Evolutionary and Continuous Stability in Asymmetric Games with Continuous Strategy Sets: The Parental Investment Conflict as an Example

In a population that is fixed at an evolutionarily stable strategy (ESS), no mutant strategy can invade and spread. If, however, the strategy set is continuous, one can ask which mutations can be established in a population that is fixed not at an ESS but, rather, at a different, nearby strategy. This question gives rise to a possible distinction between the various ESSs with respect to their dynamic stability characteristics and is treated here for the case of asymmetric games. Two distinct types of ESSs can exist in such games: ESSs that are continuously stable (CSSs) and ESSs that are not. Any strategy in the neighborhood of a continuously stable ESS can always be invaded by mutants that are closer to the ESS. In contrast, any neighborhood of an ESS that is not a CSS contains a nonzero measure set of strategies that are not immune to any mutation that is further away from the ESS. Thus, in natural situations, one can expect more frequently to find populations at (or near) an ESS that is a CSS than at (or near) an ESS that is not continuously stable. The ideas are illustrated by two examples, the parental investment conflict and the dispersal conflict between males and females.