we're playing some one random
We need to do both the calculations for "they act independently" (Assume they choose the Nash equilibrium, if we don't know anything else about their decision-making)
Why do we assume that?
In order to figure out our best move, we need to something about how the other person will move. It's generally a good idea to assume your opponent is indeed rational and utility maximizing. The Nash equilibrium strategy is the one that is stable such that neither side can do better by switching, which gives it a great deal of stablity. As such, it's often a good model for what people will do.
In this case, the full machinery isn't necessary. Defect strictly dominates Cooperate in every choice, so barring special considerations like TDT, it's what anyone with a modicum of sense will do.
Update: This post has also been superseded - new comments belong in the latest thread.
The second thread has now also exceeded 500 comments, so after 42 chapters of MoR it's time for a new thread.
From the first thread: