Saturday, September 28, 2013

99. Evolutionarily Stable Strategies

Maynard Smith (1974, 1976, 1982) established the field of evolutionary game theory, and came up with the notion of evolutionarily stable strategies (ESS). The gist of the ESS idea is this: Whenever the best strategy of an individual depends on what others are doing, the strategy actually adopted will be an ESS, meaning that two competing species will evolve to specialize on the use of different resources, even though each species may tend to be a generalist when in isolation. ‘If everyone else is eating spinach, it will pay to concentrate on cabbage; (similarly), since most forest trees put out their leaves late in spring, it pays forest herbs to put out leaves early’ (Maynard Smith 1976).

As an illustration of the validity of the ESS idea within a species, consider the question of male-female ratio. Why is it generally close to unity (50:50)? The answer was first suggested by Fisher (1930). He assumed that genes of an individual can affect the relative number of his male and female offspring. That sex ratio will tend to dominate which maximizes the number of descendents of the individual. Suppose there are more than 50% males in the population. Then, because of selection pressure, an individual with genes which favour the birth of females will stand a better chance of contributing to the gene pool of the population. If, on the other hand, there are more females in the population, then individuals which tend to produce more sons will have more grandchildren. The 1:1 ratio is the only stable ratio, and the evolutionary strategy which ensures this is called an ESS (even though no player is taking any conscious decisions regarding strategy).

Let us next discuss ‘the logic of animal conflict’ within a species. It is observed that such conflicts are usually of the ‘limited war’ type. Serious injury to one or both contestants is avoided. This must be the result of an ESS. Individuals which had a tendency to violate an ESS must have got eliminated gradually by the processes of Darwinian natural selection. I give here some details of the game-theoretic two-player non-zero-sum version of the ESS approach applied to animal contests or conflicts by Maynard Smith and Price (1973).

Suppose there is a contest between two males of a species because there is a conflict of interest. Therefore, the various possible outcomes of the contest will not be placed in the same order of preference by the two contenders. A genotype determines the pure or mixed game-theoretic strategy adopted by an animal. Suppose the first animal adopts strategy I, and the other animal adopts strategy J. Then the payoff to the player with strategy I can be written as EJ(I). It is the expected gain, namely by a change in the fitness of the first player as a result of the contest; this fitness change is the increase or decrease in the contribution of this animal to the gene pool of future generations. An ESS is defined formally as follows:

Suppose there is a population of individuals adopting strategies I or J with probabilities p and q respectively, with p + q = 1. Then the payoff or fitness of an individual adopting strategy I (we call it individual I), or strategy J is, respectively:

 Fitness of I = p.EI(I) + q.EJ(I)

Fitness of J = p.EI(J) + q.EJ(J)

Suppose a strategy, say I, is an ESS. Then it must have the property that a population of individuals playing I must be ‘protected’ against invasion by any mutant strategy, say J. If this is not the case, then I is an unstable strategy. Thus, when I is common, it must be fitter than any mutant strategy. Formally, I is an ESS if, for J ≠ I,

either EI(I) > EI(J),

or EI(I) = EI(J) and EJ(I) > EJ(J)

Under these conditions, a population playing strategy I is evolutionarily stable, and no mutant can intrude successfully into the population. This is so because, for small q (i.e. when most of the members of the population adopt strategy I) the fitness of strategy I is greater than the fitness of J.

We can state this in the language of mixed-strategy game theory. An ESS consists of a square payoff matrix A, along with a possibly mixed strategy, such that

either xAxtyAxt for all yx,

or, if xAxt = yAxt, then xAyt > yAyt

The condition 'xAxtyAxt for all yx' says that no intruder (y) will do better against the established population x than x does against itself. And the condition 'if xAxt = yAxt, then xAyt > yAyt ' says that if y does as well against x as x itself, then there would be created a new environment (mainly x, but with a small proportion of y) in which x will do better than y, thus making it an ESS.

The ESS idea is related to the notion of noncooperative equilibrium in game theory, with the proviso that the choice of strategy gets interpreted genetically, rather than as a conscious strategic choice (Shubik 1993).

In the next post I shall illustrate some of these ideas with a case study that models the various scenarios in an analysis of the logic of animal conflict.