Evolutionary Game Theory PDF

Game theory and Evolutionary Stable Strategies 1 After this lecture you should understand: game theory terminology; how to construct and interpret a pay-off matrix; different reproductive strategies; evolution of 50:50 sex ratio (and exceptions). 2 John Maynard Smith 1920-2004 Evolution and th...

Game theory and Evolutionary Stable Strategies 1 After this lecture you should understand: game theory terminology; how to construct and interpret a pay-off matrix; different reproductive strategies; evolution of 50:50 sex ratio (and exceptions). 2 John Maynard Smith 1920-2004 Evolution and the Theory of Games (1982) 3 game theory: developed to analyse economic behaviour strategy: plan of how to behave in different situations (no teleology) tactic: the behaviour used in a specific situation (exercising strategy in particular case) ESS: a strategy such that, if all members of a population adopt it, then no mutant could invade… (by) natural selection frequency dependence: central to idea of ESS fitness (“pay-off”) depends on own phenotype and on others, and their frequency 4 individuals behave selfishly 5 the Hawk – Dove game contest for resource value ‘V’ display, escalate or retreat Hawk: fight to win, retreat only if injured Dove: display only, retreat if opponent escalates 6 the Hawk – Dove game Pay off matrix playing against: H Pay off to: H D ½(V-C) V D 0 V/2 E(H,D) V=value of winning C=cost of losing E(D,D) if V=50 and C=100 H D H -25 50 D 0 25 7 the Hawk – Dove game H D H ½(V-C) V D 0 V/2 ESS analysis D is not ESS, since E(D,D)<E(H,D) H is ESS if ½(V-C)>0 or V>C if V<C, neither H nor D are ESS results in ‘mixed’ strategy fitness H = fitness D p.½(V-C) + (1-p).V = p.0 + (1-p).V/2 p=P(H) p = V/C ESS: a strategy such that, if all members of a population adopt it, then no mutant could invade… (by) natural selection 8 the Hawk – Dove game Tells us Aggression has costs and benefits (trade-off!) Selfish evolution may not result in pure aggression (Group selection is not necessary to explain restraint) FOR STABLE POLYMORPHISM STRATEGIES MUST HAVE EQUAL FITNESS 9 Reproductive strategies Alternative strategies genetic polymorphism equal fitnesses uncommon Mixed strategies probabilistic equal fitnesses rare Conditional strategies best-of-a-bad-job unequal fitnesses Gross 1996 TREE 11: 92-98 common 10 Reproductive strategies Gross 1996 TREE 11: 92-98 11 Reproductive strategies Paper-Scissors-Stone = Rock-Paper-Scissors R P S R -e -1 1 P 1 -e -1 S -1 1 -e e=(small) cost of a draw ESS: mixed strategy I = 1/3R + 1/3P + 1/3S There is no pure ESS. Polymorphism is unstable 12 Alternative reproductive strategies Orange aggressive territorial Blue less aggressive small territories Yellow sneakers non-territorial Side blotched lizard Uta stansbursiana Sinervo & Lively 1996 Nature 380: 240-243 13 Reproductive strategies There is no Evolutionary Stable State (ESSt) Sinervo & Lively 1996 Nature 380: 240-243 14 Reproductive strategies Sinervo & Lively 1996 Nature 380: 240-243 15 Reproductive strategies Alternative strategies genetic polymorphism equal fitnesses uncommon Mixed strategies probabilistic equal fitnesses rare Conditional strategies best-of-a-bad-job unequal fitnesses Gross 1996 TREE 11: 92-98 common 16 Reproductive strategies Conditional strategies ‘best-of-a-bad-job’ Long tailed tit Aegithalos caudatus 17 Reproductive strategies Conditional strategies 18 Reproductive strategies 0.3 When do helpers help? Expected success Conditional strategies Breeding Helping 0 May June Percent 100 April MacColl & Hatchwell 2002 Am. Nat. 160: 186-194 Re-nesting Helpers 0 19 April May June Reproductive strategies Conditional strategies Breeding vs. Helping Lifetime Reproductive Success N 70 Ind. fitness > 0: 55 1.10 = 0: 15 0.18 Individual fitness > 0: LRS Lifetime reproductive success LRS Helping is a ‘best-of-a-bad-job’ tactic MacColl & Hatchwell 2004 J. Anim. Ecol. 73: 1137-1148 20 21 Sex ratio evolution 22 Sex ratio evolution Sex ratio theory Most organisms have a 50:50 sex ratio at conception - genetic material is transmitted through both sons and daughters - every individual has exactly one mother and one father, therefore total male (genetic) contribution must equal total female contribution - NS favours ‘strategy’ in which total parental investment in each sex is equal - if sex ratio is not 50:50, then individuals of the rarer sex contribute more on average - the rarer sex will (on average) obtain more matings and leave more offspring than the commoner sex. 23 Sex ratio evolution Consider a population with more females than males The reproductive success of is higher than that of . Any mutation that results in its bearers producing more sons will now spread. 1 2 3 4 1 2 34 1 2 3 4 1 23 4 5 Benefit of over-producing the rare sex continues until a 50:50 sex ratio is reached. FREQUENCY DEPENDENT SELECTION 50:50 sex ratio is an example of an EVOLUTIONARY STABLE STATE 24 Sex ratio evolution Evidence Most organisms have 50:50 sex ratio Circular? No variation for sex determination in most organisms with genetic sex determination But just a consequence of Mendelian segregation of sex chromosomes? 25 Sex ratio evolution Exceptions prove the rule low temp South Carolina female High T ToC sensitive Low T New York Nova Scotia Atlantic silverside Menidia menidia ToC insensitive Conover & Van Voorhees 1990 Science 250: 1556-8 26 Sex ratio evolution Caveats 50:50 only expected when: diploid species nucleus has control of sex determination costs and benefits of sexes are equal 27 Sex ratio evolution Local mate competition Nasonia vitripennis Haplodiploid wasp: females are diploid (fertilised eggs) males are haploid (unfertilised eggs) Charnov 1982 The Theory of Sex Allocation. Princeton UP 28 Sex ratio evolution Cooperative breeders females ‘stay at home’ to help low quality territories: helpers reduce success Seychelles warbler Acrocephalus sechellensis high quality territories: helpers increase success Komdeur et al. 1997 Nature: 385: 522-525 29 The Prisoner’s Dilemma The evolution of cooperation Group selection Kin selection (within species) But e.g.: fig and fig wasp cleaner fish and clients ants and acacia 30 The Prisoner’s Dilemma Reward for mutual coop Pay off to: playing against: Coop Defect Cooperator 3 0 Defector 5 1 Temptation to defect Prisoner B Stays Silent Prisoner A Stays Silent Both serve six months Prisoner A Betrays Sucker’s payoff Punishment for mutual defection Prisoner B Betrays Prisoner A serves ten years Prisoner B goes free Prisoner A goes free Both serve two years Prisoner B serves ten years Axelrod & Hamilton 1981 Science 211: 1390-6 31 The Prisoner’s Dilemma Iterated Prisoner’s Dilemma Round-robin tournament Tit-for-tat Cooperate on first move (generous) Do whatever opponent played on previous move Memory of only one move (forgiving) 32 Axelrod & Hamilton 1981 Science 211: 1390-6 The Prisoner’s Dilemma 33 The Prisoner’s Dilemma Wilkinson 1984 Nature, Carter & Wilkinson 2014 Proc Roy Soc B 34 Game theory provides models useful when outcome depends on what others do frequency dependent not often stictly applicable in nature but ‘heuristic’ (helps us to understand) 35