Neither of those have reproductive benefits, but the competitive disposition can show up in anything a man touches. Men even create arenas in which they can compete. Women are almost absent, except for the category of crocheting. In Darwinian terms, it begins with the differences in sexual reproduction. When males compete for a mate, there is an awareness of status and a recognition of where they stand in the pack.
Women view competition more as a spectator sport.
Women want men who will give them resources, but they don't really engage in the competition themselves. Of course, the news for males is not all good. That's why you find more males with extreme intelligence who are also extreme criminals.