No.
Males of polygamous species that win mating battles have the opportunity to mate with several females, significantly increasing their genetic contribution to the next generation. Thus, natural selection promotes the development of tendencies in males to engage in such fights and qualities that help them win.
At the same time, a female's reproductive success does not depend on the number of males she has mated with.
In any case, she will produce roughly the same amount of offspring. Although females can also participate in fights, this occurs for different reasons, such as competition for food, defending their territory, or protecting their young.
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