Superrationality

The concept of superrationality is discussed in Douglas Hofstadter's book "Metamagical Themas". Superrationality is a type of rational decision making which is different than the usual game-theoretic one, since a superrational player playing against a superrational opponent in a prisoner's dilemma will cooperate while a game-theoretically rational player will defect.

Prisoner's Dilemma
The idea behind superrationality is that two logical thinkers analyzing the same problem will come up with the same, correct, answer. 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 the sum is known. In arithmetic, knowing that the two answers are going to be the same doesn't change the value of the sum, but in game theory, knowing that the answer will be the same might change the answer itself.

In a prisoner's dilemma, each player can either cooperate (C) or defect (D). The players choose without knowing what the other is going to do. If both cooperate, each get 100 dollars. If they both defect, they both get 1 dollar. If one cooperates and the other defects, the defecting player gets 101 dollars, while the cooperating player gets nothing.

One valid way for the players to reason is as follows: whatever the other player does, my payoff is increased by defecting, even if only by one dollar. So the rational thing to do is to defect. This defines game-theoretic rationality, and two game-theoretic rational players will both defect and receive one dollar.

Superrationality is an alternative way to reason. First, it is assumed that the answer to a symmetric problem will be the same for all the superrational players. Then the sameness is taken into account before knowing what the strategy will be. The strategy is found by maximizing the payoff to each player, assuming that they all use the same strategy. Since the superrational player knows that the other superrational player will do the same thing, whatever that might be, there are only two choices for two superrational players. Both will cooperate or both will defect depending on the value of the superrational answer. The two superrational players will both cooperate, since this answer maximizes the payoff to each one.

Note that a superrational player playing against a game-theoretic rational player will defect, since the strategy only assumes that the superrational players will agree. A superrational player playing against a player of uncertain superrationality will sometimes defect and sometimes cooperate.

If the two players are only superrational with probability p, and with probability 1-p they are game-theoretic rational and therefore defect, the result is as follows. Assuming that the superrational strategy is to cooperate, the expected payoff is 100p. Assuming that the superrational strategy is to defect, the expected payoff is 1. So as long as p>.01 the superrational strategy will be to cooperate.

This means that when the temptation to defect is very small, the superrational strategy is to cooperate even in the presence of a large fraction of non-superrational opponents.

Although standard game theory assumes common knowledge of rationality, it does so in a different way. The game theoretic analysis maximizes payoffs by allowing each player to change strategies independently of the others, even though in the end, it assumes that the answer in a symmetric game will be the same for all. This is the definition of a game theoretic Nash equilibrium, which defines a stable strategy as one where no player can improve the payoffs by unilaterally changing course. The superrational equilibrium is one which maximizes payoffs where all the players strategies are forced to be the same before the maximization step.

Some argue that 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 there is no communication. Hofstadter points out that the concept of "choice" doesn't apply when the player's goal is to figure something out, and that the decision does not cause the other player to cooperate, but rather same logic leads to same answer independent of communication or cause and effect. This debate is over whether it is reasonable for human beings to act in a superrational manner, not over what superrationality means.

There is no agreed upon extension of the concept of superrationality to asymmetric games.

Probabilistic Strategies
For simplicity, the foregoing account of superrationality ignored mixed strategies: the possibility that the best choice could be to flip a coin, or more generally to choose different outcomes with some probability. In the Prisoner's Dilemma, it is superrational to cooperate with probability 1 even when mixed strategies are admitted, 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 platonia 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 (if it is known that all 20 are superrational) is to send a telegram with probability p=1/20, which maximizes the probability that exactly one telegram is received.

Notice though that this is not the solution in a conventional game-theoretical analysis. Twenty game-theoretically rational players would each send in a telegram 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.