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A Cognitive Hierarchy Theory of One-Shot Games Teck H. Ho Haas School of Business University of California, Berkeley Joint work with Colin Camerer, Caltech Juin-Kuan Chong, NUS CH Model Teck-Hua Ho 1 Motivation q Nash equilibrium and its refinements: Dominant theories in economics for predicting behaviors in games. q Subjects in experiments hardly play Nash in the first round but do often converge to it eventually. q Multiplicity problem (e.g., coordination games) q Modeling heterogeneity really matters in games. CH Model Teck-Hua Ho 2 Research Goals qHow to model bounded rationality (first-period behavior)? qCognitive Hierarchy (CH) model qHow to model equilibration? qEWA learning model (Camerer and Ho, Econometrica, 1999; Ho, Camerer, and Chong, 2003) qHow to model repeated game behavior? qTeaching model (Camerer, Ho, and Chong, Journal of Economic Theory, 2002) CH Model Teck-Hua Ho 3 Modeling Principles Principle Nash Thinking Strategic Thinking í í Best Response í í Mutual Consistency í CH Model Teck-Hua Ho 4 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) CH Model Teck-Hua Ho 5 Example 1: “zero-sum game” Messick(1965), Behavioral Science CH Model Teck-Hua Ho 6 Nash Prediction: “zero-sum game” CH Model Teck-Hua Ho 7 CH Prediction: “zero-sum game” http://groups.haas.berkeley.edu/simulations/CH/ CH Model Teck-Hua Ho 8 Empirical Frequency: “zero-sum game” CH Model Teck-Hua Ho 9 The Cognitive Hierarchy (CH) Model qPeople are different and have different decision rules qModeling heterogeneity (i.e., distribution of types of players) qModeling decision rule of each type qGuided by modeling philosophy (general, precise, and empirically disciplined) CH Model Teck-Hua Ho 10 Modeling Decision Rule q f(0) step 0 choose randomly q f(k) k-step thinkers know proportions f(0),...f(k-1) q Normalize and best-respond CH Model Teck-Hua Ho 11 Example 1: “zero-sum game” CH Model Teck-Hua Ho 12 Implications qExhibits “increasingly rational expectations” q Normalized g(h) approximates f(h) more closely as kà ∞ (i.e., highest level types are “sophisticated” (or ”worldly) and earn the most qHighest level type actions converge as kà ∞ à marginal benefit of thinking harder à0 CH Model Teck-Hua Ho 13 Alternative Specifications qOverconfidence: 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 Kà ∞ qHas some odd properties (e.g., cycles in entry games) qSelf-conscious: qk-steps think there are other k-step thinkers qSimilar to Quantal Response Equilibrium/Nash qFits worse CH Model Teck-Hua Ho 14 Modeling Heterogeneity, f(k) q A1: q sharp drop-off due to increasing working memory constraint q A2: f(1) is the mode q A3: f(0)=f(2) (partial symmetry) q A4a: f(0)+f(1)=f(2)+f(3)+f(4)… q A4b: f(2)=f(3)+f(4)+f(5)… CH Model Teck-Hua Ho 15 Implications q A1à Poisson distribution with mean and variance = t qA1,A2à Poisson distribution, 1< t < 2 qA1,A3 à Poisson, t=Ö2=1.414.. q(A1,A4a,A4b) à Poisson, t=1.618..(golden ratio Φ) CH Model Teck-Hua Ho 16 Poisson Distribution q f(k) with mean step of thinking t: CH Model Teck-Hua Ho 17 Historical Roots q “Fictitious play” as an algorithm for computing Nash equilibrium (Brown, 1951; Robinson, 1951) q In our terminology, the fictitious play model is equivalent to one in which f(k) = 1/N for N steps of thinking and N à ∞ q 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 CH Model Teck-Hua Ho 18 Theoretical Properties of CH Model qAdvantages over Nash equilibrium qCan “solve” multiplicity problem (picks one statistical distribution) qSolves refinement problems (all moves occur in equilibrium) qSensible interpretation of mixed strategies (de facto purification) qTheory: qτà∞ converges to Nash equilibrium in (weakly) dominance solvable games qEqual splits in Nash demand games CH Model Teck-Hua Ho 19 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 1st period: q Consistent with Nash prediction, %(entrants) increases with D q “To a psychologist, it looks like magic”-- D. Kahneman ‘88 CH Model Teck-Hua Ho 20 Example 2: Entry games (data) CH Model Teck-Hua Ho 21 Behaviors of Level 0 and 1 Players (t =1.25) Level 1 % of Entry Level 0 Demand (as % of # of players) CH Model Teck-Hua Ho 22 Behaviors of Level 0 and 1 Players(t =1.25) Level 0 + Level 1 % of Entry Demand (as % of # of players) CH Model Teck-Hua Ho 23 Behaviors of Level 2 Players (t =1.25) Level 2 Level 0 + Level 1 % of Entry Demand (as % of # of players) CH Model Teck-Hua Ho 24 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) CH Model Teck-Hua Ho 25 Entry Games (Imposing Monotonicity on CH Model) CH Model Teck-Hua Ho 26 Estimates of Mean Thinking Step t CH Model Teck-Hua Ho 27 CH Model: CI of Parameter Estimates CH Model Teck-Hua Ho 28 Nash versus CH Model: LL and MSD CH Model Teck-Hua Ho 29 CH Model: Theory vs. Data (Mixed Games) CH Model Teck-Hua Ho 30 Nash: Theory vs. Data (Mixed Games) CH Model Teck-Hua Ho 31 CH Model: Theory vs. Data (Entry and Mixed Games) CH Model Teck-Hua Ho 32 Nash: Theory vs. Data (Entry and Mixed Games) CH Model Teck-Hua Ho 33 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, would they have made a higher payoff? CH Model Teck-Hua Ho 34 Nash versus CH Model: Economic Value CH Model Teck-Hua Ho 35 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 CH Model Teck-Hua Ho 36 A Sample of Caltech Board of Trustees § David Baltimore • David D. Ho President Director California Institute of The Aaron Diamond AIDS Research Center Technology § Donald L. Bren • Gordon E. Moore Chairman Emeritus Chairman of the Board Intel Corporation The Irvine Company • Stephen A. Ross • Eli Broad Co-Chairman, Roll and Ross Asset Mgt Corp Chairman SunAmerica Inc. • Lounette M. Dyer • Sally K. Ride Chairman President Imaginary Lines, Inc., and Silk Route Technology Hibben Professor of Physics CH Model Teck-Hua Ho 37 Results from Caltech Board of Trustees CH Model Teck-Hua Ho 38 Results from Two Other Smart Subject Pools CH Model Teck-Hua Ho 39 Results from College Students CH Model Teck-Hua Ho 40 CH Model: Parameter Estimates CH Model Teck-Hua Ho 41 Summary q CH Model: qDiscrete thinking steps qFrequency Poisson distributed q One-shot games qFits better than Nash and adds more economic value qExplains “magic” of entry games qSensible interpretation of mixed strategies qCan “solve” multiplicity problem q Initial conditions for learning CH Model Teck-Hua Ho 42

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posted: | 9/26/2013 |

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