1. 1 Teck-Hua Ho CH Model April 4, 2004 Outline  In-class Experiment and Examples  Empirical Alternative I: Model of Cognitive Hierarchy (Camerer, Ho, and Chong, QJE, 2004)  Empirical Alternative II: Quantal Response Equilibrium (McKelvey and Palfrey, GEB, 1995)  Empirical Alternative III: Model of Noisy Introspection (Goeree and Holt, AER, 2001)

2. 2 Teck-Hua Ho CH Model April 4, 2004 Example 1:  Consider the game in which two players independently and simultaneously choose integer numbers between and including 180 and 300. Both players are paid the lower of the two numbers, and, in addition, an amount R > 1 is transferred from the player with the higher number to the player with the lower number.  R = 5 versus R = 180

3. 3 Teck-Hua Ho CH Model April 4, 2004 Example 1:

4. 4 Teck-Hua Ho CH Model April 4, 2004 Example 2:  Consider a symmetric matching pennies game in which the row player chooses between Top and Bottom and the column player simultaneously chooses between Left and Right, as shown below:

5. 5 Teck-Hua Ho CH Model April 4, 2004 Example 2:

6. 6 Teck-Hua Ho CH Model April 4, 2004 Example 3:  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.

7. 7 Teck-Hua Ho CH Model April 4, 2004 Example 3:

8. 8 Teck-Hua Ho CH Model April 4, 2004 Motivation  Nash equilibrium and its refinements: Dominant theories in economics for predicting behaviors in games.  Subjects in experiments do not play Nash in most one- shot games and in some repeated games.  Multiplicity problem (e.g., coordination games).  Modeling heterogeneity really matters in games.

9. 9 Teck-Hua Ho CH Model April 4, 2004 Research Goals  How to model bounded rationality (one-shot game)?  Cognitive Hierarchy (CH) model (Camerer, Ho, and Chong, QJE, 2004)  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, JET, 2002)

10. 10 Teck-Hua Ho CH Model April 4, 2004 Modeling Principles PrincipleNash CH QRE NI Strategic Thinking     Best Response   Mutual Consistency  

11. 11 Teck-Hua Ho CH Model April 4, 2004 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)

12. 12 Teck-Hua Ho CH Model April 4, 2004 Example 1: “zero-sum game” Messick(1965), Behavioral Science

13. 13 Teck-Hua Ho CH Model April 4, 2004 Nash Prediction: “zero-sum game”

14. 14 Teck-Hua Ho CH Model April 4, 2004 CH Prediction: “zero-sum game” http://groups.haas.berkeley.edu/simulations/CH/

15. 15 Teck-Hua Ho CH Model April 4, 2004 Empirical Frequency: “zero-sum game”

16. 16 Teck-Hua Ho CH Model April 4, 2004 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)

17. 17 Teck-Hua Ho CH Model April 4, 2004 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

18. 18 Teck-Hua Ho CH Model April 4, 2004Implications  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

19. 19 Teck-Hua Ho CH Model April 4, 2004 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

20. 20 Teck-Hua Ho CH Model April 4, 2004 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)…

21. 21 Teck-Hua Ho CH Model April 4, 2004 Implications  A1  Poisson distribution with mean and variance =   A1,A2  Poisson distribution, 1<   A1,A3  Poisson,    a  b)  Poisson,  golden ratio Φ)

22. 22 Teck-Hua Ho CH Model April 4, 2004 Poisson Distribution  f(k) with mean step of thinking  :

23. 23 Teck-Hua Ho CH Model April 4, 2004 Example 1: “zero-sum game”

24. 24 Teck-Hua Ho CH Model April 4, 2004 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

25. 25 Teck-Hua Ho CH Model April 4, 2004 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

26. 26 Teck-Hua Ho CH Model April 4, 2004 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

27. 27 Teck-Hua Ho CH Model April 4, 2004 Example 2: Entry games (data)

28. 28 Teck-Hua Ho CH Model April 4, 2004 Behaviors of Level 0 and 1 Players (  =1.25) Level 0 Level 1 % of Entry Demand (as % of # of players)

29. 29 Teck-Hua Ho CH Model April 4, 2004 Behaviors of Level 0 and 1 Players(  =1.25) Level 0 + Level 1 % of Entry Demand (as % of # of players)

30. 30 Teck-Hua Ho CH Model April 4, 2004 Behaviors of Level 2 Players (  =1.25) Level 2 Level 0 + Level 1 % of Entry Demand (as % of # of players)

31. 31 Teck-Hua Ho CH Model April 4, 2004 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)

32. 32 Teck-Hua Ho CH Model April 4, 2004 Entry Games (Imposing Monotonicity on CH Model)

33. 33 Teck-Hua Ho CH Model April 4, 2004 Estimates of Mean Thinking Step 

34. 34 Teck-Hua Ho CH Model April 4, 2004 CH Model: CI of Parameter Estimates

35. 35 Teck-Hua Ho CH Model April 4, 2004 Nash versus CH Model: LL and MSD

36. 36 Teck-Hua Ho CH Model April 4, 2004 CH Model: Theory vs. Data (Mixed Games)

37. 37 Teck-Hua Ho CH Model April 4, 2004 Nash: Theory vs. Data (Mixed Games)

38. 38 Teck-Hua Ho CH Model April 4, 2004 CH Model: Theory vs. Data (Entry and Mixed Games)

39. 39 Teck-Hua Ho CH Model April 4, 2004 Nash: Theory vs. Data (Entry and Mixed Games)

40. 40 Teck-Hua Ho CH Model April 4, 2004 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?  Also a measure of disequilibration

41. 41 Teck-Hua Ho CH Model April 4, 2004 Nash versus CH Model: Economic Value

42. 42 Teck-Hua Ho CH Model April 4, 2004 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

43. 43 Teck-Hua Ho CH Model April 4, 2004 CH Model: Parameter Estimates

44. 44 Teck-Hua Ho CH Model April 4, 2004 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