Evolutionarily stable strategy

Evolutionarily stable strategy
A solution concept in game theory
Relationships
Subset of	Nash equilibrium
Superset of	Stochastically stable 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

In game theory and behavioural ecology, an evolutionarily stable strategy (ESS) is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium refinement of the Nash equilibrium -- it is a Nash equilibrium which is "evolutionarily" stable meaning that once it is fixed in a population, natural selection alone is sufficient to prevent alternative ( mutant) strategies from successfully invading.

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. The definition of an ESS excludes such Nash equilibria.

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. Such was the time taken in peer-reviewing the paper for Nature that this was preceded by 1972 essay by Maynard Smith contained in a book of essays entitled On Evolution. The 1972 essay is sometimes cited instead of the 1973 paper, but while university libraries almost certainly contain copies of Nature, the book is more obscure. Nature papers are usually short and Maynard Smith published a lengthier paper in the Journal of Theoretical Biology in 1974 and a further explanation is to be found in Maynard Smith's 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 it is assumed the reader has knowledge of it already and therefore no need for further reading.

Maynard Smith mathematically formalised a verbal argument made by Price that he came across while peer-reviewing Price's paper, offering to make Price co-author of the Nature paper when it became apparent that the somewhat disorganised Price was not ready to revise his article to make it suitable for publication.

The concept was derived from R.H. MacArthur and W.D. Hamilton's work on sex ratios, derived from Fisher's principle, especially Hamilton's (1967) 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 .

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. 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. It is assumed that players are aware of the structure of the game, are consciously attempting to maximize their payoffs, and are attempting to predict the moves of their opponents. In addition, it is presumed 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

1. E(S,S) > E(T,S), or
2. 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; 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):

1. E(S,S) ≥ E(T,S), and
2. 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

Cooperate Defect Cooperate 3, 3 1, 4 Defect 4, 1 2, 2 Prisoner's Dilemma

A B A 2, 2 1, 2 B 2, 1 2, 2 Harm thy neighbour

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 (Defect) it is also an ESS.

In some games, there may be Nash equilibria that are not ESSes. For example in Harm thy neighbour 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).

C D C 2, 2 1, 2 D 2, 1 0, 0 Harm everyone

Swerve Stay Swerve 0,0 -1,+1 Stay +1,-1 -20,-20 Chicken

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 scoring 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.. 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.

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) 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

	Cooperate	Defect
Cooperate	3, 3	1, 4
Defect	4, 1	2, 2
Prisoner's Dilemma

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 later 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 -- the mutant will be eliminated. 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.

This demonstrates the difficulties in applying the formal definiation of an ESS to games with large strategy spaces, and has motivated some to consider alternatives instead.

ESS and human behaviour

The fields of sociobiology and 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 a result of 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 can be used to explain human behaviors that lack any genetic influences.