Hofstadter’s Superrationality
We have discovered that rational agents often arrive at equilibria that are harmful to everyone involved; a counterintuitive fact that we formalized with the Prisoner’s Dilemma, Nash Equilibrium, Pareto Optimality and Social Optimality. As an alternative to rationality, Douglas Hofstadter (the author of Gödel, Escher, Bach) invented the concept of superrationality. A rational agent is defined as one that attempts to maximize its payoff, given perfect knowledge of the game and the assumption that the other players are also rational. A superrational agent, on the other hand, is defined in the exact same way but with the added condition that it knows that all superrational agents would behave the same way in any given situation. This has the interesting result of inducing cooperation for the prisoner’s dilemma.
Superrationality is demonstrated in an interesting game. 20 random people across the country receive letters promising one billion dollars to whichever person sends back a postcard requesting it; but only if exactly one person chooses to do so. Each player knows that there are 20 other people, but cannot correspond with them. A rational agent, of course, would send the postcard. Not sending the postcard is guaranteed to result in 0 payoff, so sending the postcard is a dominant strategy. But then the Nash Equilibrium is obviously for everyone to send a postcard and for no one to profit.
A superrational agent will choose a strategy such that, were everyone to follow it, its own expected profit would be maximized. The strategy must be a probabilistic mix of sending and not sending a postcard. Call the probability of sending the postcard p. The expected profit is the probability of being the only one to send a postcard, p (1-p)^19, times one billion dollars. This is maximized when p = 1/20. So a superrational agent would roll a twenty sided die, and send in the postcard only if the result is 20. By the way, the expected profit for an agent in this scenario is about 0.019 billion dollars = 19 million dollars.
A similar contest was actually carried out. The readers of the magazine in which this concept was presented were invited to send in letters with any whole number on them. The prize was to be one million dollars divided by the sum of all submitted numbers, and it was to be paid to someone randomly chosen from the participants, weighted be their number. So a higher number means a greater chance of winning, but lower payoff. The expected payoff (when the total of other submissions = m) is one million times 1/(m+n) times n/(m+n), which is maximized at m=n. So a rational agent tries to match the total of the other submissions. One can see that as long as there are more than two people, there is no Nash equilibrium; unilateral updates drive the strategies to infinity. A superrational agent, on the other hand, would choose the following familiar strategy. They would estimate the number of participants (P), and submit the number “1” with a probability of 1/P.
In the actual contest, the submissions included incalculably high numbers, demonstrating the runaway dynamics of rational strategies. It was impossible to total, much less sample from the submissions, but the prize would have been unimaginably small; certainly less than 10^(-90) cents.
Why is the concept of superrationality esoteric instead of standard? What is so far fetched about the assumption of symmetric behavior? Given the profits of superrationality, why aren’t there more instances of it in the real world?