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the (175), and (83), game (72), strategy (59), theory (50), ess (43), #stable (41), #evolution (39), for (39), evolutionary (37), equilibrium (35), nash (35), evolutionarily (32), strategies (32), that (31), selection (28), population (28), this (25), maynard (21), smith (21), are (21), not (17), edit (16), players (16), with (15), biology (15), animal (15), games (15), dilemma (15), defect (15), from (14), psychology (14), price (14), can (14), cooperate (14), social (13), but (13), two (13), always (13), equilibria (13), hypothesis (12), cognitive (12), player (12), prisoner (12), against (12), may (11), which (11), concept (11), was (10), sexual (10), history (10), first (10), second (10), tit (10), tat (10), perfect (10), doi (10), nature (9), cultural (9), biological (9), theoretical (9), human (9), number (9), behavioral (9), other (9), there (9), definition (9), mutant (9), his (9), john (8), state (8), play (8), payoff (8), also (8), has (8), esses (8), condition (8), wikipedia 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h the payoffs of the game representing reproductive success biological fitness it is imagined that alternative strategies of the game occasionally occur via a process like mutation to be an ess a strategy must be resistant to these alternatives given the radically different motivating assumptions it may come as a surprise that esses and nash equilibria often coincide in fact every ess corresponds to a nash equilibrium but some nash equilibria are not esses nash equilibrium edit an ess is a refined or modified form of a nash equilibrium see the next section for examples which contrast the two in a nash equilibrium if all players adopt their respective parts no player can benefit by switching to any alternative strategy in a two player game it is a strategy pair let e s t represent the payoff for playing strategy s against strategy t the strategy pair s s is a nash equilibrium in a two player game if and only if for both players for any strategy t e s s e t s in this definition a strategy t s can be a neutral alternative to s scoring equally well but not better a nash equilibrium is presumed to be stable even if t scores equally on the assumption that there is no long term incentive for players to adopt t instead of s this fact represents the point of departure of the ess maynard smith and price 2 specify two conditions for a strategy s to be an ess for all t s either e s s e t s or e s s e t s and e s t e t t the first condition is sometimes called a strict nash equilibrium 14 the second is sometimes called maynard smith s second condition the second condition means that although strategy t is neutral with respect to the payoff against strategy s the population of players who continue to play strategy s has an advantage when playing against t there is also an alternative stronger definition of ess due to thomas 15 this places a different emphasis on the role of the nash equilibrium concept in the ess concept following the terminology given in the first definition above this definition requires that for all t s e s s e t s and e s t e t t in this formulation the first condition specifies that the strategy is a nash equilibrium and the second specifies that maynard smith s second condition is met note that the two definitions are not precisely equivalent for example each pure strategy in the coordination game below is an ess by the first definition but not the second in words this definition looks like this the payoff of the first player when both players play strategy s is higher than or equal to the payoff of the first player when he changes to another strategy t and the second player keeps his strategy s and the payoff of the first player when only his opponent changes his strategy to t is higher than his payoff in case that both of players change their strategies to t this formulation more clearly highlights the role of the nash equilibrium condition in the ess it also allows for a natural definition of related concepts such as a weak ess or an evolutionarily stable set 15 examples of differences between nash equilibria and esses edit cooperate defect cooperate 3 3 1 4 defect 4 1 2 2 prisoner s dilemma a b a 2 2 1 2 b 2 1 2 2 harm thy neighbor in most simple games the esses and nash equilibria coincide perfectly for instance in the prisoner s dilemma there is only one nash equilibrium and its strategy defect is also an ess some games may have nash equilibria that are not esses for example in harm thy neighbor whose payoff matrix is shown here both a a and b b are nash equilibria since players cannot do better by switching away from either however only b is an ess and a strong nash a is not an ess so b can neutrally invade a population of a strategists and predominate because b scores higher against b than a does against b this dynamic is captured by maynard smith s second condition since e a a e b a but it is not the case that e a b e b b c d c 2 2 1 2 d 2 1 0 0 harm everyone swerve stay swerve 0 0 1 1 stay 1 1 20 20 chicken nash equilibria with equally scoring alternatives can be esses for example in the game harm everyone c is an ess because it satisfies maynard smith s second condition d strategists may temporarily invade a population of c strategists by scoring equally well against c but they pay a price when they begin to play against each other c scores better against d than does d so here although e c c e d c it is also the case that e c d e d d as a result c is an ess even if a game has pure strategy nash equilibria it might be that none of those pure strategies are ess consider the game of chicken there are two pure strategy nash equilibria in this game swerve stay and stay swerve however in the absence of an uncorrelated asymmetry neither swerve nor stay are esses there is a third nash equilibrium a mixed strategy which is an ess for this game see hawk dove game and best response for explanation this last example points to an important difference between nash equilibria and ess nash equilibria are defined on strategy sets a specification of a strategy for each player while ess are defined in terms of strategies themselves the equilibria defined by ess must always be symmetric and thus have fewer equilibrium points vs evolutionarily stable state edit in population biology the two concepts of an evolutionarily stable strategy ess and an evolutionarily stable state are closely linked but describe different situations in an evolutionarily stable strategy if all the members of a population adopt it no mutant strategy can invade 4 once virtually all members of the population use this strategy there is no rational alternative ess is part of classical game theory in an evolutionarily stable state a population s genetic composition is restored by selection after a disturbance if the disturbance is not too large an evolutionarily stable state is a dynamic property of a population that returns to using a strategy or mix of strategies if it is perturbed from that initial state it is part of population genetics dynamical system or evolutionary game theory this is now called convergent stability 16 b thomas 1984 applies the term ess to an individual strategy which may be mixed and evolutionarily stable population state to a population mixture of pure strategies which may be formally equivalent to the mixed ess 17 whether a population is evolutionarily stable does not relate to its genetic diversity it can be genetically monomorphic or polymorphic 4 stochastic ess edit in the classic definition of an ess no mutant strategy can invade in finite populations any mutant could in principle invade albeit at low probability implying that no ess can exist in an infinite population an ess can instead be defined as a strategy which should it become invaded by a new mutant strategy with probability p would be able to counterinvade from a single starting individual with probability p as illustrated by the evolution of bet hedging 18 prisoner s dilemma edit cooperate defect cooperate 3 3 1 4 defect 4 1 2 2 prisoner s dilemma a common model of altruism and social cooperation is the prisoner s dilemma here a group of players would collectively be better off if they could play cooperate but since defect fares better each individual player has an incentive to play defect one solution to this problem is to introduce the possibility of retaliation by having individuals play the game repeatedly against the same player in the so called iterated prisoner s dilemma the same two individuals play the prisoner s dilemma over and over while the prisoner s dilemma has only two strategies cooperate and defect the iterated prisoner s dilemma has a huge number of possible strategies since an individual can have different contingency plan for each history and the game may be repeated an indefinite number of times there may in fact be an infinite number of such contingency plans three simple contingency plans which have received substantial attention are always defect always cooperate and tit for tat the first two strategies do the same thing regardless of the other player s actions while the latter responds on the next round by doing what was done to it on the previous round it responds to cooperate with cooperate and defect with defect if the entire population plays tit for tat and a mutant arises who plays always defect tit for tat will outperform always defect if the population of the mutant becomes too large the percentage of the mutant will be kept small tit for tat is therefore an ess with respect to only these two strategies on the other hand an island of always defect players will be stable against the invasion of a few tit for tat players but not against a large number of them 19 if we introduce always cooperate a population of tit for tat is no longer an ess since a population of tit for tat players always cooperates the strategy always cooperate behaves identically in this population as a result a mutant who plays always cooperate will not be eliminated however even though a population of always cooperate and tit for tat can coexist if there is a small percentage of the population that is always defect the selective pressure is against always cooperate and in favour of tit for tat this is due to the lower payoffs of cooperating than those of defecting in case the opponent defects this demonstrates the difficulties in applying the formal definition of an ess to games with large strategy spaces and has motivated some to consider alternatives human behavior edit the fields of sociobiology and evolutionary psychology attempt to explain animal and human behavior and social structures largely in terms of evolutionarily stable strategies sociopathy chronic antisocial or criminal behavior may be a result of a combination of two such strategies 20 evolutionarily stable strategies were originally considered for biological evolution but they can apply to other contexts in fact there are stable states for a large class of adaptive dynamics as a result they can be used to explain human behaviours that lack any genetic influences see also edit antipredator adaptation behavioral ecology evolutionary psychology fitness landscape hawk dove game koinophilia sociobiology war of attrition game farsightedness game theory references edit 1 2 maynard smith j 1972 game theory and the evolution of fighting on evolution edinburgh university press isbn 0 85224 223 9 1 2 3 maynard smith j price g r 1973 the logic of animal conflict nature 246 5427 15 8 bibcode 1973natur 246 15s doi 10 1038 246015a0 maynard smith j 1974 the theory of games and the evolution of animal conflicts pdf journal of theoretical biology 47 1 209 21 bibcode 1974jthbi 47 209m doi 10 1016 0022 5193 74 90110 6 pmid 4459582 1 2 3 maynard smith john 1982 evolution and the theory of games cambridge university press isbn 0 521 28884 3 gadgil m 1972 male dimorphism as a consequence of sexual selection the american naturalist 106 951 574 580 parker g a 1970 the reproductive behaviour and the nature of sexual selection in scatophaga stercoraria l diptera scatophagidae ii the fertilization rate and the spatial and temporal relationships of each sex around the site of mating and oviposition the journal of animal ecology 205 228 shaw r f mohler j d 1953 the selective significance of the sex ratio the american naturalist 87 837 337 342 fretwell s d lucas h l 1970 on the territorial behaviour and other factors influencing habitat distribution in birds i theoretical development acta biotheor 19 16 36 doi 10 1007 bf01601953 trivers r l 1971 the evolution of reciprocal altruism the quarterly review of biology 46 1 35 57 macarthur r h 1965 waterman t horowitz h eds theoretical and mathematical biology new york blaisdell hamilton w d 1967 extraordinary sex ratios science 156 3774 477 88 bibcode 1967sci 156 477h doi 10 1126 science 156 3774 477 jstor 1721222 pmid 6021675 press release archived 2016 03 03 at the wayback machine for the 1999 crafoord prize alexander jason mckenzie 23 may 2003 evolutionary game theory stanford encyclopedia of philosophy retrieved 31 august 2007 harsanyi j 1973 oddness of the number of equilibrium points a new proof int j game theory 2 1 235 50 doi 10 1007 bf01737572 1 2 thomas b 1985 on evolutionarily stable sets j math biology 22 105 115 doi 10 1007 bf00276549 apaloo j brown j s vincent t l 2009 evolutionary game theory ess convergence stability and nis evolutionary ecology research 11 489 515 archived from the original on 2017 08 09 retrieved 2018 01 10 thomas b 1984 evolutionary stability states and strategies theor popul biol 26 1 49 67 bibcode 1984tpbio 26 49t doi 10 1016 0040 5809 84 90023 6 king oliver d masel joanna 1 december 2007 the evolution of bet hedging adaptations to rare scenarios theoretical population biology 72 4 560 575 bibcode 2007tpbio 72 560k doi 10 1016 j tpb 2007 08 006 pmc 2118055 pmid 17915273 axelrod robert 1984 the evolution of cooperation basic books isbn 0 465 02121 2 mealey l 1995 the sociobiology of sociopathy an integrated evolutionary model behavioral and brain sciences 18 3 523 99 doi 10 1017 s0140525x00039595 further reading edit weibull jörgen 1997 evolutionary game theory mit press isbn 978 0 262 73121 8 classic reference textbook hines w g s 1987 evolutionary stable strategies a review of basic theory theoretical population biology 31 2 195 272 bibcode 1987tpbio 31 195h doi 10 1016 0040 5809 87 90029 3 pmid 3296292 leyton brown kevin shoham yoav 2008 essentials of game theory a concise multidisciplinary introduction san rafael ca morgan claypool publishers isbn 978 1 59829 593 1 an 88 page mathematical introduction see section 3 8 free online archived 2000 08 15 at the wayback machine at many universities parker g a 1984 evolutionary stable strategies in behavioural ecology an evolutionary approach 2nd ed krebs j r davies n b eds pp 30 61 blackwell oxford shoham yoav leyton brown kevin 2009 multiagent systems algorithmic game theoretic and logical foundations new york cambridge university press isbn 978 0 521 89943 7 archived from the original on 2011 05 01 retrieved 2008 12 17 a comprehensive reference from a computational perspective see section 7 7 downloadable free online maynard smith john 1982 evolution and the theory of games isbn 0 521 28884 3 classic reference external links edit evolutionarily stable strategies at animal behavior an online textbook by michael d breed game theory and evolutionarily stable strategies kenneth n prestwich s site at college of the holy cross evolutionarily stable strategies knol archived https web archive org web 20091005015811 http knol google com k klaus rohde evolutionarily stable strategies and xk923bc3gp4 50 v t e game theory glossary game theorists games traditional game theory definitions asynchrony bayesian regret best response bounded rationality cheap talk coalition complete contract complete information complete mixing conjectural variation contingent cooperator coopetition cooperative game ...
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