Should females prefer to mate with low-quality males?

We analyse a model of mate choice when males differ in reproductive quality and provide care for their offspring. Females choose males on the basis of the success they will obtain from breeding with them and a male chooses his care time on the basis of his quality so as to maximise his long-term rate of reproductive success. We use this model to establish whether high-quality males should devote a longer period of care to their broods than low-quality males and whether females obtain greater reproductive success from mating with higher quality males. We give sufficient conditions for optimal care times to decrease with increasing male quality. When care times decrease, this does not necessarily mean that high-quality males are less valuable to the female because quality may more than compensate for the lack of care. We give a necessary and sufficient condition for high-quality males to be less valuable mates, and hence for females to prefer low-quality males. Females can prefer low-quality males if offspring produced and cared for by high-quality males do well even if care is short, and do not significantly benefit from additional care, while offspring produced and cared for by low-quality males do well only if they receive a long period of care.