Sperm competition games: raffles and roles

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Abstract
Evolutionary games in which two males mate with the same female are examined by using an evolutionarily stable strategy (ESS) approach. These prospective models of competitive ejaculation seek ESS sperm numbers for cases when sperm competition obeys the `raffle principle'. In a `fair raffle', each male's fertilization probability equals his sperm number divided by the total sperm number in the female tract. In a loaded raffle, one male's sperm are `devalued' relative to his competitor, e.g. a sperm from the second male to mate counts as only (say) half of a ticket in the fertilization lottery compared with one of the first male's sperm. The models assume that there is a trade-off between effort spent on sperm and effort spent on obtaining matings, and that fitness is the product of number of matings and the expected gain from each one. In a general sense, ESS sperm numbers increase with the probability that two males mate with the same female, and decrease with the degree of unfairness in the raffle. However, the ESS depends critically on the information available to the two competitors, and whether they occupy roles (of first or second to mate) randomly or non-randomly. If males `know' whether they are the first or the second to mate, but these roles are assigned randomly, sperm numbers should be equal for the two males whether the raffle is fair or unfair. If roles are not random, so that (say) a given male tends to be first to mate, sperm numbers will not be equal unless the raffle is a fair one. In a loaded raffle, the male whose sperm are devalued should compensate by expending a greater reproductive effort on sperm. If sperm costs are equal for the two males, then the male in the favoured role should produce less sperm, but nevertheless achieves higher paternity because of the loading in the raffle. If sperm costs are asymmetric for the two roles, in a fair raffle the male that can produce sperm more cheaply ejaculates more sperm (and hence experiences higher paternity) although he expends less reproductive effort on sperm production.