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Behavioral game theory* Colin F. Camerer, Caltech

Behavioral game theory* Colin F. Camerer, Caltech. BGT: How people actually play games Framing: Mental representation Feeling : Social preferences (Fehr et al) Thinking : Cognitive hierarchy (  ) Learning : Hybrid fEWA adaptive rule (  )

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Behavioral game theory* Colin F. Camerer, Caltech

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  1. Behavioral game theory*Colin F. Camerer, Caltech • BGT: How people actually play games • Framing: Mental representation • Feeling: Social preferences (Fehr et al) • Thinking: Cognitive hierarchy () • Learning: Hybrid fEWA adaptive rule () • Teaching: Bounded rationality in repeated games (, ) *Behavioral Game Theory, Princeton Press 03 (550 pp); Trends in Cog Sci, May 03 (10 pp); AmerEcRev, May 03 (5 pp); Science, 13 June 03 (2 pp) CEEL Summer Camp, Trento 07/07/03

  2. Framing:Limited planning in bargaining (Science, 03) CEEL Summer Camp, Trento 07/07/03

  3. Thinking: A one-parameter cognitive hierarchy theory of one-shot games (with Teck Ho, Berkeley; Kuan Chong, N Univ Singapore) • Model of constrained strategic thinking • Model does several things: • 1. Limited equilibration in some games (e.g., pBC) • 2. Surprisingly fast equilibration in some games (e.g. entry) • 3. De facto purification in mixed games • 4. Limited belief in noncredible threats • 5. Has economic value • 6. Can prove theorems • e.g. risk-dominance in 2x2 symmetric games • 7. Permits individual diff’s & relation to cognitive measures CEEL Summer Camp, Trento 07/07/03

  4. Unbundling equilibrium PrincipleNashCHQRE Strategic Thinking   Best Response  Mutual Consistency  CEEL Summer Camp, Trento 07/07/03

  5. The cognitive hierarchy (CH) model (I) • Discrete steps of thinking: Step 0’s choose randomly (nonstrategically) K-step thinkers know proportions f(0),...f(K-1) Calculate what 0, …K-1 step players will do Normalize beliefs g(n)=f(n)/ h=0K-1 f(h). Calculate expected payoffs and best-respond • Exhibits “increasingly rational expectations”: • Normalized g(n) approximates f(n) more closely as n ∞ • i.e., highest level types are “sophisticated”/”worldly and earn the most • Also: highest level type actions converge as K ∞ • ( marginal benefit of thinking harder 0) CEEL Summer Camp, Trento 07/07/03

  6. The cognitive hierarchy (CH) model (II) • Two separate features: • Not imagining k+1 types • Not believing there are other k types Overconfidence: K-steps think others are all one step lower (K-1) (Nagel-Stahl-CCGB) “Increasingly irrational expectations” as K ∞ Has some odd properties (cycles in entry games…) Self-conscious: K-steps believe there are other K-step thinkers “Too similar” to quantal response equilibrium/Nash (& fits worse) CEEL Summer Camp, Trento 07/07/03

  7. The cognitive hierarchy (CH) model (III) • What is a reasonable simple f(K)? • A1*: f(k)/f(k-1) ∝1/k  Poisson f(k)=e-ttk/k! mean, variance t • A2: f(1) is modal  1< t < 2 • A3: f(1) is a ‘maximal’ mode or f(0)=f(2)  t=2=1.414.. • A4: f(0)+f(1)=2f(2)  t=1.618 (golden ratio Φ) *Amount of working memory (digit span) correlated with steps of iterated deletion of dominated strategies (Devetag & Warglien, 03 J Ec Psych) CEEL Summer Camp, Trento 07/07/03

  8. Poisson distribution • Discrete, one parameter • ( “spikes” in data) • Steps > 3 are rare (working memory bound) • Steps can be linked to cognitive measures CEEL Summer Camp, Trento 07/07/03

  9. Limited equilibrationBeauty contest game • N players choose numbers xi in [0,100] • Compute target (2/3)*( xi /N) • Closest to target wins $20 CEEL Summer Camp, Trento 07/07/03

  10. 1. Limited equilibration in p-BC:Pick [0,100]; closest to (2/3)*(average) wins CEEL Summer Camp, Trento 07/07/03

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  12. Estimates of  in pBC games CEEL Summer Camp, Trento 07/07/03

  13. pBC estimation: Gory details CEEL Summer Camp, Trento 07/07/03

  14. 2. Approximate equilibration in entry games • Entry games: N entrants, capacity c Entrants earn $1 if n(entrants)<c; earn 0 if n(entrants)>c Earn $.50 by staying out All choose simultaneously • Close to equilibrium in the 1st period: Close to equilibrium prediction n(entrants) ≈ c “To a psychologist, it looks like magic”-- D. Kahneman ’88 • How? Pseudo-sequentiality of CH  “later” entrants smooth the entry function CEEL Summer Camp, Trento 07/07/03

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  17. 3. De facto purification in mixed-equilibrium games (t=1.62) row step thinker choices CH mixed L R 0 1 2 3 4... pred’n equilm data T 2,0 0,1 .5 1 1 0 0 .68 .50 .72 B 0,1 1,0 .5 0 0 1 1 .32 .50 .28 0 .5 .5 1 .5 .5 2 0 1 3 0 1 4 0 1 5 0 1 CH .26 .74 mixed .33 .67 data .33 .67 CEEL Summer Camp, Trento 07/07/03

  18. “typical” example 1: (median in LL(CH)/LL(Nash) CEEL Summer Camp, Trento 07/07/03

  19. “Typical” example 2 CEEL Summer Camp, Trento 07/07/03

  20. 5. Automatic reduction of belief in noncredible threats (subgame perfection) row level 0 1 2 3+ T 4,4 .5 1 0 0 L R B 6,3 0,1 .5 0 1 1 (T,R) Nash, (B,L) subgame perfect CH Prediction: (=1.5) 89% play L 56% play B  (Level 1) players do not have enough faith in rationality of others (Beard & Beil, 90 Mgt Sci; Weiszacker ’03 GEB) CEEL Summer Camp, Trento 07/07/03

  21. Estimates of τ game • Matrix gamesspecificτ common τ Stahl, Wilson (0, 6.5) 1.86 Cooper, Van Huyck (.5, 1.4) .80 Costa-Gomes et al (1, 2.3) 1.69 • Mixed-equil. games (.9,3.5) 1.48 • Entry games --- .70 • Signaling games (.3,1.2) --- Fits consistently better than Nash, QRE Unrestricted 6-parameter f(0),..f(6) fits only 1% better CEEL Summer Camp, Trento 07/07/03

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  23. 4. Economic Value • Treat models like consultants • If players were to hire Mr. Nash and Mr. Camhocho as consultants and listen to their advice, would they have made a higher payoff? • If players are in equilibrium, Nash advice will have zero value • if theories have economic value, players are not in equilibrium • Advised strategy is what highest-level players choose •  economic value is the payoff advantage of thinking harder CEEL Summer Camp, Trento 07/07/03

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  25. 6. Other theoretical properties of CH model • Advantages over Nash equilibrium • No multiplicity problem (picks one distribution) • No “weird” beliefs in games of incomplete info. • Theory: • τ∞ converges to Nash equilibrium in (weakly) dominance solvable games • Coincides with “risk dominant” equilibrium in symmetric 2x2 games • “Close” to Nash in 2x2 mixed games (τ=2.7  82% same-quadrant correspondence) • Equal splits in Nash demand games • Group size effects in stag hunt, beauty contest, centipede games CEEL Summer Camp, Trento 07/07/03

  26. 7. Preliminary findings on individual differences & response times • Caltech  is .53 higher than PCC • Persistent individual differences: • Estimated i (1st half) correlates .64 with i (2nd half) • Upward drift in estimated , .69 from 1st 11 games to 2nd 11 games (no-feedback “learning”?) • One step adds .85 secs to response time CEEL Summer Camp, Trento 07/07/03

  27. BGT modelling aesthetics • General (game theory) • Precise (game theory) • Progressive (behavioral econ) • Cognitively detailed (behavioral econ) • 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) CEEL Summer Camp, Trento 07/07/03

  28. Thinking: Conclusions • Discrete thinking steps, frequency Poisson distributed (mean number of stepsτ≈ 1.5) • Predicts one-shot games & initial conditions for learning • Accounts for limited convergence in dominance-solvable games and approximate convergence in mixed & entry games • Advantages: More precise than Nash: Can “solve” multiplicity problem Has economic value Can be tied to cognitive measures • Important! This is game theory It is a formal specification which makes predictions It is not ad hoc… …quite the opposite Definition of “ad hoc”: “temporary, for the purpose at hand, as in an ad hoc committee” CEEL Summer Camp, Trento 07/07/03

  29. Potential economic applications • Price bubbles • thinking steps correspond to timing of selling before a crash • Speculation • Violates “Groucho Marx” no-bet theorem* A B C D I info (A,B) (C,D) I payoffs +32 -28 +20 -16 II info A (B,C) D II payoffs -32 +28 -20 +16 *Milgrom-Stokey ’82 Ec’a; Sonsino, Erev, Gilat, unpub’d; Sovik, unpub’d CEEL Summer Camp, Trento 07/07/03

  30. Potential economic applications (cont’d) A B C D I info (A,B) (C,D) data .77 .53 CH (=1.5) .46 .89 I payoffs +32 -28 +20 -16 II info A (B,C) D data .00 .83 1.00 CH (=1.5) .12 .72 .89 II payoffs -32 +28 -20 +16 CEEL Summer Camp, Trento 07/07/03

  31. Potential economic applications (cont’d) Prediction: Betting in (C,D) and (B,C) drops when one number is changed A B C D I info (A,B) (C,D) data ? ? CH (=1.5) .46 .46 I payoffs +32 -28 +32 -16 II info A (B,C) D data ? ? ? CH (=1.5) .12 .12 .89 II payoffs -32 +28 -32 +16 CEEL Summer Camp, Trento 07/07/03

  32. 3. De facto purification in mixed games: Battle of the sexes (t=1.62) row step thinker choices CH mixed L R 0 1 2 3 4... pred’n equilm data T 0,0 1,3 .5 0 1 0 0 .36 .25 B 3,1 0,0 .5 1 0 1 1 .64 .75 0 .5 .5 1 0 1 2 1 0 3 0 1 4 0 1 5 0 1 CH .36 .64 mixed .25 .75 data CEEL Summer Camp, Trento 07/07/03

  33. Learning: fEWA • Attraction A ij (t) for strategy j updated by A ij (t) =(A ij (t-1) + (actual))/ ((1-)+1) (chosen j) A ij (t) =(A ij (t-1) +   (foregone))/ ((1-  )+1) (unchosen j) logit response function Pij(t)=exp(A ij (t)/[Σkexp(A ik (t)]* • key parameters:  imagination,  decay/change-detection • “In nature a hybrid [species] is usually sterile, but in science the opposite is often true”-- Francis Crick ’88 • Special cases: • Weighted fictitious play (=1, =0) • Choice reinforcement (=0) • EWA estimates parameters , ,  (Cam.-Ho ’99 Ec’a) *Or divide by payoff variability (Erev et al ’99 JEBO); automatically “explores” when environment changes CEEL Summer Camp, Trento 07/07/03

  34. Functional fEWA • Substitute functions for parameters Easy to estimate (only ) Tracks parameter differences across games Allows change within a game • “Change detector” for decay rate φ φ(i,t)=1-.5[k ( S-ik (t) - =1t S-ik()/t ) 2 ] φ close to 1 when stable, dips to 0 when unstable • (i,t)=φ(i,t)/W (W=support of Nash equil’m) CEEL Summer Camp, Trento 07/07/03

  35. Example: Price matching with loyalty rewards (Capra, Goeree, Gomez, Holt AER ‘99) • Players 1, 2 pick prices [80,200] ¢ Price is P=min(P1,,P2) Low price firm earns P+R High price firm earns P-R • What happens? (e.g., R=50) CEEL Summer Camp, Trento 07/07/03

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  38. A decade of empirical studies of learning: Taking stock • Early studies show models can track basic features of learning paths • McAllister, ’91 Annals OR; Cheung-Friedman ’94 GEB; Roth-Erev ’95 GEB,’98 AER • Is one model generally better?: “Horse races” • Speeds up process of single-model exploration • Fair tests: Common games & empirical methods “match races” in horse racing: Champions forced to compete • Development of hybrids which are robust (improve on failures of specific models) • EWA (Camerer-Ho ’99, Anderson-Camerer ’00 Ec Thy) • fEWA (Camerer-Ho, ’0?) • Rule learning (Stahl, ’01 GEB) CEEL Summer Camp, Trento 07/07/03

  39. New frontiers • Field applications! • Imitation learning • Experimetrics (Wilcox, ’03 big downward bias in  due to heterogeneity) • No-feedback “learning” • Trifurcation: • Rational gt: Firms, expert players, long-run outcomes • Behavioral gt: Normal people, new games • Evolutionary gt: Animals, humans imitating CEEL Summer Camp, Trento 07/07/03

  40. Conclusions • Framing: Much to do…need help from cognitive science • Thinking CH model ( mean number of steps)  is similar (≈1.5) in many games: Explains limited and surprising equilibration Easy to use empirically & do theory challenge: endogenize number of steps (a la Gabaix-Laibson) • Learning ( response sensitivity) Hybrid fits & predicts well (20+ games) One-parameter fEWA fits well, easy to estimate • Teaching ( fraction of teaching) Retains strategic foresight in repeated games with matching Fits trust, entry deterrence better than QRE • Pedagogy: A radical new way teach game theory • Start with concept of a game. Mixing, dominance • Then teach cognitive hierarchy, learning… • end with equilibrium! CEEL Summer Camp, Trento 07/07/03

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  42. Behavior in “continental divide game” CEEL Summer Camp, Trento 07/07/03

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  45. Simulations vs data: fEWA CEEL Summer Camp, Trento 07/07/03

  46. Potential applications • Thinking • price bubbles, speculation, competition neglect • Learning • evolution of institutions, new industries • Neo-Keynesian macroeconomic coordination • bidding, consumer choice • Teaching • contracting, collusion, inflation policy CEEL Summer Camp, Trento 07/07/03

  47. Framing: How are games represented? • Invisible assumption: • People represent games in matrix/tree form • Mental representations may be simplified… • analogies: `Iraq war is Afghanistan, not Vietnam’ • shrinking-pie bargaining • …or enriched • Schelling matching games • timing & “virtual observability” CEEL Summer Camp, Trento 07/07/03

  48. Framing enrichment: Timing & virtual observability • Battle-of-sexes row 1stunobserved B G simul seq’l seq’l B 0,0 1,3 .38 .10 .20 G 3,1 0,0 .62 .90 .80 Simul. .62 .38 Seq’l .80 .20 Unobs. .70 .30 CEEL Summer Camp, Trento 07/07/03

  49. Equilibrium and CH vs data in matrix games CEEL Summer Camp, Trento 07/07/03

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