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Solution concept in game theory

An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) 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 (or set of strategies) 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 (mutant) strategies from replacing it (although this does not preclude the possibility that a better strategy, or set of strategies, will emerge in response to selective pressures resulting from environmental change).

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.

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. 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.

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. 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.