Females may not always be the choosy ones. Sometimes females compete and males choose. For example, in the bark louse, females grapple with each other and sing at the tops of their lungs to impress males who might eventually condescend to pay attention.
The extent to which males and females compete and choose is directly affected (although not necessarily determined) by the mating system. Several different mating systems are quite common: monogamy, polygyny, polyandry and promiscuity. Only when one sex competes for many mates and risks getting none ­ as in polygyny and polyandry ­ does it pay for that sex to invest in fancy display.
However, these mating systems are more useful as rough classifications than as rigid divisions. For instance, until recently, many species of birds were thought to be strictly monogamous, choosing a single mate for life. But then DNA analyses came along, and showed that even in these most romantic of systems, quite a bit of cheating goes on.