On the definition of an evolutionarily stable strategy

An ESS must be able to withstand invasion by a small group. It is shown that there are (at least) two possible mathematical interpretations of this statement. In some important applications the two definitions of an ESS are equivalent, but this is not generally the case and a simple example is given to illustrate this. In the weaker form, not favoured here, an ESS may not withstand some infinitely small invasions, but in the stronger form the ESS is certain to survive all invasions up to a fixed fraction of the population. Also, when a pay-off matrix is used to define a dynamic, the stronger definition is the more convenient.