Duration of courtship effort as a costly signal

We consider a male and a female in a courtship encounter over continuous time. Both parties pay participation costs per unit time. The game ends when either one or other of the parties quits or the female accepts the male as a mate. We assume that there is a binary variable which determines whether the male is a "good" or "bad" type from the female's point of view, according to either his condition or his willingness to care for the young after mating. This variable is not directly observable by the female, but has fitness consequences for her: she gets a positive fitness payoff from mating with a "good" male but a negative fitness payoff from mating with a "bad" male. We assume also that a "good" male has a higher ratio of fitness benefit from mating to fitness cost per unit time of courtship than a "bad" male. We show that, under suitable assumptions, there are evolutionarily stable equilibrium behaviours in which time-extended courtship takes place. A "good" male is willing to court for longer than a "bad" male; in this way the duration of a male's courtship signals his type, and acts as a costly handicap. By not being willing to mate immediately the female achieves a degree of screening because the posterior probability that the male is "good", conditional on his not having quit the game, increases with the duration of courtship.