Outline Motivation Mutual Consistency CH Model Noisy BestResponse
Outline Ø Motivation Ø Mutual Consistency: CH Model Ø Noisy Best-Response: QRE Model Ø Instant Convergence: EWA Learning Duke Ph. D Summer Camp 1
Standard Assumptions in Equilibrium Analysis August 2007 Duke Ph. D Summer Camp 2
Example A: Exercise q 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: G 1 G 2 August 2007 Duke Ph. D Summer Camp 3
Example A: Exercise q 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: G 1 G 2 August 2007 Duke Ph. D Summer Camp 4
Example A: Data August 2007 Duke Ph. D Summer Camp 5
Example B: Exercise q The two players choose “effort” levels simultaneously, and the payoff of each player is given by pi = min (e 1, e 2) – c x ei q Efforts are integer from 110 to 170. August 2007 Duke Ph. D Summer Camp 6
Example B: Exercise q The two players choose “effort” levels simultaneously, and the payoff of each player is given by pi = min (e 1, e 2) – c x ei q Efforts are integer from 110 to 170. q C = 0. 1 or 0. 9. August 2007 Duke Ph. D Summer Camp 7
Example B: Data August 2007 Duke Ph. D Summer Camp 8
Motivation: CH q Model heterogeneity explicitly (people are not equally smart) q Introduce the word surprise into the game theory’s dictionary (e. g. , Next movie) q Generate new predictions (reconcile various treatment effects in lab data not predicted by standard theory) Camerer, Ho, and Chong (QJE, 2004) August 2007 Duke Ph. D Summer Camp 9
Example 1: “zero-sum game” Messick(1965), Behavioral Science August 2007 Duke Ph. D Summer Camp 10
Nash Prediction: “zero-sum game” August 2007 Duke Ph. D Summer Camp 11
CH Prediction: “zero-sum game” August 2007 Duke Ph. D Summer Camp 12
Empirical Frequency: “zero-sum game” http: //groups. haas. berkeley. edu/simulations/CH/ August 2007 Duke Ph. D Summer Camp 13
The Cognitive Hierarchy (CH) Model q. People are different and have different decision rules. q. Modeling heterogeneity (i. e. , distribution of types of players). Types of players are denoted by levels 0, 1, 2, 3, …, q. Modeling decision rule of each type. August 2007 Duke Ph. D Summer Camp 14
Modeling Decision Rule q Frequency of k-step is f(k) q Step 0 choose randomly q k-step thinkers know proportions f(0), . . . f(k-1) q Form beliefs and best-respond based on beliefs q Iterative and no need to solve a fixed point August 2007 Duke Ph. D Summer Camp 15
August 2007 Duke Ph. D Summer Camp 16
Theoretical Implications q. Exhibits “increasingly rational expectations” q Normalized g. K(h) approximates f(h) more closely as k ∞ (i. e. , highest level types are “sophisticated” (or "worldly") and earn the most. q. Highest level type actions converge as k ∞ marginal benefit of thinking harder 0 August 2007 Duke Ph. D Summer Camp 17
Alternative Specifications q. Overconfidence: qk-steps think others are all one step lower (k-1) (Stahl, GEB, 1995; Nagel, AER, 1995; Ho, Camerer and Weigelt, AER, 1998) q“Increasingly irrational expectations” as K ∞ q. Has some odd properties (e. g. , cycles in entry games) q. Self-conscious: qk-steps think there are other k-step thinkers q. Similar to Quantal Response Equilibrium/Nash q. Fits worse August 2007 Duke Ph. D Summer Camp 18
Modeling Heterogeneity, f(k) q A 1: qsharp drop-off due to increasing difficulty in simulating others’ behaviors q A 2: f(0) + f(1) = 2 f(2) August 2007 Duke Ph. D Summer Camp 19
Implications q A 1 Poisson distribution and variance = t with mean q. A 1, A 2 Poisson, t=1. 618. . (golden ratio Φ) August 2007 Duke Ph. D Summer Camp 20
Poisson Distribution q f(k) with mean step of thinking t: August 2007 Duke Ph. D Summer Camp 21
Existence and Uniqueness: CH Solution Ø Existence: There is always a CH solution in any game Ø Uniqueness: It is always unique August 2007 Duke Ph. D Summer Camp 22
Theoretical Properties of CH Model q. Advantages over Nash equilibrium q. Can “solve” multiplicity problem (picks one statistical distribution) q. Sensible interpretation of mixed strategies (de facto purification) q. Theory: qτ ∞ converges to Nash equilibrium in (weakly) dominance solvable games August 2007 Duke Ph. D Summer Camp 23
Example 2: Entry games q 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 q Experimental regularity in the 1 st period: q Consistent with Nash prediction, %(entrants) increases with D q “To a psychologist, it looks like magic”-- D. Kahneman ‘ 88 August 2007 Duke Ph. D Summer Camp 24
Example 2: Entry games (data) August 2007 Duke Ph. D Summer Camp 25
Behaviors of Level 0 and 1 Players (t =1. 25) % of Entry Level 1 Level 0 Demand (as % of # of players) August 2007 Duke Ph. D Summer Camp 26
Behaviors of Level 0 and 1 Players (t =1. 25) % of Entry Level 0 + Level 1 Demand (as % of # of players) August 2007 Duke Ph. D Summer Camp 27
Behaviors of Level 2 Players (t =1. 25) Level 2 % of Entry Level 0 + Level 1 Demand (as % of # of players) August 2007 Duke Ph. D Summer Camp 28
Behaviors of Level 0, 1, and 2 Players (t =1. 25) Level 2 % of Entry Level 0 + Level 1 + Level 2 Level 0 + Level 1 Demand (as % of # of players) August 2007 Duke Ph. D Summer Camp 29
CH Predictions in Entry Games (t = 1. 25) August 2007 Duke Ph. D Summer Camp 30
Homework Ø What value of t can help to explain the data in Example A? Ø How does CH model explain the data in Example B? August 2007 Duke Ph. D Summer Camp 31
Empirical Frequency: “zero-sum game” August 2007 Duke Ph. D Summer Camp 32
MLE Estimation August 2007 Duke Ph. D Summer Camp 33
Estimates of Mean Thinking Step t August 2007 Duke Ph. D Summer Camp 34
CH Model: CI of Parameter Estimates August 2007 Duke Ph. D Summer Camp 35
Nash versus CH Model: LL and MSD August 2007 Duke Ph. D Summer Camp 36
CH Model: Theory vs. Data (Mixed Games) August 2007 Duke Ph. D Summer Camp 37
Nash: Theory vs. Data (Mixed Games) August 2007 Duke Ph. D Summer Camp 38
Nash vs. CH (Mixed Games) August 2007 Duke Ph. D Summer Camp 39
CH Model: Theory vs. Data (Entry and Mixed Games) August 2007 Duke Ph. D Summer Camp 40
Nash: Theory vs. Data (Entry and Mixed Games) August 2007 Duke Ph. D Summer Camp 41
CH vs. Nash (Entry and Mixed Games) August 2007 Duke Ph. D Summer Camp 42
Economic Value q Evaluate models based on their value-added rather than statistical fit (Camerer and Ho, 2000) q Treat models like consultants q 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? q A measure of disequilibrium August 2007 Duke Ph. D Summer Camp 43
Nash versus CH Model: Economic Value August 2007 Duke Ph. D Summer Camp 44
Example 3: P-Beauty Contest q n players q Every player simultaneously chooses a number from 0 to 100 q Compute the group average q Define Target Number to be 0. 7 times the group average q The winner is the player whose number is the closet to the Target Number q The prize to the winner is US$20 Duke Ph. D Summer Camp 45
Results in various p-BC games August 2007 Duke Ph. D Summer Camp 46
Results in various p-BC games August 2007 Duke Ph. D Summer Camp 47
Summary q CH Model: q. Discrete thinking steps q. Frequency Poisson distributed q One-shot games q. Fits better than Nash and adds more economic value q. Explains “magic” of entry games q. Sensible interpretation of mixed strategies q. Can “solve” multiplicity problem q Initial conditions for learning August 2007 Duke Ph. D Summer Camp 48
Outline Ø Motivation Ø Mutual Consistency: CH Model Ø Noisy Best-Response: QRE Model Ø Instant Convergence: EWA Learning Duke Ph. D Summer Camp 49
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