15-382 C OLLECTIVE I NTELLIGENCE – S18 L ECTURE 32-34: G AME T HEORY 7-9 / E VOLUTIONARY G AME T HEORY 4-6 I NSTRUCTOR : G IANNI A. D I C ARO
G AMES AGAINST THE FIELD § In this population game model, each individual plays against an ‘ environment ’ and not against other individuals. The environment represents an average (isotropic) property that determines the payoffs of the individuals based on the strategy they play (i.e., that they are programmed for) 15781 Fall 2016: Lecture 22 2
G AMES AGAINST THE FIELD Payoffs are not linear functions of the population profile! § § In pairwise contest games they always are linear functions : 15781 Fall 2016: Lecture 22 𝜌 𝑡 $ ,𝑦 = 𝑦𝜌 𝑡 $ ,𝑡 ( + (1 − 𝑦) 𝜌 𝑡 $ ,𝑡 . 3
G AMES AGAINST THE FIELD 15781 Fall 2016: Lecture 22 4
G AMES AGAINST THE FIELD 15781 Fall 2016: Lecture 22 5
G AMES AGAINST THE FIELD 15781 Fall 2016: Lecture 22 6
ESS IN POLYMORPHIC POPULATIONS § In monomorphic populations: § What about ESS in polymorphic populations? § In a monomorphic population the expected reproduction rate is the same for all individuals § In polymorphic populations different phenotypes / strategies will reproduce differently (have different fitness) § à Replicator dynamics 15781 Fall 2016: Lecture 22 7
R EPLICATOR D YNAMICS 8
R EPLICATOR DYNAMICS 𝑂 individuals (phenotypes) programmed to use only pure strategies from a § finite set 𝑇 = {𝑡 ( ,𝑡 . ,…,𝑡 3 } 3 𝑜 $ = proportion of individuals using strategy 𝑡 $ , 𝑂 = ∑ 𝑜 $ § $7( 8 9 Proportion of individuals in the population using 𝑡 $ : 𝑦 $ = § : Population state: 𝒚 = (𝑦 (, 𝑦 . ,… ,𝑦 3 ) § 𝛾 = per capita birth rate of the population (independent from the game) § 𝜀 = per capita death rate of the population (independent from the game) § Rate of change of the # of individuals using pure strategy 𝑡 $ : § 𝑜̇ $ = (𝛾 − 𝜀 + 𝜌 𝑡 $ ,𝒚 )𝑜 $ § à Rate of change of the # of individuals in population: 15781 Fall 2016: Lecture 22 𝑂̇ = ∑ 3 𝑂̇ = 𝛾 − 𝜀 + 𝜌 3 ? 𝒚 = ∑ 𝑜̇ $ à ? 𝒚 𝑂 , 𝜌 𝑦 $ 𝜌 𝑡 $ ,𝒚 § $7( $7( 9
R EPLICATOR DYNAMICS 𝑂̇ = 𝐷𝑂 à 𝑂 can grow or decline exponentially § The behavior can be improved by letting 𝛾 and 𝜀 depending on 𝑂 … § Game-theoretic interesting question: how proportions 𝑜 $ change over time? § 8 9 : , by differentiating: 𝑜̇ $ = 𝑂𝑦̇ $ + 𝑦 $ 𝑂̇ à … From 𝑦 $ = § Replicator dynamics : 𝑦̇ $ = 𝜌 𝑡 $ ,𝒚 − 𝜌 ? 𝒚 𝑦 $ § The proportion of individuals with phenotype / strategy 𝑡 $ increases (decreases) if its payoff is bigger (smaller) than the average payoff in the population § We said that an ESS is an end-point in evolution A fixed point of the replicator dynamics is a population that satisfies § 𝑦̇ $ = 0, ∀𝑗 At a fixed point the population is not evol ving anymore 15781 Fall 2016: Lecture 22 10
E XAMPLE R EPLICATOR DYNAMICS Pairwise contest population with action set 𝐵 = {𝐹, 𝐺} and payoffs: § 𝜌 𝐹,𝐹 = 1, 𝜌 𝐹, 𝐺 = 1, 𝜌 𝐺, 𝐹 = 2, 𝜌 𝐺, 𝐺 = 0 à 𝜌 𝐹,𝒚 = 𝑦 ( + 𝑦 . 𝜌 𝐺,𝒚 = 2𝑦 ( § ? 𝒚 = 𝑦 ( 𝑦 ( + 𝑦 . + 𝑦 . 2𝑦 ( = 𝑦 (. + 3𝑦 ( 𝑦 . 𝜌 § § Replicator dynamics: 𝑦̇ ( = 𝑦 ( 𝑦 ( + 𝑦 . − 𝑦 (. − 3𝑦 ( 𝑦 . § 𝑦̇ . = 𝑦 . 2𝑦 ( − 𝑦 (. − 3𝑦 ( 𝑦 . § Fixed points § 𝑦 ( = 0,𝑦 . = 1 , 𝑦 ( = 1,𝑦 . = 1 , 𝑦 ( = 1/2,𝑦 . = 1/2 § ( 𝑦 ( = 0, 𝑦 . = 0) is also a fixed point but not an interesting one since § 15781 Fall 2016: Lecture 22 the population disappears 11
T WO - STRATEGY PAIRWISE CONTESTS Let’s consider the case of only two pure strategies 𝑇 = 𝑡 ( ,𝑡 . , population state § 𝒚 = 𝑦 ( ,𝑦 . , representing the fraction of individuals using strategies 𝑡 ( and 𝑡 . , and let’s use pairwise contest games for evolving population § The general payoff matrix for each pairwise game: Π = 𝜌 𝑡 ( ,𝑡 ( ,𝜌 𝑡 ( ,𝑡 ( 𝜌 𝑡 ( ,𝑡 . ,𝜌 𝑡 . ,𝑡 ( 𝜌 𝑡 . ,𝑡 ( ,𝜌 𝑡 ( ,𝑡 . 𝜌 𝑡 . ,𝑡 . ,𝜌(𝑡 . ,𝑡 . ) because of symmetry it can be simplified considering only row player: Π = 𝜌 𝑡 ( ,𝑡 ( 𝜌 𝑡 ( ,𝑡 . 𝜌 𝑡 . ,𝑡 ( 𝜌 𝑡 . ,𝑡 . Expected payoff for playing pure strategies 𝑡 ( and 𝑡 . in population 𝒚 = (𝑦 ( ,𝑦 . ) § 𝜌 𝑡 ( ,𝒚 = 𝑦 ( 𝜌 𝑡 ( ,𝑡 ( + 𝑦 . 𝜌 𝑡 ( ,𝑡 . , 𝜌 𝑡 . ,𝒚 = 𝑦 ( 𝜌 𝑡 . ,𝑡 ( + 𝑦 . 𝜌 𝑡 . ,𝑡 . Average payoff in the population : 𝜌 ? 𝒚 = 𝑦 ( 𝜌 𝑡 ( ,𝒚 + 𝑦 . 𝜌 𝑡 . ,𝒚 § Replicator dynamics: 𝑦̇ $ = 𝜌 𝑡 $ ,𝒚 − 𝜌 ? 𝒚 𝑦 $ § § We can simplify the notation and use only one variable: 15781 Fall 2016: Lecture 22 𝑦 = 𝑦 ( 𝑦 . = 1 − 𝑦 𝑦̇ . = −𝑦̇ ( 𝑦̇ = 𝑦(1 − 𝑦)(𝜌 𝑡 ( ,𝒚 − 𝜌 𝑡 . ,𝒚 ) à 12
T WO - STRATEGY PAIRWISE CONTEST § Let’s consider the case of only two pure strategies pairwise contest population with action set 𝑇 = 𝐵,𝐶 Let 𝑦 = 𝑦 L 𝑦 M = 1 − 𝑦 à 𝑦̇ M = −𝑦̇ L § Dynamics of population state : 𝑦̇ = 𝑦(1 − 𝑦)(𝜌 𝐵, 𝒚 − 𝜌 𝐶, 𝒚 ) § Fixed points: § Fitness of Fitness of (𝑦 = 0, 𝑦 = 1, 𝜌 𝐵,𝒚 = 𝜌 𝐶, 𝒚 strategy 𝐵 strategy 𝐶 If 𝑔 L 𝑦 > 𝑔 M 𝑦 → 𝑦̇ > 0 § à 𝐵 increases If 𝑔 L 𝑦 < 𝑔 M 𝑦 → 𝑦̇ < 0 § à 𝐵 decreases If 𝑔 L 𝑦 = 𝑔 M 𝑦 → 𝑦̇ = 0 § à Slope of 𝑔 L 𝑦 − 𝑔 M 𝑦 15781 Fall 2016: Lecture 22 defines the stability 13
T WO - STRATEGY PAIRWISE CONTEST § Let’s consider the pairwise contest Prisoner’s dilemma: 𝑇 = 𝐷,𝐸 , 𝜌 𝐷, 𝐷 = 3 𝜌 𝐷,𝐸 = 0 𝜌 𝐸,𝐷 = 5 𝜌 𝐸,𝐸 = 1 Let 𝑦 be the proportion of individuals using 𝐷 : § à 𝜌 𝐷, 𝒚 = 3𝑦 + 0 1 − 𝑦 = 3𝑦 𝜌 𝐸, 𝒚 = 5𝑦 + 1 1 − 𝑦 = 1 + 4𝑦 𝑦̇ = 𝑦 1 − 𝑦 𝜌 𝐷,𝒚 − 𝜌 𝐸, 𝒚 = 𝑦 1 − 𝑦 3𝑦 − (1 + 4𝑦) § = 𝑦 1 − 𝑦 (1 + 𝑦) à Fixed points: 𝑦 ∗ = 0 𝐸,𝐸 , 𝑦 ∗ = 1 (𝐷, 𝐷) § Nash equilibrium: (𝐸, 𝐸) § Since 𝑦̇ < 0 for all 𝑦 except the fixed points, § any population that is not at a fixed point evolves toward the Nash equilibrium, and away from the other equilibrium 𝑦 . Phase portrait with 𝑦 ( (playing C), and 𝑦 ( § (playing D) . The grayed area doesn’t correspond to any feasible strategy. Indeed only 15781 Fall 2016: Lecture 22 the points on the diagonal between (1,0) and 𝑦 ( 14 (0,1) represent feasible population profiles
NE AND F IXED POINTS Theorem: In a two-player games, every symmetric NE corresponds to a § fixed point in the replicator dynamics § The reverse is not necessarily true: a fixed point of the dynamics is not necessarily a NE § Can we identify a consistent relation between NE and fixed points? 𝑇 = 𝐵,𝐶 , 𝜌 𝐵, 𝐵 = 3 𝜌 𝐶,𝐶 = 1 𝜌 𝐵,𝐶 = 𝜌 𝐶, 𝐵 = 0 § ESS are 𝜏 = 1,0 , or = 0,1 everyone either plays 𝐵 or 𝐶 § ( X The mixed strategy 𝜏 = ( W , W ) that is a fixed point is not an ESS § If 𝑦 corresponds to the portion of the population playing 𝐵 : § 𝑦̇ = 𝑦 1 − 𝑦 𝜌 𝐵,𝒚 − 𝜌 𝐶,𝒚 = 𝑦 1 − 𝑦 3𝑦 − 1 − 𝑦)) § = 𝑦(1 − 𝑦)(4𝑦 − 1) § Fixed points are 𝑦 ∗ = 0, 𝑦 ∗ = 1, 𝑦 ∗ = 1/4 15781 Fall 2016: Lecture 22 15
NE AND F IXED POINTS Fixed points for the prisoners’ dilemma: 𝑦 ∗ = 0, 𝑦 ∗ = 1, 𝑦 ∗ = 1/4 § 𝑦̇ > 0 if 𝑦 > 1/4 , and 𝑦̇ < 0 if 𝑦 < 1/4 , that is the slope changes in 𝑦 ∗ = 1/4 , such § that only the pure strategies population behaviors are stable equilibria à Evolutionary end points If population starts in a state with more than 25% individuals playing 𝐵 , it evolves § until everyone uses 𝐵 , less than 25% à everyone will use 𝐶 In the phase portrait, 𝑦 ( is the population playing 𝐵, and 𝑦 . the fraction playing 𝐶 § Only the diagonal represents feasible population profiles 𝑦 ( + 𝑦 . = 1 § 15781 Fall 2016: Lecture 22 16
ESS AND EVOLUTIONARY END POINTS § Do all ESSs have a corresponding evolutionary end point and do all end points have a corresponding ESS? § A fixed point of the replicator dynamics is asymptotically stable if any small deviations from that state are eliminated by the dynamics as t → ∞ § Theorem : For any two-strategy pairwise contest, a strategy 𝜏 = (𝑦 ( ,𝑦 . ) is an ESS if and only if the corresponding fixed point in the replicator dynamics is asymptotically stable § Theorem: For any two-strategy pairwise game, the following relations hold for a strategy 𝜏 and the corresponding population state 𝒚 : § 𝜏 ∈ Set of ESSs in the symmetric game corresponding to replicator dynamics ⇔ 𝒚 ∈ Set of asymptotically stable fixed points in replicator dynamics § 𝒚 ∈ Set of asymptotically stable fixed points in the replicator dynamics ⟹ 𝜏 ∈ Set of symmetric Nash equilibrium strategies § 𝜏 ∈ Set of symmetric Nash equilibrium strategies ⟹ 𝒚 ∈ Set of fixed points 15781 Fall 2016: Lecture 22 17
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