On sperm competition games: raffles and roles revisited

In principle there are two approaches to modelling a tradeoff between the p ositive and negative outcomes of a behavior: after suitably defining a valu e for the behavior in the absence of any trade-off, one can either multiply that value by an appropriate discount or subtract an appropriate cost. In a prospective analysis of sperm competition, Parker (Proc. Roy. Sec. Lond. B (1990) 242, 120-126) adopted the multiplicative approach to model the tra de-off between the value of a mating and the cost of its acquisition. He ob tained two paradoxical results. First, if two males 'know' whether they are first or second to mate, but these roles are assigned randomly, then sperm numbers should be the same for both males whether the 'raffle' for fertili zation is fair or unfair. Second, if mating order is constant, then a favor ed male should expend less on sperm. His results are puzzling not only in t erms of intuition about nature, but also in terms of his model's consistenc y. In other words, they present both an external and an internal paradox. P arker assumed the fairness of the raffle to a disfavored male to be indepen dent of how much sperm a favored male deposits. This article both generaliz es Parker's analysis by allowing fairness to decrease with sperm expenditur e by the favored male and compares Parker's results to those obtained by th e additive approach. In many respects, results are similar. Nevertheless, i f the costs of mating are assumed to increase with sperm expenditure but no t to depend on the role in which sperm is expended, as Parker assumed, then the additive approach is more fundamentally correct. In particular, Parker 's constant-role paradox is an artifact of his approach. His random-role pa radox is internally rationalized in terms of standard microeconomic When fairness decreases, however slightly, with sperm expenditure by the fa vored male, both models demonstrate that the evolutionarily stable strategy is for more sperm to be deposited during a favored mating than during a di sfavored mating. The lower the costs, the greater the divergence. Thus a po ssible resolution of the external paradox is that fairness is not constant in nature.