On the interpretation of evolutionarily stable sets

Balkenborg, D. and Schlag, K.H. (1997) On the interpretation of evolutionarily stable sets (Discussion Papers in Economics and Econometrics, 9710) Southampton, UK. University of Southampton

Abstract

We introduce notions of evolutionary stability for sets of strategies based on the following requirements: After every sufficiently small mutation of a population playing a single strategy in the set: a) No single mutant strategy can spread. b) A single mutant strategy not in the set will be driven out. Depending on the precise interpretation of "a sufficiently small mutation" in these requirements we distinguish "simple ES sets", "pointwise uniform ES sets" and "uniform ES set". In contrast to the original definition of an ES set by Thomas (1985d) our definitions do not require the sets to be closed

We show: 1) A uniform ES set is always an ES set as defined by Thomas. 2) For analytic fitness functions, and hence for all symmetric normal form games, the notions of pointwise uniform ES set and ES set coincide. 3) All four definitions of evolutionary stability for sets coincide in symmetric bimatrix games.

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