However, to stick with your Prisoner's Dilemma theme: Part of the point of the feminist idea of "patriarchy" is that the deal that women get out of the traditional arrangement is not a nice strategy like tit-for-tat. The strategy that patriarchy teaches men to follow in their relations with women is more like an extortionate strategy, where the expected penalties for non-cooperation are much greater for women. ("Men are afraid women will laugh at them; women are afraid men will kill them.")

Men are also afraid of being killed by men. US statistics here for which sex has more to fear from men. "The battle of the sexes" is not a two player game; I think analogies to the prisoner's dilemma are misleading.

One thing I wonder about genetic essentialism on sexual behavior is how many generations it would take for moral changes at the social level to be driven into the genome. Here's a nasty, nasty thought-experiment: Suppose there is (or can be) genetic variation in the tendency of men to rape women. In a society where rape victims are made to bear their assailants' children, "rape genes" could be quite favored by evolution. In a society where rape victims reliably get abortions, and rapists are castrated or executed, "rape genes" would be extinguished — and a mutation that made rape less likely would be favored.

If you replace "rape" by more general antisocial behavior, there is some discussion about whether this has been happening in Pinker's "Angels."