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May 08, 2004

Evolutionarily Stable Strategies and the Strategy Set

I see that the selection of major terms in evolutionary biology (right side of the home page) now includes an introduction to the concept of the Evolutionarily Stable Strategy (ESS). This is useful as far as it goes, but like many brief accounts of ESS theory I think it is unclear on one important point...

In evolutionary game theory, like game theory generally, a game is defined primarily by the set of available strategies and payoffs. (Other important aspects are the number of players, whether plays are single or repeated, and the state of knowledge of the players.)

A strategy is essentially an option available to a player. It may either prescribe a single action (a pure strategy) or more than one action, each to be played with a certain probability (a mixed strategy). The set of strategies available to the players is the strategy set of the game in question. For every possible combination of strategies the players may adopt, there is a definite payoff for each player. If the strategy set or payoffs are altered, this is not a move in the game but a change of the game.

As usually defined, an ESS is a strategy such that, if all the members of a population adopt it, no mutant strategy can invade (John Maynard Smith, Evolution and the Theory of Games, p. 204). What is not always clearly stated (even by JMS) is that the mutant strategy must be selected from the strategy set. For game theorists this may be taken for granted, but for the non-expert it may lead to misunderstandings.

For example, suppose the ‘game’ is a contest between two stags, and the strategy set (based on observation of stag behaviour) contains two pure strategies: (A) lock antlers and push, or (B) try to stab the opponent in the neck. The strategy set also contains the mixed strategies consisting of actions (A) and (B) to be performed with probabilities p and (1 - p), for any p from 0 to 1. Payoffs for the matrix of pure strategies are estimated from observations. It may then be calculated that there is one or more ESS, e.g. there may be an ESS with a mixed strategy of performing (A) with probability .6 and (B) with probability .4. This means that the population will be in evolutionary equilibrium if all stags have genes inducing them to adopt this mixed strategy. If this is the case, any mutation which causes a stag to adopt some other strategy in the available strategy set (e.g. performing (A) with probability .7 and (B) with probability .3) will be selected against. In this sense the population cannot be ‘invaded’ by any mutant gene.

This does not mean that no conceivable mutant could invade. For example, a mutation which improved the efficiency of technique (B) would change the payoffs of the game, which might well invalidate the previous ESS. Or there might be a mutation for a completely different fighting technique, such as biting or kicking. If these proved more effective than (A) and (B), the mutant gene would rapidly invade the population.

The essential point is that the ESS is always relative to a given strategy set and payoffs. For game theorists this may be so obvious as hardly to need stating, but for the rest of us it may be worth emphasising.

Posted by David B at 05:08 AM