EVOLUTIONARILY STABLE STRATEGY
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In game theory and behavioural ecology, an 'evolutionarily stable strategy' (or 'ESS'; also 'evolutionary stable strategy') is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy. An ESS is an equilibrium refinement to a Nash equilibrium (a strategy set which — if adopted by all players in a game — results in everyone doing at least as well as they could by unilaterally deviating). An ESS is a Nash equilibrium which is "evolutionarily" stable in that, once fixed in a population, it cannot be altered by the evolutionary forces of mutation and natural selection.
The ESS was developed in order to define a class of solutions to game theoretic problems, equivalent to the Nash equilibrium, but which could be applied to the evolution of social behaviour in animals. Nash equilibria may sometimes exist due to the application of rational foresight, which would be inappropriate in an evolutionary context. Teleological forces such as rational foresight cannot explain the outcomes of trial-and-error processes, such as evolution, and thus have no place in biological applications. An ESS excludes Nash equilibria dependent upon foresight, and includes only those equilibria which are stable if natural selection is assumed to be the only force acting on strategy choice.
First developed in 1973, the ESS has come to be widely used in behavioural ecology and economics, and has been used in anthropology, evolutionary psychology, philosophy, and political science.
History
Evolutionarily stable strategies were defined and introduced 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 [5] and Charles Darwin (1859)[6]. Maynard Smith was jointly awarded the 1999 Crafoord Prize for his development of the concept of evolutionary stable strategies, and the application of game theory to the evolution of behaviour [7].
The ESS was first used in the social sciences by Robert Axelrod in his 1984 book ''The Evolution of Cooperation''. Since that time, there has been widespread use in the social sciences, including work in anthropology, economics, philosophy, and political science. In these fields the primary interest is not in an ESS as the end of biological evolution, but as an end point in the process of cultural evolution or individual learning.[8] In contrast, the ESS is used in evolutionary psychology primarily as a model for human biological evolution.
Motivation
The Nash equilibrium is the traditional solution concept in game theory. It is traditionally underwritten by appeals to the cognitive abilities of the players. Authors often presume that individuals are aware of the structure of the game, are consciously attempting to maximize their returns, and are attempting to predict the moves of their opponents. In addition, they presume that all of this is common knowledge between the players. These facts are then used to explain why players will choose Nash equilibrium strategies.
Evolutionarily stable strategies are motivated entirely differently. Here, it is presumed that the players are individuals with biologically encoded, heritable strategies. The individuals have no control over the strategy they play and need not even be capable of being aware of the game. The individuals reproduce and are subject to the forces of natural selection (with the payoffs of the game representing biological fitness). It is imagined that the alternative strategies of the game occasionally occur, via a process like mutation, and in order to be an ESS a strategy must be resistant to these mutations.
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 there are some Nash equilibria that are not ESSes.
Nash equilibria and ESS
An ESS is a refined, which is to say modified form of, a Nash equilibrium (see next section for examples which contrast the two). A Nash equilibrium is a strategy set where, if all players adopt their respective parts, no player can ''benefit'' by switching to any alternative strategy. Let E(''S'',''T'') represent the payoff for playing strategy ''S'' against strategy ''T''. The strategy set (''S'', ''S'') is a Nash equilibrium (in a two player game) if and only if the following holds for both players:
:E(''S'',''S'') ≥ E(''T'',''S'') for all ''T''≠''S''
This equilibrium definition allows for the possibility that strategy ''T'' is a neutral alternative to ''S'' (it scores 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 specify two conditions for a strategy ''S'' to be an ESS. Either
# E(''S'',''S'') > E(''T'',''S''), 'or'
# E(''S'',''S'') = E(''T'',''S'') and E(''S'',''T'') > E(''T'',''T'')
for all ''T''≠''S''.
The first condition is sometimes called a ''strict'' Nash equilibrium;[9] the second is sometimes referred to as "Maynard Smith's second condition". The meaning of this second condition is that although the adoption of strategy ''T'' is neutral with respect to the payoff against strategy ''S'', the population of players who continue to play strategy ''S'' have an advantage when playing against ''T''.
There is also an alternative definition of ESS which, 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):[10]
# E(''S'',''S'') ≥ E(''T'',''S''), 'and'
# E(''S'',''T'') > E(''T'',''T'')
for all ''T''≠''S''.
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.
One advantage to this alternative formulation is that the role of the Nash equilibrium condition in the ESS is more clearly highlighted. It also allows for a natural definition of related concepts such as a weak ESS or an evolutionarily stable set.
Examples of differences between Nash Equilibria and ESSes
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 the strategy which composes it is also an ESS, since (''Defect'', ''Defect'') is a strong Nash.
In some games, there may be Nash equilibria that are not ESSes. For example in In ''Harm thy neighbor'' 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, B can neutrally invade a population of A strategists, whereupon it will come to predominate since B scores higher against A 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'').
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. While D strategists may temporarily invade a population of C strategists by scoreing equally well against C, 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 the case 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. A third Nash equilibrium exists, 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 immediately reducing the possible equilibrium points.
ESS vs. Evolutionarily Stable State
In population biology, the two concepts of an ''evolutionarily stable strategy'' (ESS) and an ''evolutionarily stable state'' are closely-linked but describe different situations. An ESS is a strategy such that, if all the members of a population adopt it, no mutant strategy can invade.[11]. This idea is distinct from when a population is in an evolutionarily stable state, as this is when its genetic composition will be restored by selection after a disturbance, provided the disturbance is not too large. Whether a population has this property does not relate to genetic diversity, as the population can either be genetically monomorphic or polymorphic.[11]
An ESS is a strategy with the property that, once virtually all members of the population use it, then no 'rational' alternative exists. On the other hand, 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. The former concept fits within classical game theory, whereas the latter is a population genetics, dynamical system, or evolutionary game theory concept.
Thomas (1984)[13] 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[14]).
ESS and human behavior
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/criminal behavior) has been suggested to be a result of a combination of two such strategies.[15]
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 can be used to explain human behaviors that lack any genetic influences.
See also
★ Evolutionary game theory
★ Hawk-Dove game
★ War of attrition (game)
References
1. John Maynard Smith and George R. Price (1973), The logic of animal conflict. ''Nature'' '246': 15-18.
2. 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 (1930) ''The Genetical Theory of Natural Selection''. Clarendon Press, Oxford.
6. Charles Darwin (1859). ''On the Origin of Species''
7. The 1999 Crafoord Prize press release
8. Evolutionary Game Theory Jason McKenzie Alexander
9. Harsanyi, J (1973) Oddness of the number of equilibrium points: a new proof. ''Int. J. Game Theory'' '2': 235–250.
10. Thomas, B. (1985) On evolutionarily stable sets. ''J. Math. Biology'' '22': 105–115.
11. filler
12. filler
13. Thomas, B. (1984) Evolutionary stability: states and strategies. ''Theor. Pop. Biol.'' '26' 49-67.
14. Robert Axelrod (1984) ''The Evolution of Cooperation'' ISBN 0-465-02121-2
15. Mealey, L. (1995). The sociobiology of sociopathy: An integrated evolutionary model. ''Behavioral and Brain Sciences'' '18': 523-599. [1]
Further reading
★ 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
External links
★ 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.
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