# Evolutionarily stable strategy

Evolutionarily stable strategy
A solution concept in game theory
Relationships
Subset of: Nash 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 (aka Game of chicken)

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.

The term was introduced and defined by John Maynard Smith and George R. Price in a 1973 Nature paper and is central to Maynard Smith's (1982) book Evolution and the Theory of Games. The concept was derived from R.H. MacArthur and W.D. Hamilton's work on sex ratios, especially Hamilton's (1967) concept of an unbeatable strategy. The idea can be traced back to Ronald Fisher (1930) and Charles Darwin (1859), (see Edwards, 1998).

## Nash equilibria and ESS

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,

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) specify 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 IJ, 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'.

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):

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 despite the difference in formulation, the two definitions are actually equivalent.

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).

### An example

Consider the following payoff matrix, describing a coordination game:

 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):

 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).

### Bishop-Cannings theorem

Just as Nash equilibria can be either a pure strategy, or probabalistic mixtures of pure strategies (a mixed strategy), evolutionarily stable strategies can be either pure of mixed.

The Bishop-Cannings theorem (Bishop & Cannings, 1978) 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. 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.

## 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.

Thomas (1984) 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).

## ESS and human behaviour

The recent, controversial sciences of sociobiology and now evolutionary psychology attempt to explain animal and human behaviour and social structures, largely in terms of evolutionarily stable strategies. Sociopathy (chronic antisocial/criminal behaviour) has been suggested to be best 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 behaviour is necessarily determined by genes.