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SuperrationalityThe concept of superrationality is discussed in Douglas Hofstadter's book "Metamagical Themas". Superrationality is based on the idea that two perfect logicians will come up with the same, correct, answer to a logical or mathematical problem. For example, if two persons are both good at arithmetic, and both have been given the same complicated sum to do, it can be predicted that both will get the same answer before it is known. Applying superrationality to the prisoner's dilemma provides an explanation of how cooperation can be a rational choice even in a one-shot game. If two superrational people are playing the Prisoner's Dilemma, they will realise that because the situation is symmetrical, they will both eventually come up with the same choice. Either they will both choose to cooperate, or they will both choose to defect. Because they will be better off if they both choose to cooperate, that is what they will do. For simplicity, the above argument ignores the possibility that the best choice could be to flip a coin; or, more generally, to cooperate with probability p and defect otherwise. In the Prisoner's Dilemma, it turns out that it is still best to cooperate with probability 1 (because the average payoff when one player cooperates and the other defects is less than when both cooperate). In other situations, though, using a randomising device can be essential. One example discussed by Hofstadter is the plutonia dilemma: an eccentric trillionaire gathers 20 people together, and tells them that if one and only one of them sends him a telegram by noon the next day, that person will receive a billion dollars. If he receives more than one telegram, or none at all, no one will get any money, and cooperation between players is forbidden. In this situation, the superrational thing to do is to send a telegram with probability 1/20.
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