Tom Crispin: The utility-theoretic answer would be that all of the randomness can be wrapped up into a single number, taking account not merely of the expected value in money units but such things as the player's attitude to risk, which depends on the scatter of the distribution. It can also wrap up a player's ignorance (modelled as prior probabilities) about the other player's utility function.

For that to be useful, though, you have to be a utility-theoretic decision-maker in possession of a prior distribution over other people's decision-making processes (including processes such as this one). If you are, then you can collapse the payoff matrix by determining a probability distribution for your opponent's choices and arriving at a single number for each of your choices. No more Prisoners' Dilemma.

I suspect (but do not have a proof) that adequately formalising the self-referential arguments involved will lead to a contradiction.