Evolutionarily stable strategy


An evolutionarily stable strategy is a strategy that is impermeable when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy which may be novel or initially rare. Introduced by John Maynard Smith and George R. Price in 1972/3, it is an important concept in behavioural ecology, evolutionary psychology, mathematical game theory and economics, with applications in other fields such as anthropology, philosophy and political science.
In game-theoretical terms, an ESS is an equilibrium refinement of the Nash equilibrium, being a Nash equilibrium that is also "evolutionarily stable." Thus, once fixed in a population, natural selection alone is sufficient to prevent alternative strategies from replacing it.

History

Evolutionarily stable strategies were defined and introduced by John Maynard Smith and George R. Price in a 1973 Nature paper. Such was the time taken in peer-reviewing the paper for Nature that this was preceded by a 1972 essay by Maynard Smith in a book of essays titled On Evolution. The 1972 essay is sometimes cited instead of the 1973 paper, but university libraries are much more likely to have copies of Nature. Papers in Nature are usually short; in 1974, Maynard Smith published a longer paper in the Journal of Theoretical Biology. Maynard Smith explains further in his 1982 book Evolution and the Theory of Games. Sometimes these are cited instead. In fact, the ESS has become so central to game theory that often no citation is given, as the reader is assumed to be familiar with it.
Maynard Smith mathematically formalised a verbal argument made by Price, which he read while peer-reviewing Price's paper. When Maynard Smith realized that the somewhat disorganised Price was not ready to revise his article for publication, he offered to add Price as co-author.
Maynard Smith on p. 174 of the Postscript in his, 1982 work Evolution and the Theory of Games argues that the concept of an ESS is itself polyphyletic. In this he not only mentions Price but also list various other sources for the concept. These include the following:
  • variable 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, and Geoff Parker's 1970 work on The reproductive behaviour and the nature of sexual selection in Scatophaga stercoraria.
  • equilibrium by equalising the payoffs for producing sons and daughters, which he sources to Richard F. Shaw and James Dawson Mohler
  • 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.
  • 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.
But the concept proper was derived from R. H. MacArthur and W. D. Hamilton's work on sex ratios, derived from Fisher's principle, especially Hamilton's 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.
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.
  • In evolutionary psychology, ESS is used primarily as a model for human biological evolution.

    Motivation

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. 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. 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

An ESS is a refined or modified form of a Nash equilibrium. 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 represent the payoff for playing strategy S against strategy T. The strategy pair is a Nash equilibrium in a two player game if and only if for both players, for any strategy T:
In this definition, a strategy T≠''S can be a neutral alternative to S''.
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. For all T≠''S, either
  1. E > E, or
  2. E = E and E > E
The first condition is sometimes called a strict Nash equilibrium. 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. 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
  1. E ≥ E, and
  2. E > E
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 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.

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 its strategy is also an ESS.
Some games may have Nash equilibria that are not ESSes. For example, in harm thy neighbor both and are Nash equilibria, since players cannot do better by switching away from either. However, only B is an ESS. 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 = E, but it is not the case that E > E.
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 = E, it is also the case that E > E. 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 and. 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.
This last example points to an important difference between Nash equilibria and ESS. Nash equilibria are defined on strategy sets, 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

In population biology, the two concepts of an evolutionarily stable strategy 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. 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.
B. Thomas 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.
Whether a population is evolutionarily stable does not relate to its genetic diversity: it can be genetically monomorphic or polymorphic.