PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 A Nash equilibrium of a dynamic game is subgame-perfect if the strategies of the Nash equilibrium constitute or induce a Nash equilibrium in every subgame of the game. Subgame-perfect Nash equilibrium is a Nash equilibrium. Subgame-perfect Nash equilibrium PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Games for Social Psych Game theory is two different enterprises: (1) Using games as a language or taxonomy to parse the social world; (language for theory construction) (2) deriving precise predictions about how players will play in a game by assuming that players maximize expected utility (personal valuation) of consequences, plan ahead, and form beliefs about other players likely actions. (This is one theory expressed in the language) Changing the assumptions at (2) allows for modeling what people actually do, using a precise theoretical language. PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Eliciting Social Preferences But they are MY preferences Don t care about the payoffs of others Preference Models I m a team player Self interested Not Self interested Altruism Cares about the payoffs of others Apparent interest in others really self interest disguised. Any payment is in expectation of a larger self benefit later. Equality Reciprocity Willing to pay Willing to pay to punish for other s violations of equality and benefit reciprocity, and pay to reward obediance to norms PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Games for eliciting social preferences PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 More Games PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Public Good Games (Tragedy) Public goods games: Every player is best off by contributing nothing to the public good, but contributions from everyone would make everyone better off. Example: n subjects per group, each with an endowment of $y. Each Each contributes $0-$y to a group project. Common payoff of $m per $1 in group project (share in the investment). In n addition, mn > 1=(thei G g group return for one i=1 more dollar > $1). A dollar saved is a dollar earned, so: Payoff for player i: pi = y gi + mG, PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 $g = i s investment, Ultimatum Observed offer: ~40%, relatively independent of stake size Predicted offer: smallest increment weak or unreplicated effects: gender, major (econ majors offer and accept less), physical attractiveness (women offers >50% to attractive men), age (young children accept lower offers), and autism (autistic adults offer very little; see Hill and Sally, 2002), sense of entitlement PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Ultimatum with competition Competing receivers- lower offers ~20% Competing proposers- higher offers ~75% Why? Altruism (a preference for sharing equally) Non self-interested Strategic fairness (a fear that low offers will be rejected) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 self-interested Dictator game: Dictator 1 $x accept ($y x,$x) ($y x,$x) Proposer division of $y between self and other player Self interested prediction Propose: $0 Students: ~10 25%, Kansas workers/Chaldeans ~50% same as in Ultimatum 2 reject PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Modeling Social Preferences Two model flavors have been proposed Inequality-aversion: players prefer more money and also prefer that allocations be more equal. Fehr and Schmidt (1999) xi = payoff of player i Ui(x) = xi - ai(xj - xi) if player i is worse off than player j (xj xi 0), and Ui(x) = xi - bi(xj - xi) if player i is better off than player j (xj xi 0). Envy: ai measures player i s dislike of disadvantageous inequality Guilt: bi measures player i s dislike of advantageous inequality Models of reciprocity. Rabin Utility model Ui($,q=personality) = Ui ($) + w Upi (qi)*Upk (qk) Upi ( niceness )>0, Upi ( meanness )<0, Thus, if the other player is nice (positive niceness) they want to be nice too, so the product of nicenesses will be positive. But if the other player is mean (negative niceness) they PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 S1 = S2 = {Mum, Confess} Sets of strategies: Utility functions are now on both players payoffs Players Strategies Prisoner 1 Mum Confess Prisoner 2 Mum Confess U1(-1,-1), U1(-9,0), U2(-9,0) U2(-1,-1) U1(0,-9), U2(0,-9) U1(-6,-6), U2(-6,-6) Modeling social preferences via utilities on opponent s outcomes Set of players: {Prisoner 1, Prisoner 2} Utilities PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Fairness seeking PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Fig.2. Activation related to Feeling: This is your brain on unfairness the presentation of an unfair (Sanfey et al, Sci 13 March 03)offer. (A) Map of the t statistic for the contrast [unfair human offer fair human offer] showing activation of bilateral anterior insula and anterior cingulate cortex. Areas in orange showed greater activation following unfair as compared with fair offers (P _ 0.001). (B) Map of the t statistic for the contrast [unfair human offer fair human offer] showing activation of right dorsolateral prefrontal cortex. (C) Event related plot for unfair and fair offers in right anterior insula. The offer was revealed at t _ 0 on the x axis. (D) Event related plot for unfair and fair offers in left anterior insula. (E) Event related plot for different human unfair and fair offers in subset of left anterior insula. PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Fig.1. (A) Time line for a single round of the Ultimatum Game, each lasting 36 s. Each round began with a 12 s preparation interval. The participant then saw the photograph and name of their partner in that trial for 6 seconds. A picture of a computer was shown if it was a computer trial, or a roulette wheel if it was a control trial. Next, participants saw the offer proposed by the partner for a further 6 s, after which they indicated whether they accepted or rejected the offer by pressing one of two buttons on a button box. (B) Behavioral results from the Ultimatum Game. These are the offer acceptance rates averaged over all trials. Each of 19 participants saw five $5:$5 offers, one $7:$3 offer, two $8:$2 offers, and two $9:$1 offers from both human and computer partners (20 offers in total). PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Ultimatum offer experimental sites PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 The Machiguenga independent families cash cropping slash & burn gathered foods fishing hunting PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 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 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Researcher: Mike Alvard Ultimatum offers across societies (mean shaded, mode is largest circle ) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Israeli subject (autistic?) complaining post-experiment (Zamir, 2000) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Behavioral game theory BGT: How people actually play games Key extensions over traditional Game Theory Framing: Mental representation Feeling: Social preferences (Fehr et al) Thinking: Cognitive hierarchy ( ) Learning: Hybrid fEWA (Experience-weighted attraction) adaptive rule ( ) Teaching: Bounded rationality in repeated games ( , ) BGT Notes based on notes from Colin F. Camerer, Caltech http://www.hss.caltech.edu/~camerer/camerer.html 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) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Thinking: A one-parameter cognitive hierarchy theory of one-shot games (Camerer, Ho, Chong) Model of constrained strategic thinking Model does several things: 1. 2. 3. 4. 5. 6. Limited equilibration in some games (e.g., pBC) Surprisingly fast equilibration in some games (e.g. entry) De facto purification in mixed games Limited belief in noncredible threats Has economic value Can prove theorems e.g. risk-dominance in 2x2 symmetric games 7. Permits individual diff s & relation to cognitive measures PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Principle QRE Strategic Thinking Different equilibrium notions Nash CH Best ResponseNash: Everyone s Mutual Consistency QRE: the same, ideal and make best self-interested response CH: Everyone s NOT the same, but makes best response given values QRE: Everyone s the same, but NOT best response given values quantal-response equilibrium. Players do not choose the best response with probability one (as in Nash equilibrium). Instead, they better-respond , choosing responses with higher expected payoffs with higher probability. CH: PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Camerer-Ho 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 gK(n)=f(n)/ h=0K 1 f(h). Calculate expected payoffs and best-respond (can t imagine what smarter people would do, but can for simpler) Exhibits increasingly rational expectations : Normalized gK(n) approximates f(n) more closely as K 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) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 The cognitive hierarchy (CH) model (II) Two separate features: Not imagining k+1 types Not believing there are many other k types Models 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 ) What if self-conscious?: Then K-steps believe there are other K-step thinkers Predictions Too similar to quantal response equilibrium/Nash (& fits worse) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 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 /k! mean, variance With additional assumptions, it is possible to pin down the parameter 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 ) *Amount of working memory (digit span) correlated with steps of iterated deletion of dominated strategies (Devetag & Warglien, 03 J Ec Psych) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Poisson distribution Discrete, one parameter ( spikes in data) Steps > 3 are rare (working memory bound) Steps can be linked to cognitive measures Poisson distributions for various =1 =1.5 =2 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Beauty contest game N players choose real numbers xi in [0,100] Compute target (2/3)*( xi /N) Closest to target wins $20 Nash Eq? Real? (2/3)n mean, n inf * Integers? PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 1. Limited equilibration in p-BC: Pick [0,100]; closest to (2/3)*(average) wins Beauty contest results (Expansion, Financial Times, Spektrum) average 23.07 0.20 0.15 0.10 0.05 relative 0.00 frequencies numbers 0 22 33 50 100 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Beauty contest results (Expansion, Financial Times, Spektrum) average 23.07 0.20 0.15 0.10 0.05 relative 0.00 frequencies numbers 0 22 33 50 100 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 predicted frequency 0 1 9 17 25 33 41 49 57 65 73 81 89 97 number choices PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 Estimates of in pBC games Table 1: Data and estimates of PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 pBC estimation: Gory details PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 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 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 0-Step and 1-Step Entry 100 90 80 70 60 50 40 Percentage Entry 30 20 10 0 1 11 21 21 31 31 41 41 51 51 61 61 71 71 81 81 91 91 101 101 Percentage Capacity Percentage Capacity Capacity Capacity 0-Level 0+1 Level 1-Level PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 0-Step + 1-Step1-Step Entry 0-Step and + 2 Step Entry 100 90 80 70 60 60 50 50 40 40 Percentage Entry Percentage Entry 30 30 20 20 10 10 0 0 0 0 10 10 10 20 20 20 30 30 30 40 40 50 50 60 70 40 50 6060 70 70 Percentage Capacity Percentage Capacity Percentage Capacity 8080 9090 100 100 100 80 90 ` `` Capacity Capacity Capacity 0+1 Level 0+1+2 Level 0+1 Level 2-Level PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 1 0.9 0.8 0.7 0.6 0.5 0.4 frequency 0.3 0.2 0.1 0 2 total entry Nash equilibrium CH fit (tau=1.5) 4 6 8 10 capacity (out of 12) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2004 The Mating Game Mate-for-life Game Female Male Commit Pass (search) Commit (Mm,Mf ) (Sm, Sf Rf ) Pass (search) (Sm Rm, Sf ) (Sm, Sf ) Mate Payoff related to mate quality M f = f q (qm ) qm = ...