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.)