The concept of super-rational thinking is explained by Douglas Hofstadter in his book Metamagical Themas. It provides you with a logical reason to cooperate in a one-shot game of the Prisoner's Dilemma, even though whatever the other person does you would be better off defecting.

The basic idea is that super-rational thinkers recognise that two reasoning beings will come up with the same, correct, answers to logical or mathematical problems. For example, if you and your friend are both good at arithmetic, and you both have the same complicated sum to do, you can tell that you'll both get the same answer before you know what the answer is.

Now, if you're applying super-rational thinking to the Prisoner's Dilemma, you reason that the correct answer must be either "cooperate" or "defect", but whichever it is, the other player will do the same. Because you're better off if you both cooperate than if you both defect, you choose to cooperate. The other player also cooperates, having followed the same line of reasoning, and you both get the reward for cooperation.

Actually, the above paragraph simplified matters by ignoring the possibility that the correct answer might be to cooperate with probability p and defect otherwise. However, the Prisoner's Dilemma is set up so that if the players get locked into out-of-phase alternation, one cooperating and the other defecting, taking turns, they'll do worse than if they both keep cooperating. Hence it turns out that the best strategy in a one-shot game is to cooperate with probability 1.

Super-rational thinking can be applied to any dilemma which is symmetrical with respect to the participants. Hofstadter explained how to apply it to the Plutonia Dilemma, where an eccentric billionaire tells 20 people that if just one of them sends him a telegram within the next day, that person will win a billion dollars, but if more than one person sends a telegram, or no-one sends a telegram, then no-one will win anything. (This is a case where you have to choose randomly.)