Stable equilibrium strategies and penalty functions in a game of attrition.

Maynard-Smith (1974) has presented a game of attrition model for animal conflict. He assumed that the penalty function, giving the cost in terms of fitness, of displaying for a given time period, is a linear function of the time of display. Under this assumption he shows that an evolutionary stable strategy (ESS) always exists: one such is a mixed strategy for which display times have a negative exponential distribution. Given the diversity of reproductive strategies and patterns of agonistic behavior in nature, it is reasonable to consider games with different types of cost functions. In this paper it is shown that if more general cost functions are allowed (not necessarily linear), then ESS’s still exist and give a great variety of distributions of dispIay time. Supporting data are presented to suggest that these distributions may be found in nature. It is suggested that the interrelations between an animal’s fitness budget and the game’s penalty function will determine the nature of an ESS for different kinds of games.