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Encyclopedia > Evolutionarily stable strategy
Evolutionarily stable strategy
A solution concept in game theory
Relationships
Subset of: Nash equilibrium
Superset of: Stochastically stable equilibrium
Intersects with: Subgame perfect equilibrium, Trembling hand perfect equilibrium, Perfect Bayesian equilibrium
Significance
Proposed by: John Maynard Smith and George R. Price
Used for: Biological modeling and Evolutionary game theory
Example: Hawk-dove
This box: view  talk  edit

In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. The concept is an equilibrium refinement to a Nash equilibrium. The difference between a Nash equilibrium and an ESS is that a Nash equilibrium may sometimes exist due to the assumption that rational foresight prevents players from playing an alternative strategy with no short term cost, but which will eventually be beaten by a third strategy. An ESS is defined to exclude such equilibria, and assumes that natural selection is the only force which selects against using strategies with lower payoffs. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... Game theory is most often described as a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... A stochastically stable equilibrium is a refinement of the evolutionarily stable strategy in evolutionary game theory, proposed by Dean Foster and Peyton Young. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... The trembling hand perfection is a notion that eliminates actions of players that are unsafe because they were chosen through a slip of the hand. ... In game theory, a Bayesian game is one in which information is incomplete. ... Professor John Maynard Smith[1], F.R.S. (6 January 1920 â€“ 19 April 2004) was a British evolutionary biologist and geneticist. ... George R. Price (1922 - January 6, 1975) was a American population geneticist. ... This article or section does not cite its references or sources. ... Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ... The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious damage unless one of them backs down. ... Game theory is most often described as a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... Homo economicus, or Economic man, is the concept in some economic theories of man as both rational and It is a term used for an approximation or model of Homo sapiens that acts to obtain the highest possible well-being for himself given available information about opportunities and other constraints... The GalÃ¡pagos Islands hold 13 species of finches that are closely related and differ most markedly in the shape of their beaks. ... A payoff matrix or payoff function is a concept in game theory which shows what payoff each player will receive at the outcome of the game. ...

The term was introduced and defined by John Maynard Smith and George R. Price in a 1973 Nature paper[1] and is central to Maynard Smith's (1982) book Evolution and the Theory of Games[2]. The concept was derived from R.H. MacArthur[3] and W.D. Hamilton's[4] work on sex ratios, especially Hamilton's (1967) concept of an unbeatable strategy. The idea can be traced back to Ronald Fisher (1930)[5] and Charles Darwin (1859)[6], (see Edwards, 1998). Professor John Maynard Smith[1], F.R.S. (6 January 1920 â€“ 19 April 2004) was a British evolutionary biologist and geneticist. ... George R. Price (1922 - January 6, 1975) was a American population geneticist. ... First title page, November 4, 1869 Nature is one of the oldest and most reputable scientific journals, first published on 4 November 1869. ... Book cover Evolution and the Theory of Games is a 1982 book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. ... Robert Helmer MacArthur (April 7, 1930 â€“ November 1, 1972) was an American ecologist who made a major impact on many areas of community and population ecology. ... This article is about the British biologist Bill Hamilton. ... Sex ratio by country for total population. ... In game theory, an unbeatable strategy is defined by W.D. Hamilton in his 1967 paper on sex ratios in Science. ... Sir Ronald Aylmer Fisher, FRS (17 February 1890 â€“ 29 July 1962) was a British statistician, evolutionary biologist, and geneticist. ... Charles Robert Darwin (12 February 1809 â€“ 19 April 1882) was an eminent English naturalist who achieved lasting fame by convincing the scientific community that species develop over time from a common origin. ...

## Nash equilibria and ESS GA_googleFillSlot("encyclopedia_square");

A Nash equilibrium is a strategy in a game such that if all players adopt it, no player will benefit by switching to play any alternative strategy. If a player choosing strategy J in a population where all other players play strategy I receives a payoff of E(J,I), then strategy I is a Nash equilibrium if, In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ...

E(I,I) ≥ E(J,I) for any J

This equilibrium definition allows for the possibility that strategy J is a neutral alternative to I (it scores equally, but not better). A Nash equilibrium is presumed to be stable even if J scores equally, on the assumption that players do not play J

Maynard Smith and Price (1973)[1] specify two conditions for a strategy I to be an ESS. Either Professor John Maynard Smith[1], F.R.S. (6 January 1920 â€“ 19 April 2004) was a British evolutionary biologist and geneticist. ... George R. Price (1922 - January 6, 1975) was a American population geneticist. ...

1. E(I,I) > E(J,I), or
2. E(I,I) = E(J,I) and E(I,J) > E(J,J)

must be true for all JI, where E(I,J) is the expected payoff to strategy I when playing against strategy J.

The first condition is sometimes called a 'strict Nash' equilibrium (Harsanyi, 1973)[7], the second is sometimes referred to as 'Maynard Smith's second condition'.

There is also an alternative definition of ESS which, though it maintains functional equivalence, 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, we have (adapted from Thomas, 1985)[8]:

1. E(I,I) ≥ E(J,I), and
2. E(I,J) > E(J,J)

In this formulation, the first condition specifies that the strategy be a Nash equilibrium, and the second specifies that Maynard Smith's second condition be 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.

One advantage to this change is that the role of the Nash equilibrium in the ESS is more clearly highlighted. It also allows for a natural definition of other concepts like a weak ESS or an evolutionarily stable set (Thomas, 1985)[8].

### An example

Consider the following payoff matrix, describing a coordination game: It has been suggested that this article or section be merged with normal form game. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...

 A B A 1,1 0,0 B 0,0 1,1 Coordination game

Both strategies A and B are ESS, since a B player cannot invade a population of A players nor can an A player invade a population of B players. Here the two pure strategy Nash equilibria correspond to the two ESS. In this second game, which also has two pure strategy Nash equilibria, only one corresponds to an ESS:

 C D C 1,1 0,0 D 0,0 0,0 Simple game

Here (D, D) is a Nash equilibrium (since neither player will do better by unilaterally deviating), but it is not an ESS. Consider a C player introduced into a population of D players. The C player does equally well against the population (she scores 0), however the C player does better against herself (she scores 1) than the population does against the C player. Thus, the C player can invade the population of D players.

Even if a game has pure strategy Nash equilibria, it might be the case that none of the strategies are ESS. Consider the following example (known as Chicken): The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious harm unless one of them backs down. ...

 E F E 0,0 -1,+1 F +1,-1 -20,-20 Chicken

There are two pure strategy Nash equilibria in this game (E, F) and (F, E). However, in the absence of an uncorrelated asymmetry, neither F nor E are ESSes. A third Nash equilibrium exists, a mixed strategy, which is an ESS for this game (see Hawk-dove game and Best response for explanation). In game theory an uncorrelated asymmetry is an informational asymmetry in a game which is otherwise symmetrical. ... The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious damage unless one of them backs down. ... In game theory, the best response, is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players strategies as given. ...

### Bishop-Cannings theorem

Just as Nash equilibria can be either a pure strategy, or probabilistic mixtures of pure strategies (a mixed strategy), evolutionarily stable strategies can be either pure or mixed. A pure strategy is a term used to refer to strategies in Game theory. ... In game theory a mixed strategy is a strategy which chooses randomly between possible moves. ...

The Bishop-Cannings theorem (Bishop & Cannings, 1978)[9] proves that all members of a mixed evolutionarily stable strategy have the same payoff, and that none of these can also be a pure evolutionarily stable strategy[10]. The same logic also applies to Nash equilibria and so the same will hold true for members of a mixed Nash as for members of a mixed ESS. In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ...

## ESS vs. Evolutionarily Stable State

An ESS or evolutionarily stable strategy is a strategy such that, if all the members of a population adopt it, no mutant strategy can invade. --Maynard Smith (1982)[2].
A population is said to be in an evolutionarily stable state if its genetic composition is restored by selection after a disturbance, provided the disturbance is not too large. Such a population can be genetically monomorphic or polymorphic. --Maynard Smith (1982)[2].

An ESS is a strategy with the property that, once virtually all members of the population use it, then no 'rational' alternative exists. An evolutionarily stable state is a dynamical property of a population to return to using a strategy, or mix of strategies, if it is perturbed from that strategy, or mix of strategies. The former concept fits within classical game theory, whereas the latter is a population genetics, dynamical system, or evolutionary game theory concept. Professor John Maynard Smith[1], F.R.S. (6 January 1920 â€“ 19 April 2004) was a British evolutionary biologist and geneticist. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In general, polymorphism describes multiple possible states for a single property (it is said to be polymorphic). ... Professor John Maynard Smith[1], F.R.S. (6 January 1920 â€“ 19 April 2004) was a British evolutionary biologist and geneticist. ... A population is said to be in an evolutionarily stable state if its genetic composition is restored by selection after a disturbance, provided the disturbance is not too large. ... Game theory is most often described as a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... Population genetics is the study of the distribution of and change in allele frequencies under the influence of the four evolutionary forces: natural selection, genetic drift, mutation, and migration. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ...

Thomas (1984)[11] 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.

## Prisoner's dilemma and ESS

Consider a large population of people who, in the iterated prisoner's dilemma, always play Tit for Tat in transactions with each other. (Since almost any transaction requires trust, most transactions can be modelled with the prisoner's dilemma.) If the entire population plays the Tit-for-Tat strategy, and a group of newcomers enter the population who prefer the Always Defect strategy (i.e. they try to cheat everyone they meet), the Tit-for-Tat strategy will prove more successful, and the defectors will be converted or lose out. Tit for Tat is therefore an ESS, with respect to 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. (see Robert Axelrod's The Evolution of Cooperation[12]). Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma is a type of non-zero-sum game in which two players can cooperate with... Tit for Tat is a highly-effective strategy in game theory for the iterated prisoners dilemma. ... This article is about a political scientist. ... The Evolution of Cooperation is a book by political science professor Robert Axelrod which explores the conditions under which fundamentally selfish agents will spontaneously cooperate. ...

## ESS and human behavior

The recent, controversial sciences of sociobiology and now evolutionary psychology attempt to explain animal and human behavior and social structures, largely in terms of evolutionarily stable strategies. Sociopathy (chronic antisocial/criminal behavior) has been suggested[13] to be best explained as a combination of two such strategies. Sociobiology is a synthesis of scientific disciplines that explains behaviour in all species by considering the evolutionary advantages of social behaviours. ... Evolutionary psychology (abbreviated ev-psych or EP) is a theoretical approach to psychology that attempts to explain certain mental and psychological traitsâ€”such as memory, perception, or languageâ€”as evolved adaptations, i. ... Antisocial personality disorder (APD) is a personality disorder which is often characterised by antisocial and impulsive behaviour. ...

Although ESS were originally considered as stable states for biological evolution, it need not be limited to such contexts. In fact, ESS are stable states for a large class of adaptive dynamics. As a result ESS are used to explain human behavior without presuming that the behavior is necessarily determined by genes. Adaptive Dynamics is a set of techniques for studying long-term phenotypical evolution developed during the 1990s. ... For other meanings of this term, see gene (disambiguation). ...

• Hawk-Dove game
• War of attrition (game)

The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious damage unless one of them backs down. ... In game theory the War of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while accumulating costs at a constant rate c. ...

## References

1. ^ a b John Maynard Smith and George R. Price (1973), The logic of animal conflict. Nature 246: 15-18.
2. ^ a b c John Maynard Smith. (1982) Evolution and the Theory of Games. ISBN 0-521-28884-3
3. ^ MacArthur, R. H. (1965). in: Theoretical and mathematical biology T. Waterman & H. Horowitz, eds. Blaisdell: New York.
4. ^ W.D. Hamilton (1967) Extraordinary sex ratios. Science 156, 477-488.
5. ^ Ronald Fisher The Genetical Theory of Natural Selection. Clarendon Press, Oxford.
6. ^ Charles Darwin (1859). On the Origin of Species
7. ^ Harsanyi, J (1973) Oddness of the number of equilibrium points: a new proof. Int. J. Game Theory 2: 235-250.
8. ^ a b Thomas, B. (1985) On evolutionarily stable sets. J. Math. Biology 22: 105-115.
9. ^ Bishop, D.T. and C. Cannings. 1978. A generalized war of attrition. Journal of Theoretical Biology 70:85-124.
10. ^ Prestwich, K. The Bishop-Cannings Theorem (an annotated version of Maynard Smith's exposition of the The Bishop-Cannings Theorem) at the College of the Holy Cross Game Theory website
11. ^ Thomas, B. (1984) Evolutionary stability: states and strategies. Theor. Pop. Biol. 26 49-67.
12. ^ Robert Axelrod (1984) The Evolution of Cooperation ISBN 0-465-02121-2
13. ^ Mealey, L. (1995). The sociobiology of sociopathy: An integrated evolutionary model. Behavioral and Brain Sciences 18: 523-599. [1]

• 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.
• Hines, WGS (1987) Evolutionary stable strategies: a review of basic theory. Theoretical Population Biology 31: 195-272.
• John Maynard Smith. (1982) Evolution and the Theory of Games. ISBN 0-521-28884-3

Professor Geoffrey Alan Parker FRS (born 24 May 1944) is a professor of biology at the University of Liverpool. ... Sir John Richard Krebs (born 1945) is a British biologist and a Fellow of the Royal Society. ... Professor John Maynard Smith[1], F.R.S. (6 January 1920 â€“ 19 April 2004) was a British evolutionary biologist and geneticist. ... Book cover Evolution and the Theory of Games is a 1982 book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. ...

Results from FactBites:

 Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) (7258 words) In order for a strategy to be evolutionarily stable, it must have the property that if almost every member of the population follows it, no mutant (that is, an individual who adopts a novel strategy) can successfully invade. Roughly, if only two pure strategies exist, then given a (possibly mixed) evolutionarily stable strategy, the corresponding state of the population is a stable state under the replicator dynamics. This representation of strategy selection clearly presupposes hyperrational players and fails to represent the process by which one player observes his opponent's behavior, learns from these observations, and makes the best move in response to what he has learned (as one might expect, for there is no need to model learning in hyperrational individuals).
 NationMaster - Encyclopedia: Evolutionarily stable strategy (4740 words) The difference between a Nash equilibrium and an ESS is that a Nash equilibrium may sometimes exist due to the assumption that rational foresight prevents players from playing an alternative strategy with no short term cost, but which will eventually be beaten by a third strategy. An evolutionarily stable state is a dynamical property of a population to return to using a strategy, or mix of strategies, if it is perturbed from that strategy, or mix of strategies. ESS · Risk dominance In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally.
More results at FactBites »

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