"A large population mate choice game"Ramsey, David MarkThis talk presents a game-theoretical model of mate choice. At the beginning of the breeding season each member of a large population searches for a mate of the opposite sex. Mutual acceptance is required for a partnership to be formed. On forming a partnership, a male and a female leave the pool of searchers. The length of time allowed for search is assumed to be finite. Given that an individual finds a partner, then his/her reward from search is assumed to be the value of his/her partner. The sex ratio is equal to one (i.e. there are the same number of males and females) and the distribution of the value of an individual is assumed to be discrete and independent of sex. The possible values of mates are v_1,v_2,… ,v_n, where v_1>v_2>?>v_n>0. An individual of value v_i is said to be of type i. If an individual does not find a partner, then his/her reward from search is assumed to be zero. It follows that the game is symmetric with respect to sex. The rate at which prospective partners are found is assumed to be non-decreasing in the proportion of individuals still searching for a partner, i.e. as time progresses the rate at which prospective partners are found is non-increasing. At one extreme of such a spectrum, the rate at which prospective partners are found is assumed to be proportional to the fraction of individuals still searching for a mate (the random mixing model). At the other end of this spectrum, the rate at which prospective partners are found is assumed to be constant (the singles bar model). When a searcher meets a prospective partner, then the value of this partner is taken from the distribution of the value of searchers in the current mating pool. Hence, the pressure on an individual to accept a partner results from the risk of not finding a partner (which increases as time progresses) and the fact that the pool of potential partners tends to become less attractive on average as times passes. We look for a symmetric equilibrium (according to sex) at which the strategy used by an individual only depends on his/her value (i.e. type). In the case where the reward of an individual is discounted according to the time spent searching, it has been shown that multiple equilibria can exist. This talk shows that when the reward of searchers is not discounted, then there exists a unique symmetric equilibrium. Given the expected reward of a type 1 searcher at this equilibrium, then the equilibrium can be derived by an inductive procedure. Based on this, a value induction procedure is defined to approximate the equilibrium. |
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