Sperm competition games: a general approach to risk assessment.

We investigate game theory models of ejaculate expenditure by males in a species where there is a risk probability, q, that females will mate twice before laying a given set of eggs. With frequency 1-q females mate just once; then males optimally ejaculate an arbitrary minimum amount of sperm. The paper extends the analysis of Parker et al. (1997) in which males have limited information about the three risk states of the female: 0 (virgins which will mate now but not again), 1 (virgins which will mate now and then once more), and 2 (once-mated females who will mate now but not again). We derive a general structure for finding ESSs under imperfect information about states, and examine two special cases in which males know the overall risk probability q, but have imperfect knowledge of the states (0, 1, 2). In Case 1, males cannot distinguish between states 0 and 1 but have limited information about state 2. As their information increases, so does the difference in sperm allocation, with more going to females assessed as mated (state 2) than to females assessed as virgins (0, 1). This difference decreases with q in a species, but the average ejaculate expenditure increases with q. Even for small amounts of information, the behaviour as q-->0 is different from that predicted for zero information. In Case 2, males have perfect information about state 2, but limited information about states 0, 1. This has a major effect if q is small: males effectively behave as if they had perfect information by giving equal amounts of sperm to females assessed as 1 as those assessed to be 2, while giving a minimum amount to females assessed as 0. Ejaculate expenditures generally increase with the overall species-level risk q. A result common to both cases is that even a small amount of information allows more strategic choice than the zero information case and hence qualitatively different behaviour.