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1 University of Michigan, Ann Arbor A Cognitive Hierarchy (CH) Model of Games

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2 University of Michigan, Ann Arbor Motivation Nash equilibrium and its refinements: Dominant theories in economics for predicting behaviors in competitive situations. Subjects do not play Nash in many one-shot games. Behaviors do not converge to Nash with repeated interactions in some games. Multiplicity problem (e.g., coordination games). Modeling heterogeneity really matters in games.

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3 University of Michigan, Ann Arbor Main Goals Provide a behavioral theory to explain and predict behaviors in any one-shot game Normal-form games (e.g., zero-sum game, p- beauty contest) Extensive-form games (e.g., centipede) Provide an empirical alternative to Nash equilibrium (Camerer, Ho, and Chong, QJE, 2004) and backward induction principle (Ho, Camerer, and Chong, 2005)

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4 University of Michigan, Ann Arbor Modeling Principles PrincipleNash CH Strategic Thinking Best Response Mutual Consistency

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5 University of Michigan, Ann Arbor Modeling Philosophy Simple(Economics) General(Economics) Precise(Economics) Empirically disciplined(Psychology) “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)

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6 University of Michigan, Ann Arbor

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7 Example 1: “zero-sum game” Messick(1965), Behavioral Science

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8 University of Michigan, Ann Arbor Nash Prediction: “zero-sum game”

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9 University of Michigan, Ann Arbor CH Prediction: “zero-sum game”

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10 University of Michigan, Ann Arbor Empirical Frequency: “zero-sum game” http://groups.haas.berkeley.edu/simulations/CH/

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11 University of Michigan, Ann Arbor 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

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12 University of Michigan, Ann Arbor Modeling Decision Rule Proportion 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

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13 University of Michigan, Ann Arbor

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14 University of Michigan, Ann Arbor 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

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15 University of Michigan, Ann Arbor Modeling Heterogeneity, f(k) A1: sharp drop-off due to increasing difficulty in simulating others’ behaviors A2: f(0) + f(1) = 2f(2)

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16 University of Michigan, Ann Arbor Implications A1 Poisson distribution with mean and variance = A1,A2 Poisson, golden ratio Φ)

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17 University of Michigan, Ann Arbor La loi de Poisson a été introduite en 1838 par Siméon Denis Poisson (1781– 1840), dans son ouvrage Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Le sujet principal de cet ouvrage consiste en certaines variables aléatoires N qui dénombrent, entre autres choses, le nombre d'occurrences (parfois appelées « arrivées ») qui prennent place pendant un laps de temps donné. Si le nombre moyen d'occurrences dans cet intervalle est λ, alors la probabilité qu'il existe exactement k occurrences (k étant un entier naturel, k = 0, 1, 2,...) est: Où: e est la base de l'exponentielle (2,718...) k! est la factorielle de k λ est un nombre réel strictement positif. On dit alors que X suit la loi de Poisson de paramètre λ. Par exemple, si un certain type d'évènements se produit en moyenne 4 fois par minute, pour étudier le nombre d'évènements se produisant dans un laps de temps de 10 minutes, on choisit comme modèle une loi de Poisson de paramètre λ = 10×4 = 40.

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18 University of Michigan, Ann Arbor Poisson Distribution f(k) with mean step of thinking :

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19 University of Michigan, Ann Arbor

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20 University of Michigan, Ann Arbor 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

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21 University of Michigan, Ann Arbor Estimates of Mean Thinking Step

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22 University of Michigan, Ann Arbor Nash: Theory vs. Data

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23 University of Michigan, Ann Arbor CH Model: Theory vs. Data

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24 University of Michigan, Ann Arbor 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?

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25 University of Michigan, Ann Arbor Nash versus CH Model: Economic Value

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26 University of Michigan, Ann Arbor Application: Strategic IQ http://128.32.67.154/siq13/default1.asp A battery of 30 "well-known" games Measure a subject's strategic IQ by how much money she makes (matched against a defined pool of subjects) Factor analysis + fMRI to figure out whether certain brain region accounts for superior performance in "similar" games Specialized subject pools Soldiers Writers Chess players Patients with brain damages

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27 University of Michigan, Ann Arbor Example 2: 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 closest to the Target Number The prize to the winner is US$20 Ho, Camerer, and Weigelt (AER, 1998)

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28 University of Michigan, Ann Arbor A Sample of CEOs David Baltimore President California Institute of Technology Donald L. Bren Chairman of the Board The Irvine Company Eli Broad Chairman SunAmerica Inc. Lounette M. Dyer Chairman Silk Route Technology David D. Ho Director The Aaron Diamond AIDS Research Center Gordon E. Moore Chairman Emeritus Intel Corporation Stephen A. Ross Co-Chairman, Roll and Ross Asset Mgt Corp Sally K. Ride President Imaginary Lines, Inc., and Hibben Professor of Physics

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29 University of Michigan, Ann Arbor Results in various p-BC games

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30 University of Michigan, Ann Arbor Summary CH Model: Discrete thinking steps Frequency Poisson distributed One-shot games Fits better than Nash and adds more economic value Sensible interpretation of mixed strategies Can “solve” multiplicity problem Application: Measurement of Strategic IQ

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31 University of Michigan, Ann Arbor Research Agenda Bounded Rationality in Markets Revised Utility Functions Empirical Alternatives to Nash Equilibrium A New Taxonomy of Games Neural Foundation of Game Theory

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32 University of Michigan, Ann Arbor Bounded Rationality in Markets: Revised Utility Function

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33 University of Michigan, Ann Arbor Bounded Rationality in Markets: Alternative Solution Concepts

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34 University of Michigan, Ann Arbor Neural Foundations of Game Theory Neural foundation of game theory

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35 University of Michigan, Ann Arbor Strategic IQ: A New Taxonomy of Games

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36 University of Michigan, Ann Arbor Nash versus CH Model: LL and MSD (in-sample)

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37 University of Michigan, Ann Arbor Economic Value: Definition and Motivation “A normative model must produce strategies that are at least as good as what people can do without them.” (Schelling, 1960) A measure of degree of disequilibrium, in dollars. If players are in equilibrium, then an equilibrium theory will advise them to make the same choices they would make anyway, and hence will have zero economic value If players are not in equilibrium, then players are mis-forecasting what others will do. A theory with more accurate beliefs will have positive economic value (and an equilibrium theory can have negative economic value if it misleads players)

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38 University of Michigan, Ann Arbor 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

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39 University of Michigan, Ann Arbor Example 3: Centipede Game 1 2 2 2 11 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 25.60 6.40 Figure 1: Six-move Centipede Game

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40 University of Michigan, Ann Arbor CH vs. Backward Induction Principle (BIP) Is extensive CH (xCH) a sensible empirical alternative to BIP in predicting behavior in an extensive-form game like the Centipede? Is there a difference between steps of thinking and look-ahead (planning)?

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41 University of Michigan, Ann Arbor BIP consists of three premises Rationality: Given a choice between two alternatives, a player chooses the most preferred. Truncation consistency: Replacing a subgame with its equilibrium payoffs does not affect play elsewhere in the game. Subgame consistency: Play in a subgame is independent of the subgame’s position in a larger game. Binmore, McCarthy, Ponti, and Samuelson (JET, 2002) show violations of both truncation and subgame consistencies.

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42 University of Michigan, Ann Arbor Truncation Consistency VS. 1 2 2 2 11 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 25.60 6.40 Figure 1: Six-move Centipede game 1 2 2 1 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 Figure 2: Four-move Centipede game (Low-Stake)

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43 University of Michigan, Ann Arbor Subgame Consistency 1 2 2 2 11 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 25.60 6.40 VS. 2 2 11 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 25.60 6.40 Figure 1: Six-move Centipede game Figure 3: Four-move Centipede game (High-Stake (x4))

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44 University of Michigan, Ann Arbor Implied Take Probability Implied take probability at each stage, p j Truncation consistency: For a given j, p j is identical in both 4-move (low-stake) and 6-move games. Subgame consistency: For a given j, p n-j (n=4 or 6) is identical in both 4-move (high-stake) and 6-move games.

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45 University of Michigan, Ann Arbor Prediction on Implied Take Probability Implied take probability at each stage, p j Truncation consistency: For a given j, p j is identical in both 4-move (low-stake) and 6-move games. Subgame consistency: For a given j, p n-j (n=4 or 6) is identical in both 4-move (high-stake) and 6-move games.

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46 University of Michigan, Ann Arbor Data: Truncation & Subgame Consistencies

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47 University of Michigan, Ann Arbor K-Step Look-ahead (Planning) 1 2 2 2 11 0.40 0.10 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 12.80 25.60 6.40 1 2 0.40 0.10 0.20 0.80 V1V2V1V2 Example: 1-step look-ahead

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48 University of Michigan, Ann Arbor Limited thinking and Planning X k ( k ), k=1,2,3 follow independent Poisson distributions X 3 =common thinking/planning; X 1 =extra thinking, X 2 =extra planning X (thinking) =X 1 +X 3 ; Y (planning) =X 2 +X 3 follow jointly a bivariate Poisson distribution BP( 1, 2, 3 )

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49 University of Michigan, Ann Arbor Estimation Results Thinking steps and steps of planning are perfectly correlated

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50 University of Michigan, Ann Arbor Data and xCH Prediction: Truncation & Subgame Consistencies

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