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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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ble behaviour being explained in terms of frequency dependent selection which he sources to madhav gadgil s 1972 work on male dimorphism as a consequence of sexual selection 5 and geoff parker s 1970 work on the reproductive behaviour and the nature of sexual selection in scatophaga stercoraria 6 equilibrium by equalising the payoffs for producing sons and daughters which he sources to richard f shaw and james dawson mohler 1953 7 the analysis of animal dispersal which uses the observation that fitness should be equal at an equilibrium if we are to achieve an ideal free distribution is sourced to stephen d fretwell and henry l lucas 1970 8 the idea that reciprocal altruism can be understood for the same two opponents in a repeated game as an ess being sourced to robert trivers 1971 9 but the concept proper was derived from r h macarthur 10 and w d hamilton s 11 work on sex ratios derived from fisher s principle especially hamilton s 1967 concept of an unbeatable strategy maynard smith was jointly awarded the 1999 crafoord prize for his development of the concept of evolutionarily stable strategies and the application of game theory to the evolution of behaviour 12 uses of ess the ess was a major element used to analyze evolution in richard dawkins bestselling 1976 book the selfish gene the ess was first used in the social sciences by robert axelrod in his 1984 book the evolution of cooperation since then it has been widely used in the social sciences including anthropology economics philosophy and political science in the social sciences the primary interest is not in an ess as the end of biological evolution but as an end point in cultural evolution or individual learning 13 in evolutionary psychology ess is used primarily as a model for human biological evolution motivation edit the nash equilibrium is the traditional solution concept in game theory it depends on the cognitive abilities of the players it is assumed that players are aware of the structure of the game and consciously try to predict the moves of their opponents and to maximize their own payoffs in addition it is presumed that all the players know this see common knowledge these assumptions are then used to explain why players choose nash equilibrium strategies evolutionarily stable strategies are motivated entirely differently here it is presumed that the players strategies are biologically encoded and heritable individuals have no control over their strategy and need not be aware of the game they reproduce and are subject to the forces of natural selection with 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 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