The second theory is similar and is known as "evolutionary neuroandrogenic (ENA) theory of male aggression".   Testosterone and other androgens have evolved to masculinize a brain in order to be competitive even to the point of risking harm to the person and others. By doing so, individuals with masculinized brains as a result of pre-natal and adult life testosterone and androgens enhance their resource acquiring abilities in order to survive, attract and copulate with mates as much as possible.  The masculinization of the brain is not just mediated by testosterone levels at the adult stage, but also testosterone exposure in the womb as a fetus. Higher pre-natal testosterone indicated by a low digit ratio as well as adult testosterone levels increased risk of fouls or aggression among male players in a soccer game.  Studies have also found higher pre-natal testosterone or lower digit ratio to be correlated with higher aggression in males.     
Another point I’d like to make for people worried about a link between high testosterone and prostate cancer is that it just doesn’t make sense. Prostate cancer becomes more prevalent in men as they age, and that’s also when their testosterone levels decline. We almost never see it in men in their peak testosterone years, in their 20s for instance. We know from autopsy studies that 8% of men in their 20s already have tiny prostate cancers, so if testosterone really made prostate cancer grow so rapidly — we used to talk about it like it was pouring gasoline on a fire — we should see some appreciable rate of prostate cancer in men in their 20s. We don’t. So, I’m no longer worried that giving testosterone to men will make their hidden cancer grow, because I’m convinced that it doesn’t happen.