Saturday, 28 September 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.

Saturday, 21 September 2013

98. Coevolution of Species

'Just when you think you’ve won the rat race, along come faster rats' (Anon.).

Living organisms interact, not only with the inanimate surroundings, but also with organisms of the same or different species. Therefore, evolution of a species cannot occur in isolation from the evolution of other species with which it interacts. They coevolve. Coevolution of species is an important aspect of biological evolution.

Darwin’s theory of evolution says that a species evolves to become better and better for the task of survival in a given set of conditions. Genes of individuals in the population that are not good enough for the task of survival tend to get eliminated in successive generations. All this happens through an interplay of the blind forces of Nature, but the end effect appears to be as if the genes have the thinking power of being 'selfish' at the job of survival and propagation. Richard Dawkins’ (1989) phrase the selfish gene sums up the situation well.

This may seem to indicate that a species will always evolve strategies (consciously or 'purely chemically') that are entirely selfish when it comes to dealing with other species. In fact, even within a species, it is conceivable that the genotype, and thence the phenotype, of an individual member may exhibit selfishness, without regard for other members of the species. But what happens in reality can actually be far from this simple-minded speculation. There can be evolutionarily stable strategies, involving both competition and cooperation, and there can be evolutionary arms races. There can also be symbioses of species.

Before I discuss some of these processes, it is interesting to take note of an analysis by Douglas Caldwell (cf. Margulis and Sagan 2002), according to which the following terms were never used by Charles Darwin in his Origin of Species: association; affiliation; cooperate, cooperation; collaborate, collaboration; community; intervention; symbiosis. What I am now going to describe in this and the next few posts are some of the post-Darwinian developments.

Coevolution of species is an important ingredient of the succession of turmoil and relative stability (stasis), so that what really unfolds is punctuated equilibrium.


Although species tend to evolve towards a state of stable adaptation to the existing environment, it is also true that many of them are extinct, as the fossil records show. This is because there is never a state of permanent, static equilibrium in Nature. There is a never-ending input of energy, climatic changes, terrestrial upheavals, as also mutations and a coexistence with other competing or cooperating species. Rather than evolving towards a state of permanent equilibrium and adaptation, the entirety of species (in fact, the Earth as a whole) evolves towards a state of self-organized criticality (SOC). This state of complexity is poised at the edge of order and chaos. The never-ending inputs just mentioned result in minor or major ‘avalanches’ (catastrophic events of various magnitudes), which sometimes lead to the extinction of species, or the emergence of new ones.

SOC can explain the finding by Eldredge and Gould (1972) in fossil records that there are long periods of stasis, followed by quick bursts of evolutionary change; i.e. there is punctuated equilibrium. Imagine an ecosystem in which the various species have reached more or less a state of equilibrium with one another. Suppose there is a random genetic crossover in one of the species that is beneficial to it and thus survives and gets propagated to other members of the population. The propagation is not linear; it is more likely to be avalanche-like or ‘explosive’ with time, rather like the avalanches in the sandpile experiment I described in Part 82. In due course, things stop changing, but then some other member of the population may mutate. There is thus a steady drizzle of mutations, resulting in periods of avalanches and relative equilibrium.

As I have explained in the previous few posts, in conventional game theory it is usually presumed that each player is a rational being who expects that other players are also rational beings. Each player therefore adopts a strategy to minimize his losses, assuming that other players will work out strategies to maximize the losses of other players as they try to minimize their own losses (the minimax strategy). In the evolutionary context, if each member of a population did this, the end result may be an extinction of that species. The fact that a species has survived can imply that an evolutionarily stable strategy (ESS) must have evolved. In an ESS, the survival of an individual, though not completely subservient to the survival of the species, is determined by a strategy that ensures that contests between individuals, though leading to an improvement of the gene pool of the population, do not result in an excessive annihilation of the contest-losers. The ESS idea, which entailed a modification of conventional game theory, was put forward by Maynard Smith (1974, 1976, 1982), and it can be operative even in the coevolution of two or more species. In fact, the ESS concept is also relevant to the evolution of entire ecosystems.

I shall discuss the ESS notion in the next two posts.

Sunday, 15 September 2013

97. Cooperative Games






1. It must lie in the feasible set S.







The core


The stable set


The value




The bargaining set




