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

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

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 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 Duke PhD Summer Camp August 2007 Example A: Data

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 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 Duke PhD Summer Camp August 2007 Example B: Data

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

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

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

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

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 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 Duke PhD Summer Camp August 2007

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

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

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 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 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 Duke PhD Summer Camp August 2007 Example 2: Entry games (data)

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

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

33 Duke PhD Summer Camp August 2007 MLE Estimation

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

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

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

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

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

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

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

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

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

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

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

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

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