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Learning and teaching in games: Statistical models of human play in experiments Colin F. Camerer, Social Sciences Caltech (camerer@hss.caltech.edu) Teck Ho, Berkeley (Haas Business School) Kuan Chong, National Univ Singapore How can bounded rationality be modelled in games? Theory desiderata: Precise, general, useful (game theory), and cognitively plausible, empirically disciplined (cog sci) Three components: –Cognitive hierarchy thinking model (one parameter, creates initial conditions) –Learning model (EWA, fEWA) - Sophisticated teaching’ model (repeated games) - Sophisticated teaching’ model (repeated games) Shameless plug: Camerer, Behavioral Game Theory (Princeton, Feb ’03) or see website hss.caltech.edu/~camerer

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Behavioral models use some game theory principles, and weaken other principles Principle equilibrium Thinking LearningTeaching concept of a game strategic thinking best response mutual consistency learning strategic foresight

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(Typical) experimental economics methods Repeated matrix stage game (Markov w/ 1 state) Repeated with “one night stand” (“stranger”) rematching protocol & feedback (to allow learning without repeated-game reputation- building) Game is described abstractly, payoffs are public knowledge (e.g., read out loud) Subjects paid $ according to choices (~$12/hr) Why this style? Basic question is whether S’s can “compute” equiilibrium *, not meant to be realistic Establish regularity across S’s, different game structures Statistical fitting: Parsimonious (1+ parameters) models, fit (in sample) & predict (out of sample) & compute economic value * Question now answered (No): Would be useful to move to low- information MAL designs

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Beauty contest game: Pick numbers [0,100] closest to (2/3)*(average number) wins

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“Beauty contest” game (Ho, Camerer, Weigelt Amer Ec Rev 98): Pick numbers x i [0,100] Closest to (2/3)*(average number) wins $20

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Table: Data and estimates of in pbc games (equilibrium = 0) datasteps of subjects/gamemeanstd devthinking game theorists1921.83.7 Caltech2311.13.0 newspaper2320.23.0 portfolio mgrs2416.12.8 econ PhD class2718.72.3 Caltech g=32225.71.8 high school3318.61.6 1/2 mean2719.91.5 70 yr olds3717.51.1 Germany3720.01.1 CEOs3818.81.0 game p=0.73924.71.0 Caltech g=22229.90.8 PCC g=34829.00.1 game p=0.94924.30.1 PCC g=25429.20.0 mean1.56 median1.30

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EWA learning Attraction A i j (t) for strategy j updated by A i j (t) =( A i j (t-1) + i [s i (t),s -i (t)]/ ( (1- )+1) (chosen j) A i j (t) =( A i j (t-1) + i [s i j,s -i (t)]/ ( (1- )+1) (unchosen j) logit response (softmax) P i j (t)=e^{A i j (t)}/[Σ k e^{A i k (t)}] key parameters: imagination (weight on foregone payoffs) decay (forgetting) or change-detection growth rate of attractions ( =0 averages; =1 cumulations; =1 “lock-in” after exploration) “In nature a hybrid [species] is usually sterile, but in science the opposite is often true”-- Francis Crick ’88 Weighted fictitious play ( =1, =0) Simple choice reinforcement ( =0)

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Studies comparing EWA and other learning models

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20 estimates of learning model parameters 20 estimates of learning model parameters

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Functional EWA learning (“EWA Lite”) Use functions of experience to create parameter values (only free parameter ) i (t) is a change detector: i (t) is a change detector: i (t)=1-.5[ k ( s -i k (t) - =1 t s s -i k ( )/t ) 2 ] i (t)=1-.5[ k ( s -i k (t) - =1 t s s -i k ( )/t ) 2 ] Compares average of past freq’s s -i (1), s -i (2)…with s -i (t) Decay old experience (low ) if change is detected =1 when other players always repeat strategies =1 when other players always repeat strategies falls after a “surprise” falls after a “surprise” falls more if others have been highly variable falls less if others have been consistent = /( of Nash strategies) (creates low in mixed games) = /( of Nash strategies) (creates low in mixed games)Questions: (now) Do functional values pick up differences across games? (Yes.) (later) Can function changes create sensible, rapid switching in stochastic games?

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Example: Price matching with loyalty rewards (Capra, Goeree, Gomez, Holt AER ‘99) Players 1, 2 pick prices [80,200] ¢ Price is P=min(P 1,,P 2 ) Low price firm earns P+R High price firm earns P-R What happens? (e.g., R=50)

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Teaching in repeated (partner) games Finitely-repeated trust game (Camerer & Weigelt Econometrica ‘88) borrower action repaydefault lenderloan40,60-100,150 no loan 10,10 1 borrower plays against 8 lenders A fraction (p(honest)) borrowers prefer to repay (controlled by experimenter)

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Empirical results (conditional frequencies of no loan and default)

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Teaching in repeated trust games (Camerer, Ho, Chong J Ec Theory 02) Some ( =89%) borrowers know lenders learn by fEWA Actions in t “teach” lenders what to expect in t+1 (=.93) is “peripheral vision” weight E.g. entering period 4 of sequence 17 Seq.period 16 1 2 3 4 5 6 7 8 Repay Repay Repay Default..... Repay Repay Repay Default..... look “peripherally” ( weight) look “peripherally” ( weight) 17 1 2 3 look back Repay No loan Repay Teaching: Strategies have reputations Bayesian-Nash equilibrium: Borrowers have reputations (types)

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Heart of the model: Attraction of sophisticated Borrower strategy j after sequence k before period t J t+1 is possible sequence of choices by borrower First term is expected (myopic) payoff from strategy j Second term is summation of expected payoffs in the future (undiscounted) given effect of j and optimal planned future choices (J t+1 )

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Empirical results (top) and teaching model (bottom)

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Conclusions Learning ( response sensitivity) Hybrid fits & predicts well (20+ games) One-parameter fEWA fits well, easy to estimate Well-suited to Markov games because Φ means players can “relearn” if new state is quite different? Teaching ( fraction of teaching) Retains strategic foresight in repeated games with partner matching Fits trust, entry deterrence better than softmax Bayesian-Nash (aka QRE) Next? Field applications, explore low-information Markov domains…

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Parametric EWA learning (E’metrica ‘99) free parameters , , , , N(0) Functional EWA learning functions for parameters parameter ( ) Strategic teaching (JEcTheory ‘02) Reputation-building w/o “types” Two parameters ( , ) Thinking steps (parameter )

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