# Evolutionarily stable strategy

The evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a central concept in game theory introduced by John Maynard Smith and George R. Price in 1973 (a full account is given by Maynard Smith's 1982 book Evolution and the Theory of Games). The basic idea is that if a population of organisms is playing an evolutionarily stable strategy, a mutant allele that causes organisms to adopt a different strategy cannot invade the population, but will instead be eliminated by natural selection.

The concept was based on W.D. Hamilton's (1967) unbeatable strategy; the difference is that an unbeatable strategy is resistant to large migrations of different strategies. Through Hamilton's work on sex ratios the concept can be traced back through Ronald Fisher (1930) and Charles Darwin (1859) (see Edwards, 1998).

## Definition

An ESS depends on the idea of invasion, where a population of strategy-X players is visited by a strategy-Y player. The new player is said to invade if and only if either (1) following strategy Y, he scores better than the average strategy-X player or (2) following strategy Y, he scores equally well against the population as the average strategy-X player but does better against Y types than the average strategy-X player does against a Y player.

A strategy X is evolutionarily stable if and only if there is no strategy Y that can invade it. That is, anybody bringing a new strategy into a population of strategy-X players will fare no better on average than the X players are already doing. (See the closely-related Nash equilibrium) ESS is stable in respect to randomly and occasionally occurring invading strategies, but not necessarily stable with respect to mass counts of invaders. Similarly, ESS stability is only guaranteed under the assumption of infinitely large populations. In finite populations, stochastic effects of genetic drift may force the ESS to become unstable.

## Nash equilibria and ESS

Maynard Smith and Price specify (Maynard Smith & Price, 1973; Maynard Smith 1982) two conditions for a strategy I to be an ESS. Either

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 ${\displaystyle I\neq J}$, 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), the second is sometimes referred to as 'Maynard Smith's second condition'.

Consider the following payoff matrix, describing a coordination game such as the Stag hunt, or Battle of the sexes:

A B 1, 1 0, 0 0, 0 1, 1

Both strategies A and B are ESS, since a B player cannot invade a population of A players nor can a 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 1, 1 0, 0 0, 0 0, 0

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):

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

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).

## 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).
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).

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.

## 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, or more briefly here.

## 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. For example, in one well-known 1995 paper by Linda Mealey, sociopathy (chronic antisocial/criminal behavior) is explained as a combination of two such strategies.

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.