While men dominate in industry and government, women dominate in personal arguments. The man who is comfortable offending women will get fewer genes into the cooker, and have fewer offspring. The man who is stressed when a woman is upset, and concedes, placates, or avoids the arguments, will stay on better terms with women and will have more surviving offspring. BUT females can take food away from males and shove males aside to sit in the shade. So as in humans, males dominate in public standing, while females dominate in areas most important to their comfort and survival.
Males compensate females for romantic favors.
In our own society, men pay for the dates. Among the uninhibited Trobriand Islanders, separated from other cultures for tens of thousands of years. The custom implies that sexual intercourse, is a service rendered by the female to the male. Among chimpanzees, males may give females food to win their cooperation for clandestine matings. Among hanging flies, the males capture prey for the females in exchange for mating.