Mutual Consistency 

5. 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)

7. Example 1: “zero-sum game” Messick(1965), Behavioral Science

8. Nash Prediction: “zero-sum game”

9. CH Prediction: “zero-sum game”

10. Empirical Frequency: “zero-sum game” http://groups.haas.berkeley.edu/simulations/CH/

11. 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

12. 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 and best-respond based on beliefs • Iterative and no need to solve a fixed point

14. 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

15. Modeling Heterogeneity, f(k) • A1: • sharp drop-off due to increasing difficulty in simulating others’ behaviors • A2: f(0) + f(1) = 2f(2)

16. Implications • A1 Poisson distribution with mean and variance = t • A1,A2  Poisson, t=1.618..(golden ratio Φ)

17. 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 k λ 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.

18. Poisson Distribution • f(k) with mean step of thinking t:

20. 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

21. Estimates of Mean Thinking Step t

22. Nash: Theory vs. Data

23. CH Model: Theory vs. Data

24. 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?

25. Nash versus CH Model: Economic Value

26. 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 • Soldiers • Writers • Chess players • Patients with brain damages

27. 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 closest to the Target Number • The prize to the winner is US$20 Ho, Camerer, and Weigelt (AER, 1998)

28. A 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

29. Results in various p-BC games

30. 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

31. Research Agenda • Bounded Rationality in Markets • Revised Utility Functions • Empirical Alternatives to Nash Equilibrium • A New Taxonomy of Games • Neural Foundation of Game Theory

32. Bounded Rationality in Markets: Revised Utility Function

33. Bounded Rationality in Markets: Alternative Solution Concepts

34. Neural Foundations of Game Theory • Neural foundation of game theory

35. Strategic IQ: A New Taxonomy of Games

36. Nash versus CH Model: LL and MSD (in-sample)

37. 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)

38. 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

39. 2 1 1 1 2 2 25.60 6.40 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 Example 3: Centipede Game Figure 1: Six-move Centipede Game

40. 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)?

41. 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.

42. 2 1 1 1 2 2 25.60 6.40 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 2 1 1 2 6.40 1.60 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 Truncation Consistency Figure 1: Six-move Centipede game VS. Figure 2: Four-move Centipede game (Low-Stake)

43. 2 1 1 1 2 2 25.60 6.40 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 2 1 1 2 25.60 6.40 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 Subgame Consistency Figure 1: Six-move Centipede game VS. Figure 3: Four-move Centipede game (High-Stake (x4))

44. 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.

45. 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.

46. Data: Truncation & Subgame Consistencies

47. 2 1 1 1 2 2 25.60 6.40 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 K-Step Look-ahead (Planning) Example: 1-step look-ahead 1 2 V1 V2 0.40 0.10 0.20 0.80

48. 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, l2, l3)

49. Estimation Results  Thinking steps and steps of planning are perfectly correlated

50. Data and xCH Prediction: Truncation & Subgame Consistencies