Yes, cooperation wins. Well, as long as no one cheats and manages to "free ride" on the backs of cooperators. Once that happens, cooperation collapses back to selfishness. This ground has been plowed many times. One of the greatest mysteries of Biology continues to be the existence of human altruism. That is, how can a gene that encodes for "die for your country" behavior, survive that organism's... death for its country. The better such behavior works, the more likely the cooperator dies and leaves no children.
And yet, we all know that altruism works, we just don't know how yet.