On sperm competition games: incomplete fertilization risk and the equity paradox

Evolutionary game theory has been used to predict the effect on sperm expenditure of a trade-off between the value of a mating and the cost of its acquisition. In particular, G. A. Parker has predicted that 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 fertilization is fair or unfair. This prediction relies on the assumption that, in the absence of sperm competition, ejaculates would always contain enough sperm to ensure complete fertilization after mating. The slightest risk of incomplete fertilization, however, is enough to ensure that favoured males expend more than disfavoured males in the presence of sperm competition, unless the competition is perfectly fair. Divergence of expenditures increases with unfairness until unfairness reaches a critical value, beyond which a disfavoured male should no longer compete. The higher the fertilization risk, the lower the critical unfairness. All predictions are independent of the probability of mating first or second. Implications are discussed for the mechanisms that underlie sperm competition.     

Lanchester's attrition models and fights among social animals

Lanchester's models of attrition during warfare have servedas the basis for several predictions about conflicts betweengroups of animals. These models and their extensions describerates of mortality during battles as functions of the numberand fighting abilities of individuals in each group, allowinganalysis of the determinants of group strength and of the cumulativenumbers of casualties. We propose modifications to Lanchester'smodels to improve their applicability to social animals. Inparticular, we suggest that the per-capita mortality rate ofa group is a decreasing function of the fighting abilities ofits members, that the mortality rate is an increasing functionof the number of individuals in both groups, and that therewill often be diminishing returns for increasing numerical advantage.Models incorporating these assumptions predict that the abilityof social animals to win fights depends less on group size andmore on individual prowess than under Lanchester's originalmodels. We discuss how data on casualties can be used to distinguishamong alternative attrition models.     

Social Eavesdropping: A Game-Theoretic Analysis

Here we extend the classic Hawk-Dove model of animal conflict to allow for continuous variation in fighting strengths. Whereas the winner of a fight is chosen at random in the discrete game, in our continuous game, the winner of any fight is the stronger individual, and costs are higher for more evenly matched opponents. We identify the evolutionary stable strength threshold beyond which an animal should be prepared to engage in aggressive behaviour and show that this threshold increases with variance in fighting strength when the costs of aggression are insensitive to the level of strength asymmetry, but decreases with variance when the costs are sensitive to the level of asymmetry. In contrast to the classic discrete game, population-wide aggressive behaviour occurs only when the costs of fighting are zero. It is now known that animals can eavesdrop on the outcome of contests between neighbours and modify their behaviour towards observed winners and losers. We therefore further extend our model to allow for social eavesdropping within networks comprising three individuals. Whereas earlier work showed that eavesdropping increases the frequency of mutually aggressive contests in the discrete game by enhancing the value of victory, here we show that aggression thresholds in the continuous game are always higher with eavesdropping than without it: for sufficiently weak animals, avoiding the costs of challenging an observed winner over-rides the potential benefit of winning, so that eavesdropping reduces the frequency of aggressive encounters. Thus, even though strength is not directly observable, information is extracted from the variation in fighting ability that the classic Hawk-Dove game ignores.     

The effect of differential survivorship on the stability of reproductive queueing

Queues, in which individuals inherit resources in a predictable, temporally stable order, are widespread in animal social groups. We develop an analytic model to explore the effect of differential survivorship on the stability of a reproductive queue. We show that unless fighting for dominance is potentially fatal, future direct benefits are not alone sufficient to stabilize a queue of non-relatives under constant (age-independent) mortality rates, regardless of whether a dominant becomes an isolate or remains a dominant on the death of the first subordinate. In the absence of fatal fighting, stabilization of such a queue by future direct benefits alone requires either the dominant or the subordinate to have age-dependent mortality rates. Even when the queue is stabilized by present direct reproduction, however, the shape of the lifespan distribution can make a significant difference to the size of the required incentive. In contrast to non-relatives, queues of relatives can be stable without age-dependent mortality, so long as relatedness exceeds a critical value; however, age-dependent mortality can lower this critical value.     

The influence of contests on optimal clutch size: a game-theoretic model

We develop a game-theoretic model to predict the effect of size-dependent contest outcomes on optimal-clutch-size decisions. We consider the case where larger individuals develop from smaller clutches and, as adults, are advantaged in competition for limiting resources. The relationship between fitness and size thus depends on the sizes of other members of the population. We show that clutch-size optima are decreased by body-size-dependent contest outcomes, with larger effects when body size is most affected by clutch size, when prior resource ownership has less influence on contest outcome and when contests occur more frequently. We also show the existence of polymorphisms in clutch-size optima and that clutch-size driven changes in population density can, via an effect on the probability of host finding, further influence optimal clutch size. Our model is formulated to match the life history of a parasitoid wasp, in which clutch size affects offspring size and females engage in direct contests for host ownership, which larger females tend to win; we confirm that female-female competition is likely to influence clutch size in this species. However, the model is also relevant to clutch size in other taxa and supports recent suggestions concerning reproductive decisions in great tits.     

The Hawk—Dove game revisited: Effects of continuous variation in resource-holding potential on the frequency of escalation

Summary The classic Hawk—Dove game is extended to deal with continuous variation in resource-holding potential or RHP, when RHP is observable (via any sensory modality) but RHP difference is less than perfectly reliable as a predictor of the outcome of an escalated contest. The relationship between sensory and physical magnitudes of RHP is assumed to be governed by Fechner's psychophysical law, whose effect is that contestants interact as if they had perfect information about their relative RHP (as opposed to RHP difference). Thus, an animal is aggressive if its RHP exceeds a certain fraction, called its threshold, of its opponent's RHP and otherwise is non-aggressive; and the classic Hawk and Dove strategies correspond to zero and infinite thresholds, respectively. For RHPs drawn at random from an arbitrary Gamma distribution there is a unique evolutionarily stable strategy or ESS, which depends on a parameter measuring the reliability of RHP as a predictor of the outcome of a fight, on the ratio of the valueV of winning to the costC of losing (both measured in units of reproductive fitness) and on the mean µ and variance ² of the RHP distribution. In a population at this ESS, ifV/C < 1 then the threshold is 1 and there is no fighting. AsV/C increases beyond 1 to a second critical value , however, the threshold decreases steadily from 1 to 0 and remains 0 forV/C > ; is an increasing function of , but a decreasing function of ². That a lower variance of RHP can imply a lower escalation frequencyp is a novel insight of the analysis. The prediction is at first counterintuitive, because if the aggression threshold were fixed then larger variance would imply lowerp (dispersion effect of variance). When natural selection acts on the threshold, however, increasing the variance not only reduces the probability that an animal with larger RHP will be attacked by an animal with lower RHP at the existing threshold, but also reduces the expected costs of adopting that particular threshold, so that a mutant with a somewhat lower threshold can invade the population (selection effect of variance). Forp, the selection effect dominates toward the upper end of the interval 1 V/C .     

An escape from ‘the prisoner's dilemma‘

Conventional escapes from the paradox that noncooperation between two organisms may be rational, even when cooperation would yield a higher reward to each, are based on the mechanism of reciprocity; but an analytical model of foraging among oviposition sites reveals a more immediate rationale, namely, equivalence of selfishness and altruism. The resulting game is unconditionally the prisoner's dilemma if the players have perfect recognition; however, in the absence thereof and for three different parameter regimes, it yields either the prisoner's dilemma, or two evolutionarily stable strategies, or a unique cooperative ESS. Thus unrecognition can favor cooperation; and environments can exist in which cooperation persists, or even invades, without reciprocity. The results suggest that different mechanisms for cooperation may operate at different levels of neural complexity.     

Victory displays: a game-theoretic analysis

Two rationales have been proposed verbally for the functionof victory displays, which are performed by the winners of contestsbut not by the losers. The "advertising" rationale is that victorydisplays are attempts to communicate victory to other membersof a social group that do not pay attention to contests or cannototherwise identify the winner. The "browbeating" rationale isthat victory displays are attempts to decrease the probabilitythat the loser of a contest will initiate a future contest withthe same individual. We formally explore the logic of theserationales with game-theoretic models. The models show thatboth rationales are logically sound; however, all other thingsbeing equal, the intensity of victory displays will be highestthrough advertising in groups where the reproductive advantageof dominance is low and highest through browbeating in groupswhere the reproductive advantage of dominance is high.