Excerpts from

Evolution and the Theory of Games

John Maynard Smith

Explanation of Main Terms

A ‘strategy’ is a specification of what an individual will do in any situation in which it may find itself. A strategy may be ‘pure’ or ‘mixed’; in the latter case there is a random element in the specification.
An ‘action’ is a behaviour performed in a particular situation.
An ‘asymmetric’ contest is one in which there is a difference in ‘role’ between the contestants, of a kind which enables either of them to adopt a strategy ‘in role 1, do A; in role 2, do B.’ Clearly, roles must be perceived by the contestants; otherwise, they could not affect behaviour. Examples are male and female, or owner and intruder. It is assumed that the circumstances, genetic or environmental, which decide in which role an individual finds itself, act independently of genes determining its strategy.
A ‘symmetric’ contest is one with no role differentiation.
A ‘payoff,’ written E (A, B), is the expected change of fitness of an individual adopting a strategy A against an opponent adopting B.
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.
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. For the extended model of games against the field, an ESS must satisfy equations (2.9):
  The strategy I is an ESS if, for all JI, (2.9)
either W (J, I ) < W (I, I )  
or W (J, I ) = W (I, I )  
  and, for small q,
  W (J, Pq, J, I ) < W (I, Pq, J, I )  
  with Pq, J, I defined as the population P = qJ + (1 − q) I.

For pairwise contests in an infinite asexual population, an ESS must satisfy conditions (2.4a,b):

  The strategy I is an ESS if, for all JI, (2.4)
either E (I, I ) > E (J, I ) (2.4a)
or E (I, I ) = E (J, I ) and E (I, J ) > E (J, J ) (2.4b)
  Two points about this definition should be noted. First, if stability requires a mixture of pure strategies, then individuals must adopt the appropriate mixed strategy; a genetically polymorphic population may be in an evolutionarily stable state, but, strictly, no individual is adopting an ESS. Secondly, I have preferred to define an ESS as an uninvadable strategy, rather than as a strategy satisfying any particular mathematical conditions. In some cases, however, it is convenient to use the term ESS for any strategy satisfying conditions (2.4a,b); I hope that the context will make it clear in which sense the term is being used.

John Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982, pp. 14, 24, 204.

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