1. 1 Duke PhD Summer Camp August 2007 Outline  Motivation  Mutual Consistency: CH Model  Noisy Best-Response: QRE Model  Instant Convergence: EWA Learning

2. 2 Duke PhD Summer Camp August 2007 Standard Assumptions in Equilibrium Analysis

3. 3 Duke PhD Summer Camp August 2007 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. 4 Duke PhD Summer Camp August 2007 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. 5 Duke PhD Summer Camp August 2007 Example A: Data

6. 6 Duke PhD Summer Camp August 2007 Example B: Exercise  The two players choose “effort” levels simultaneously, and the payoff of each player is given by  i = min (e 1, e 2 ) – c x e i  Efforts are integer from 110 to 170.

7. 7 Duke PhD Summer Camp August 2007 Example B: Exercise  The two players choose “effort” levels simultaneously, and the payoff of each player is given by  i = min (e 1, e 2 ) – c x e i  Efforts are integer from 110 to 170.  C = 0.1 or 0.9.

8. 8 Duke PhD Summer Camp August 2007 Example B: Data

9. 9 Duke PhD Summer Camp August 2007 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. 10 Duke PhD Summer Camp August 2007 Example 1: “zero-sum game” Messick(1965), Behavioral Science

11. 11 Duke PhD Summer Camp August 2007 Nash Prediction: “zero-sum game”

12. 12 Duke PhD Summer Camp August 2007 CH Prediction: “zero-sum game”

13. 13 Duke PhD Summer Camp August 2007 Empirical Frequency: “zero-sum game” http://groups.haas.berkeley.edu/simulations/CH/

14. 14 Duke PhD Summer Camp August 2007 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. 15 Duke PhD Summer Camp August 2007 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

16. 16 Duke PhD Summer Camp August 2007

17. 17 Duke PhD Summer Camp August 2007 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

18. 18 Duke PhD Summer Camp August 2007 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. 19 Duke PhD Summer Camp August 2007 Modeling Heterogeneity, f(k)  A1:  sharp drop-off due to increasing difficulty in simulating others’ behaviors  A2: f(0) + f(1) = 2f(2)

20. 20 Duke PhD Summer Camp August 2007Implications  A1  Poisson distribution with mean and variance =   A1,A2  Poisson,  golden ratio Φ)

21. 21 Duke PhD Summer Camp August 2007 Poisson Distribution  f(k) with mean step of thinking  :

22. 22 Duke PhD Summer Camp August 2007 Existence and Uniqueness: CH Solution  Existence: There is always a CH solution in any game  Uniqueness: It is always unique

23. 23 Duke PhD Summer Camp August 2007 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. 24 Duke PhD Summer Camp August 2007 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. 25 Duke PhD Summer Camp August 2007 Example 2: Entry games (data)

26. 26 Duke PhD Summer Camp August 2007 Behaviors of Level 0 and 1 Players (  =1.25) Level 0 Level 1 % of Entry Demand (as % of # of players)

27. 27 Duke PhD Summer Camp August 2007 Behaviors of Level 0 and 1 Players (  =1.25) Level 0 + Level 1 % of Entry Demand (as % of # of players)

28. 28 Duke PhD Summer Camp August 2007 Behaviors of Level 2 Players (  =1.25) Level 2 Level 0 + Level 1 % of Entry Demand (as % of # of players)

29. 29 Duke PhD Summer Camp August 2007 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)

30. 30 Duke PhD Summer Camp August 2007 CH Predictions in Entry Games (  = 1.25)

31. 31 Duke PhD Summer Camp August 2007 Homework  What value of  can help to explain the data in Example A?  How does CH model explain the data in Example B?

32. 32 Duke PhD Summer Camp August 2007 Empirical Frequency: “zero-sum game”

33. 33 Duke PhD Summer Camp August 2007 MLE Estimation

34. 34 Duke PhD Summer Camp August 2007 Estimates of Mean Thinking Step 

35. 35 Duke PhD Summer Camp August 2007 CH Model: CI of Parameter Estimates

36. 36 Duke PhD Summer Camp August 2007 Nash versus CH Model: LL and MSD

37. 37 Duke PhD Summer Camp August 2007 CH Model: Theory vs. Data (Mixed Games)

38. 38 Duke PhD Summer Camp August 2007 Nash: Theory vs. Data (Mixed Games)

39. 39 Duke PhD Summer Camp August 2007 Nash vs. CH (Mixed Games)

40. 40 Duke PhD Summer Camp August 2007 CH Model: Theory vs. Data (Entry and Mixed Games)

41. 41 Duke PhD Summer Camp August 2007 Nash: Theory vs. Data (Entry and Mixed Games)

42. 42 Duke PhD Summer Camp August 2007 CH vs. Nash (Entry and Mixed Games)

43. 43 Duke PhD Summer Camp August 2007 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. 44 Duke PhD Summer Camp August 2007 Nash versus CH Model: Economic Value

45. 45 Duke PhD Summer Camp August 2007 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. 46 Duke PhD Summer Camp August 2007 Results in various p-BC games

47. 47 Duke PhD Summer Camp August 2007 Results in various p-BC games

48. 48 Duke PhD Summer Camp August 2007 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. 49 Duke PhD Summer Camp August 2007 Outline  Motivation  Mutual Consistency: CH Model  Noisy Best-Response: QRE Model  Instant Convergence: EWA Learning