Report

Behavioral game theory* Colin F. Camerer, Caltech [email protected] Behavioral game theory: – How people actually play games – Uses concepts from psychology and data – It is game theory: Has formal, replicable concepts Framing: Feeling: Thinking: Learning: Teaching: Mental representation Social preferences Cognitive hierarchy () Hybrid fEWA adaptive rule 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) BGT modelling aesthetics General Precise Progressive Cognitively detailed Empirically disciplined (game theory) (game theory) (behavioral econ) (behavioral econ) (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) Thinking: A one-parameter cognitive hierarchy theory of one-shot games* (with Teck Ho, Berkeley; Kuan Chong, NUSingapore) Model of constrained strategic thinking Model does several things: – – – – – – 1. Limited equilibration in some games (e.g., pBC) 2. Instant 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 – *Q J Econ August ‘04 Unbundling equilibrium Principle Nash Strategic Thinking Best Response Mutual Consistency CH QRE The cognitive hierarchy (CH) model (I) Selten (1998): – “The natural way of looking at game situations…is not based on circular concepts, but rather on a step-by-step reasoning procedure” 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 Choose best responses Exhibits “increasingly rational expectations”: – Normalized beliefs approximate f(n) as n ∞ i.e., highest level types are “sophisticated”/”worldly and earn the most Easy to calculate (see website “calculator” http://groups.haas.berkeley.edu/simulations/ch/default.asp) The cognitive hierarchy (CH) model (II) What is a reasonable simple f(K)? – A1*: f(k)/f(k-1) ∝1/k Poisson f(k)=e-k/k! mean, variance – A2: f(1) is modal 1< < 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) Poisson distribution Discrete, one parameter – ( “spikes” in data) Steps > 3 are rare (tight working memory bound) Steps can be linked to cognitive measures frequency Poisson distributions for various 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 =1 =1.5 =2 0 1 2 3 4 number of steps 5 6 1. Limited equilibration Beauty contest game N players choose numbers xi in [0,100] Compute target (2/3)*( xi /N) Closest to target wins $20 relative frequencies Beauty contest results (Expansion, Financial Times, Spektrum) average 23.07 0.20 0.15 0.10 0.05 0.00 numbers 22 100 50 33 num be r choice s 97 89 81 73 65 57 49 41 33 25 17 9 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 predicted frequency 0 Estimates of in pBC games Table 1: Data and estimates of in pbc games (equilibrium = 0) Mean Steps of subjects/game Data CH Model Thinking game theorists 19.1 19.1 3.7 Caltech 23.0 23.0 3.0 newspaper 23.0 23.0 3.0 portfolio mgrs 24.3 24.4 2.8 econ PhD class 27.4 27.5 2.3 Caltech g=3 21.5 21.5 1.8 high school 32.5 32.7 1.6 1/2 mean 26.7 26.5 1.5 70 yr olds 37.0 36.9 1.1 Germany 37.2 36.9 1.1 CEOs 37.9 37.7 1.0 game p=0.7 38.9 38.8 1.0 Caltech g=2 21.7 22.2 0.8 PCC g=3 47.5 47.5 0.1 game p=0.9 49.4 49.5 0.1 PCC g=2 54.2 49.5 0.0 mean 1.56 median 1.30 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 n(entrants) ≈ c in the 1st period: “To a psychologist, it looks like magic”-- D. Kahneman ’88 How? Pseudo-sequentiality of CH “later”-thinking entrants smooth the entry function frequency 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 total entry Nash equilibrium CH fit (tau=1.5) 2 4 6 8 capacity (out of 12) 10 0-Step and 1-Step Entry 100 90 Percentage Entry 80 70 60 Capacity Capacity 0-Level 0+1 Level 1-Level 50 40 30 20 10 0 1 11 21 21 31 31 41 41 51 51 61 61 Percentage Percentage Capacity Capacity 71 71 81 81 91 91 101 101 0-Step + 1-Step + 2 Step Entry 0-Step and 1-Step Entry 100 90 Percentage Entry Entry Percentage 80 70 60 60 50 50 ` `` 40 40 30 30 20 20 10 10 0 0 0 0 10 10 10 20 20 20 30 30 30 40 40 50 100 40 50 50 6060 60 7070 70 8080 80 9090 90 100 100 Percentage Percentage Capacity Capacity Percentage Capacity Capacity Capacity Capacity 0+1 Level 0+1+2 Level 0+1 Level 2-Level 3. Purification and partial equilibration in mixed-equilibrium games (=1.62) row step thinker choices T B 0 1 2 3 4 5 L 2,0 0,1 .5 .5 0 0 0 0 R 0,1 1,0 .5 .5 1 1 1 1 0 .5 .5 1 1 0 2 1 0 3 4... 0 0 1 1 3. Purification and partial equilibration in mixed-equilibrium games (=1.62) row step thinker choices L T 2,0 B 0,1 0 .5 1 .5 2 0 3 0 4 0 5 0 CH .26 mixed .33 data .33 R 0,1 1,0 .5 .5 1 1 1 1 .74 .67 .67 0 .5 .5 1 1 0 2 1 0 3 4... 0 0 1 1 CH mixed pred’n equilm .68 .50 .32 .50 data .72 .28 Estimates of τ game Matrix games Stahl, Wilson Cooper, Van Huyck Costa-Gomes et al Mixed-equil. games Entry games Signaling games specific τ common τ (0, 6.5) 1.86 (.5, 1.4) .80 (1, 2.3) 1.69 (.9,3.5) --(.3,1.2) 1.48 .70 --- Fits consistently better than Nash, QRE Unrestricted 6-parameter f(0),..f(6) fits only 1% better CH fixes errors in Nash predictions Figure 2: Mean Absolute Deviation for Matrix Games: Nash vs Cognitive Hierarchy (Common ) 1.00 0.90 MAD(Cognitive Hierarchy) 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.00 0.10 0.20 0.30 0.40 0.50 MAD(Nash) 0.60 0.70 0.80 0.90 1.00 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 (selection pressure in replicator dynamics) Table 7: Economic Value: Cross-dataset Estimation CH 580 1277 9% 9% QRE 542 1277 2% 9% Nash 513 1277 -3% 9% 573 8% 484 -9% 556 5% 460 40% 427 30% 355 8% 134 14% 98 -17% 121 2% 103 1% 111 9% 95 -7% 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 7. Preliminary findings on individual differences & response times Caltech is .53 higher than PCC Individual differences: – Estimated i (1st half) correlates .64 with i (2nd half) Upward drift in , .69 from 1st half to 2nd half of game (no-feedback “learning” ala Weber ExEc 03?) One step adds .85 secs to response time Thinking: Conclusions Discrete thinking steps (mean τ ≈ 1.5) Predicts one-shot games & initial conditions for learning Accounts for limited convergence in dominancesolvable 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 Feeling in ultimatum games: How much do you offer out of $10? Proposer has $10 Offers x to Responder (keeps $10-x) What should the Responder do? – Self-interest: Take any x>0 – Empirical: Reject x=$2 half the time What are the Responders thinking? – Look inside their brains… Feeling: This is your brain on unfairness (Sanfey et al, Sci 13 March ’03) Ultimatum offers of children who failed/passed false belief test Israeli subject (autistic?) complaining postexperiment (Zamir, 2000) Ultimatum offer experimental sites The Machiguenga independent families cash cropping slash & burn gathered foods fishing hunting African pastoralists (Orma in Kenya) Whale Hunters of Lamalera, Indonesia High levels of cooperation among hunters of whales, sharks, dolphins and rays. Protein for carbs, trade with inlanders. Carefully regulated division of whale meat Researcher: Mike Alvard Fair offers correlate with market integration (top), cooperativeness in everyday life (bottom) New frontiers Field applications! Imitation learning Trifurcation: – Rational gt: Firms, expert players, long-run outcomes – Behavioral gt: Normal people, new games – Evolutionary gt: Animals, humans imitating Conclusions 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 Feeling Ultimatum rejections are common, vary across culture fairness correlated with market integration (cf. Adam Smith) Unfair offers activate insula, ACC, DLPFC U-shaped rejections common Dictators offer less when threatened with 3rd-party punishment Pedagogy: A radical new way to teach game theory – Start with concept of a game. – Building blocks: Mixing, dominance, foresight. – Then teach cognitive hierarchy, learning… – end with equilibrium! 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 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” Framing enrichment: Timing & virtual observability Battle-of-sexes row 1st unobserved B G Simul. Seq’l Unobs. B 0,0 3,1 .62 .80 .70 G 1,3 0,0 .38 .20 .30 simul seq’l .38 .10 .62 .90 seq’l .20 .80 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 Potential economic applications (cont’d) A B I info (A,B) data .77 CH (=1.5) .46 C D (C,D) .53 .89 I payoffs +32 II info A data .00 CH (=1.5) .12 -28 +20 (B,C) .83 .72 -16 D 1.00 .89 II payoffs -32 +28 +16 -20 Potential economic applications (cont’d) Prediction: Betting in (C,D) and (B,C) drops when one number is changed A B I info (A,B) data ? CH (=1.5) .46 C D (C,D) ? .46 I payoffs +32 II info A data ? CH (=1.5) .12 -28 +32 -16 (B,C) D ? ? .12 .89 II payoffs -32 +28 -32 +16 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) Framing: Limited planning in bargaining (JEcThry ‘02; Science, ‘03) Learning: fEWA Attraction A j i A ij (t) =(A A ij (t) =(A j) (t-1) + (actual))/ ((1-)+1) (chosen j) (t-1) + (foregone))/ ((1- )+1) (unchosen j i j i (t) for strategy j updated by logit response function Pij(t)=exp(A j i (t)/[Σkexp(A k i (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 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 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) Ultimatum offers across societies (mean shaded, mode is largest circle…) 191~200 181~190 171~180 161~170 151~160 141~150 131~140 121~130 111~120 101~110 91~100 81~90 80 1 3 5 Period 7 9 Empirical Frequency 0.9 0.8 0.7 0.6 0.5 Prob 0.4 0.3 0.2 0.1 0 Strategy 191~200 181~190 171~180 161~170 151~160 141~150 131~140 121~130 111~120 101~110 191~200 181~190 171~180 161~170 151~160 141~150 131~140 9 0 91~100 81~90 80 1 3 5 Period 7 9 5 121~130 111~120 101~110 91~100 81~90 1 80 3 Period 7 Empirical Frequency 0.9 0.7 0.8 0.6 0.5 0.4 Prob 0.2 0.3 0.1 Thinking fEWA Strategy 0.9 0.8 0.7 0.6 0.5 0.4 Prob 0.3 0.2 0.1 0 Strategy 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) 5. Automatic reduction of belief in noncredible threats (subgame perfection) T B 4,4 L 6,3 R 0,1 0 .5 row level 1 2 3+ 1 0 0 .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)