Evolutionary Process. Evolutionary Game Theory. Relationship to Game Theory. How Game Theory Differs from Evolutionary Biology

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1 Game Theor and the Social Sciences Evolutionar Game Theor Mong-Hun Chang Evolutionar Process Genotpe: Combination of genes Behavioral Phenotpe Fitness: A quantitative measure of the success of a phenotpe What do we mean b success? Reproductive success: Success with which an animal passes on its genes to the net generation and perpetuate its phenotpe The fitness depends on the relationship of the individual organism to its environment (including its interactions with other members of its species). 2 Selection: The fitter phenotpes become relativel more numerous in the net generation than the less fit phenotpes. Selection Dnamics Stable State Mutation (driven b chance) Most mutations are deleterious. A rare mutation that leads to a new and fitter phenotpe will successfull invade a population through rapid diffusion. A configuration of a population and its current phenotpes are said to be evolutionaril stable if the population cannot be invaded successfull b an mutant. 3 Relationship to Game Theor The behavior of a phenotpe A strateg of the animal Not chosen through a purposive calculation, but geneticall pre-determined. Fied course of action (embedded strateg) for an individual and a population of individuals with diverse set of embedded strategies. Reproductive fitness Individual paoffs Games plaed through pair-b-pair matches from a population containing a mi of phenotpes. 4 How Game Theor Differs from Evolutionar Biolog Communication and Information Transmission b the plaers Observation and Imitation Some purposive thinking and reinforcement learning (over the rules of thumb) Conscious eperimentation with new strategies rather than the accidental mutation in biological games. Evolutionar Stable Configurations Monomorphism A single phenotpe (strateg) proves fitter than others and the population comes to consist of it alone = evolutionar stable strateg (ESS) Polmorphism Two or more phenotpes ma be equall fit and ma coeist in some proportions Similar to the game-theoretic notion of a mied strateg Difference: each individual pursues a pure strateg but the population contains a miture of individuals with different pure strategies. 5 6

2 Prisoners Dilemma Plaed b a Population The Basic Prisoners Dilemma A population consists of two phenotpes. proportion of cooperators: Alwas cooperate regardless of the partner 2. (-) proportion of defectors: Alwas defect regardless of the partner Plaer 2 Plaer Defect Cooperate Defect Cooperate 2, 2 8, 0 0, 8 4, 4 8 Epected paoff for a tpical cooperator 4 + 0(-) = 4 Epected paoff for a tpical defector 8 + 2(-) = > 4 A defector is fitter than cooperator. An increase in the proportion of defectors (a decrease in ) over time Long run configuration: The population will consist entirel of defectors (monomorphic configuration) What if = 0 in the beginning? Can a mutant cooperator survive? No! A defector population cannot be invaded successfull b mutant cooperators. Defect is the evolutionar stable strateg. Theorem: If a game has a strictl dominant strateg, that strateg will also be the ESS. 9 0 Repeated Prisoners Dilemma Each individual plaer plas just one strateg. The strateg must be a complete plan of action. Onl two tpes of strategies eist: Alwas Defect (A) Tit-for-Tat (T) Pairs randoml selected from the population Each selected pair plas the game a specified number of times, n. When two A tpes meet Both alwas defect. Each earns 2n over the n plas. When two T tpes meet Both alwas cooperate. Each earns 4n over the n plas. When an A tpe meets a T tpe First encounter A defects and T cooperates Remaining (n-) encounters Both defect each time A tpe earns: 8 + 2(n-) = 6 + 2n T tpe earns: 0 + 2(n-) = n 2

3 Suppose there are proportion of T tpes and (-) proportion of A tpes. Average paoff for the A tpe (6+2n) + (-)(2n) = 2n + 6 Average paoff for the T tpe (4n) + (-)(-2+2n) = (2n-2) + (2n+2) Compare the fitness T tpe is fitter if (2n-2) + (2n+2) > 2n + 6 => (n-2) > T tpe is fitter if and onl if: (2n-2) + (2n+2) > 2n + 6 [Inequalit R] => (n-2) > For n = or 2, A tpe is alwas fitter: monomorphism with the eventual population containing entirel of A tpe. For n > 2 T tpe is fitter if and onl if > /(n-2): Monomorphism with all T. A tpe is fitter, if < /(n-2): Monomorphism with all A. 3 4 What if = /(n-2)? Both tpes are equall fit. Polmorphism? This polmorphic configuration is unstable. It cannot be sustained in the face of a mutant of either tpe. The arrival of a mutant will tip the balance in favor of the mutant tpe. n = 4 (2n-2) + (2n+2) 2n Etensions to Other Games Game of Chicken Assurance Game Battle of Sees Game Hawk-Dove Game /2 5 6 Battle of the Sees Interactions Across Species Battle-of-the-Sees Game Need to agree on the dating place Two equilibria: Plaers have heterogeneous preferences over the two equilibria - different from the Assurance Game The plaers belong to different species. Hard-liner MEN Compromiser Hard-liner WOMEN Compromiser 0, 0 2,, 2 0, 0 8

4 : proportion of hard-liners among men : proportion of hard-liners among women Epected paoff for a hard-liner man: *0 + (-)*2 = 2(-) Epected paoff for a compromiser man: * + (-)*0 = Epected paoff for a hard-liner woman: *0 + (-)*2 = 2(-) Epected paoff for a compromiser woman: * + (-)*0 = Among men, hardliners are fitter than compromisers when 2(-) > => < 2/3. Among women, hardliners are fitter than compromisers when 2(-) > => < 2/3. The fitness of each tpe within a given species depends on the proportion of tpes found in other species. Hardliners of each species do better when the other species does not have too man hardliners of its own. The get to meet compromisers more often! /3 Onl two ESS (, ) = (0, ) (, ) = (, 0) (, ) = (2/3, 2/3) is unstable. 0 2/ The Hawk-Dove Game The Basic Game Game plaed b two animals of the same species Hawk strateg: aggressive in fighting to get the resource of value V Dove strateg: offer to share but shirk from a fight Hawk-Hawk contest: Each is equall likel to win and get V or to lose, be injured, and get C. Dove-Dove contest: The share without a fight, each getting V/2. Hawk-Dove contest: Dove retreats with 0, while Hawk gets V. 24

5 Paoff Table Hawk A Dove B Hawk Dove (V-C)/2, (V-C)/2 V, 0 0, V V/2, V/2 Rational Choice and Equilibrium V > C Prisoners Dilemma Hawk is the dominant strateg. V < C Game of Chicken Two pure strateg Nash equilibria (Hawk, Dove) and (Dove, Hawk) Mied-strateg equilibrium p = V/C, where p is B s probabilit of choosing Hawk Evolutionar Stabilit V > C Consider an initial population which is predominantl of Hawks Can it be invaded b mutant Doves? Let d be the population proportion of the mutant Doves. A Hawk s fitness = d*v + (-d)*(v-c)/2 A mutant Dove s fitness = d*(v/2) + (-d)*0 Given V > C, d*v + (-d)*(v-c)/2 > d*(v/2) + (-d)*0. The Hawk tpe is fitter: The Hawk strateg is evolutionar stable (population is monomorphic with all Hawk). What if the initial population were all Doves? Hawks can invade and take over. 27 V < C Can the population be monomorphic? Can we have a population composed of all Hawk or all Dove?. Suppose the initial population is predominantl Hawks Let d be the population proportion of the mutant Doves. A Hawk s fitness = d*v + (-d)*(v-c)/2 A mutant Dove s fitness = d*(v/2) + (-d)*0 Given ver small d (and V < C), d*v + (-d)*(v-c)/2 < d*(v/2) + (-d)*0. Dove mutants can successfull invade the population that is composed predominantl of Hawks. Can the dominate the population? Suppose now the initial population is predominantl Doves Let h be the population proportion of the mutant Hawks. A Hawk mutant s fitness = h*[(v-c)/2]+ (-h)*v A Dove s fitness = h*0 + (-h)*(v/2) Given ver small h (and V < C), h*[(v-c)/2]+ (-h)*v > h*0 + (-h)*(v/2). Hawk mutants can successfull invade the population that is composed predominantl of Doves. The population can not be monomorphic! What are the alternative possibilities? Polmorphic Equilibrium: The population has a stable mi of plaers following different strategies. Ever plaer ma use mied strateg. Stable Polmorphic Population Suppose the population proportion of Hawks is h. The fitness of a Hawk is h*[(v-c)/2]+ (-h)*v. The fitness of a Dove is h*0 + (-h)*(v/2). Hawk tpe is fitter if and onl if: h*[(v-c)/2]+ (-h)*v > h*0 + (-h)*(v/2) => h < V/C Dove tpe is fitter if and onl if h > V/C or (-h) < V/C => (-h) < [(C-V)/C] Each tpe is fitter when it is rarer. A stable polmorphic equilibrium is at the balancing point: h = V/C

6 What if each plaer uses a mied strateg? Introduce an additional tpe into the population who uses mied strategies. Three tpes in the population H tpe: Pure Hawk D tpe: Pure Dove M tpe: Hawk with probabilit p = V/C and Dove with probabilit -p = V/C = (C -V)/C Now, if each tpe confronts the M tpe, it needs to compute the epected paoff based on p. Epected paoff for a H tpe meeting an M tpe: p[(v-c)/2] + (-p)v = (V/C)[(V-C)/2] + [(C-V)/C]V =V[(C-V)/2C] Epected paoff for a D tpe meeting an M tpe: p*0 + (-p)[v/2] = [(C-V)/C][V/2] = [V(C-V)/2C] Epected paoff for an M tpe meeting an M tpe is the same. Let K = V(C-V)/2C 3 32 Test the stabilit of M tpe against H invasion Population predominantl M tpe with a few H tpe mutants (h proportion of the total population, where h is ver small) Mutant H s epected paoff = h[(v-c)/2] + (-h)k M tpe s epected paoff = h[p{(v-c)/2} + (-p)*0] + (-h)k = hp[(v-c)/2] + (-h)k Given V < C, M tpe will alwas do better than H tpe. Hawks can not invade the M population Test the stabilit of M tpe against D invation Same as above M is an ESS! When V < C, there are two evolutionaril stable outcomes. A stable polmorphism: A population with a miture of tpes. A monomorphism with a single tpe that mies the strategies in the same proportions that define the polmorphism General Theor Available Tpes (Strategies): {A, B} Let E(i, j) be the paoff to an i-tpe plaer in a single encounter with a j-tpe plaer, where i, j is an element in {A, B}. Let W(i) be the fitness of an i-tpe (epected paoff from meeting randoml picked opponents, given a distribution of tpes in the population). Test for the monomorphism of the population with A-tpe. Suppose the population consists entirel of A- tpe. Let there be m proportion of mutant B-tpes. Fitness of an A-tpe is: W(A) = m*e(a, B) + (-m)*e(a, A) Fitness of a mutant B-tpe is: W(B) = m*e(b, B) + (-m)*e(b, A) 35 36

7 A-tpe is evolutionaril stable if W(A) W(B) > 0 => m[e(a, B) E(B, B)] + (-m)[e(a, A) E(B, A)] > 0. If m is sufficientl small (m -> 0), then W(A) > W(B) iff E(A, A) > E(B, A) --- primar criterion What if E(A, A) = E(B, A)? For a given m, W(A) > W(B) iff E(A, B) > E(B, B) --- secondar criterion Evolutionar Game with 3-Tpes Equilibrium in the rational choice game? Column Rock Scissor Paper Rock 0, 0, - -, Row Scissor -, 0, 0, - Paper, - -, 0, Let q r be the proportion of Rock-tpe. Let q s be the proportion of Scissor-tpe. - q r -q s : proportion of Paper-tpe. A Rock-tpe s epected paoff = q r *0 + q s + (- q r -q s )(-) =2 q s + q r > 0? => 2 q s + q r >? A Scissor-tpe s epected paoff = q r (-) + q s *0 + (- q r -q s ) = - 2q r -q s > 0? => 2q r +q s <? A Paper-tpe s epected paoff = q r + q s *(-) + (- q r -q s )* 0 = q r -q s q s /2 /3 Population Dnamics q s +2q r = Polmorphic Equilibrium q r +2q s = q r /3 / Compare to the rational mied-strateg equilibrium => (/3, /3, /3) 4