A two-step game model of female mate preference and paternal care is examined, with a particular focus on the case of two females and two males. In a mating season, females choose their mates, and in the following breeding season males invest in paternal care, knowing the likelihood of their paternity in chicks. If parental ability is the same between individuals of each sex, the evolutionarily stable mating pattern is always monogamy. If fe males differ in fecundity and males differ in paternal care capacity, monogamy with assortative mating is likely to be evolutionarily stable. If the male cost function increases at a strongly accelerating rate, however, polyandry is evolutionarily stable when the difference of female fecundity is very large, but the game may have no evolutionarily stable state when the difference of female fecundity is small. The care graph (in which females are connected to males giving paternal care to their chicks) is often much simpler than the mating graph (in which females are connected to males they accepted). To be exact, no 'loop' should be included in the evolutionarily stable care graph for the general case of n females and m males. This prediction is in accord with the observed prevalence of social monogamy in spite of genetic promiscuity among altricial birds.
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