Superrationality
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The 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 a possible 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. According to Hofstadter this assumes that each person knows that the other will act in an equally superrational manner. This, though, is not enough to generate the cooperative outcome. Standard game theory also assumes common knowledge of rationality yet yields the non cooperative solution (if the other player cooperates my payoff is higher from defection). Arguably, superrationality implies a kind of magical thinking in which each player supposes that his decision to cooperate will cause the other player to cooperate, despite the fact that no communication occurs between the players.
For simplicity, the foregoing account of superrationality ignores mixed strategies: 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 contacts 20 people, and tells them that if one and only one of them sends him a telegram (assumed to cost nothing) 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.
Notice though that this is not the solution under standard assumptions. A conventional game-theoretical analysis would result in all players sending in telegrams and therefore receiving nothing. This is because sending the telegram is the dominant strategy; if an individual player sends a telegram he has a chance of receiving money, but if he sends no telegram he cannot get anything.
