From Behavioral Ecology Online
Mating games: the evolution of human mating transaction
This is but one of several models that the authors have expressed as mathematical equations. This is the type of monstrosity that results when psychologists actually learn to do math: Bonkers Math.
Model 2: two females bidding for one male (1 × 2 model)
In our second model, there are again three interacting potential mates; however, this time our players are two females and one male. As will be further discussed below, this model is rarely expected to occur in typical human mating systems, except in cases in which there is a female-biased sex ratio or marked social inequality leading to hypergyny. The 1 × 2 model is solved by examining the outcome of an opportune male assessing the bids of the two highest valued females available to him. The bidding females have genetic qualities qf and the male has genetic quality qm. We again will specify that the “primed” variables refer to the losing female. As in all previous models, the male can generate and invest a given amount of resources, Rm, in either female at a cost, zRm, to himself. As before, a male choosing to invest amount Rm of his resources in a female has the potential to increase offspring success by amount vRm. In addition to this, the females possess amounts of resource Rf and R respectively, all of which are also available to invest in the male’s offspring.
In the 1 × 2 model, the resources offered to the winning female will be determined by a bidding contest between the two females. Here, each female will reduce the magnitude of the resources demanded until the losing female can no longer afford to accept a lower resource incentive from the male. As was the case with our 2 × 1 model, it is not necessary for the females to know each other in order for such a bidding game to take place, although each is necessarily aware of the presence of the other. This information is available to the females either by monitoring the behavior of the male or by actually empirically observing the male with the counter-bidder.
The fitness of the first (winning) female if she mates with the male is equal to and that of the second (losing) female if she were to mate with the male is equal to If the second female instead leaves the male, her expected fitness is The parameter y describes the female’s probability of finding another mate should she leave.
Thus, to predict who the winning female will be, we must first determine the minimum amount of resources, Rm, that the male can provide that will cause the second female to break even. We can obtain this amount by equating Expressions 11 and 12 and solving for Rm. In doing so, we obtain Next, we substitute the latter minimum resource incentive of the second female into the expression for the male’s fitness if he chooses to mate with her, that is, to obtain The first female must be willing to reduce her resource demand enough to make the male’s fitness for mating with her equal to his fitness as given by Expression 15. The male’s fitness for mating with the first female is Thus, we find the resource incentive given by the winning male by equating Expressions 15 and 16 and solving for Rm. This resource incentive is equal to… Aye, yai, yai! Wow! What does this have to do with REAL PEOPLE!