Outline

1. Outline • Motivation • Mutual Consistency: CH Model • Noisy Best-Response: QRE Model • Instant Convergence: EWA Learning

2. Standard Assumptions in Equilibrium Analysis

3. Example A: Exercise • Consider matching pennies games in which the row player chooses between Top and Bottom and the column player simultaneously chooses between Left and Right, as shown below: G1 G2

4. Example A: Exercise • Consider matching pennies games in which the row player chooses between Top and Bottom and the column player simultaneously chooses between Left and Right, as shown below: G1 G2

5. Example A: Data

6. Example B: Exercise • The two players choose “effort” levels simultaneously, and the payoff of each player is given by pi = min (e1, e2) – c x ei • Efforts are integer from 110 to 170.

7. Example B: Exercise • The two players choose “effort” levels simultaneously, and the payoff of each player is given by pi = min (e1, e2) – c x ei • Efforts are integer from 110 to 170. • C = 0.1 or 0.9.

8. Example B: Data

9. Motivation: CH • Model heterogeneity explicitly (people are not equally smart) • Introduce the word surprise into the game theory’s dictionary (e.g., Next movie) • Generate new predictions (reconcile various treatment effects in lab data not predicted by standard theory) Camerer, Ho, and Chong (QJE, 2004)

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

11. Nash Prediction: “zero-sum game”

12. CH Prediction: “zero-sum game”

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

14. 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.

15. 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

17. 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

18. 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

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

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

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

22. Existence and Uniqueness:CH Solution • Existence: There is always a CH solution in any game • Uniqueness: It is always unique

23. 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

24. 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

25. Example 2: Entry games (data)

26. Behaviors of Level 0 and 1 Players (t =1.25) Level 1 % of Entry Level 0 Demand (as % of # of players)

27. Behaviors of Level 0 and 1 Players (t =1.25) Level 0 + Level 1 % of Entry Demand (as % of # of players)

28. Level 2 Level 0 + Level 1 Behaviors of Level 2 Players (t =1.25) % of Entry Demand (as % of # of players)

29. 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)

30. CH Predictions in Entry Games (t = 1.25)

31. Homework • What value oft can help to explain the data in Example A? • How does CH model explain the data in Example B?

32. Empirical Frequency: “zero-sum game”

33. MLE Estimation

34. Estimates of Mean Thinking Step t

35. CH Model: CI of Parameter Estimates

36. Nash versus CH Model: LL and MSD

37. CH Model: Theory vs. Data (Mixed Games)

38. Nash: Theory vs. Data (Mixed Games)

39. Nash vs. CH (Mixed Games)

40. CH Model: Theory vs. Data (Entry and Mixed Games)

41. Nash: Theory vs. Data (Entry and Mixed Games)

42. CH vs. Nash (Entry and Mixed Games)

43. 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? • A measure of disequilibrium

44. Nash versus CH Model: Economic Value

45. 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

46. Results in various p-BC games

47. Results in various p-BC games

48. 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

49. Outline • Motivation • Mutual Consistency: CH Model • Noisy Best-Response: QRE Model • Instant Convergence: EWA Learning