U. Motro, EVOLUTIONARY AND CONTINUOUS STABILITY IN ASYMMETRIC GAMES WITH CONTINUOUS STRATEGY SETS - THE PARENTAL INVESTMENT CONFLICT AS AN EXAMPLE, The American naturalist, 144(2), 1994, pp. 229-241
In a population that is fixed at an evolutionarily stable strategy (ES
S), no mutant strategy can invade and spread. If, however, the strateg
y 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 b
etween the various ESSs with respect to their dynamic stability charac
teristics and is treated here for the case of asymmetric games. Two di
stinct types of ESSs can exist in such games: ESSs that are continuous
ly stable (CSSs) and ESSs that are not. Any strategy in the neighborho
od 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 i
mmune to any mutation that is further away from the ESS. Thus, in natu
ral 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 co
ntinuously stable. The ideas are illustrated by two examples, the pare
ntal investment conflict and the dispersal conflict between males and
females.