Evolutionarily stable strategies for a finite population and a variable contest size

Evolutionarily stable in-group favoritism and out-group spite in intergroup conflict

“…Within a …nite population, an individual can improve its own relative standing not only by behavior that increases the individual's own material payo¤, but also by activities that reduce the material payo¤ of other players. Intuitively speaking, this observation is the driving force in Scha¤er's (1988) framework, and it is also the driving force for our results.…”

“…That is, starting from a homogenous population in which all players follow the strategy x E , a mutant player may appear who chooses a di¤erent e¤ort x M . We can now provide a de…nition for a monomorphic equilibrium in evolutionarily stable strategies that rests on the de…nition of stability introduced by Scha¤er (1988). In this de…nition, the set of material payo¤s of players i 2 N is the determinant of evolutionary success as follows:…”

“…This one other player may belong to the same group as player i, which happens with a probability n 1 2n 1 and to the rival group with a probability n 2n 1 . De…nition 1 is taken directly from Scha¤er (1988). It formalizes the standard idea of evolutionary stability, but for a …nite population.…”

“…We apply a modi…cation of the equilibrium concept of evolutionary stability for …nite populations, introduced and developed by Scha¤er (1988). We use and extend this concept in the context of the "indirect approach", i.e., in an environment in which individuals are characterized by their preference types, rather than hard wired actions.…”

“…If (a i ; s i ) 6 = (a E ; s E ) we call this individual i a mutant. Evolutionary stability of preferences in the line of reasoning of Scha¤er (1988) is a property about the relative advantages of this preference type given the uniform preferences of all other players. Suppose that 2n 1 players follow subjective utility maximization according to a given type characterized by (a E ; s E ).…”