Superrational decision making is a type of rational decision making in which the players cooperate in a one-shot prisoner's dilemma without coordination, punishment, or magical thinking.

The idea is that when playing a symmetric prisoner's dilemma, one assumes that there is a unique solution to the mathematical problem of the optimal strategy, that this solution will be found and played by all superrational players, and that assuming the players are perfectly correlated, you maximize your utility.

The result in a one shot prisoner's dilemma is that two superrational players cooperate with each other, as opposed to two Nash-rational (or economically rational) players who defect.

A superrational player playing a Nash-rational economist will defect, and in general, in the absence of other superrational players, will play according to the Nash-rational strategy. It is only when there is a community of superrational players that one finds new types of rational behavior.

I have two closely related questions about the literature on this:

  1. Douglas Hofstadter expounds this idea at great length in a series of Scientific American articles, reprinted in his collection: "Metamagical Themas", one of which is "Dilemmas for Superrational Thinkers, Leading Up to a Luring Lottery" (Scientific American, June 1983). I believe the idea, at least in its mathematically precise form, is original to him, and I credit him whenever I mention it.

Is the mathematically precise definition of superrationality in symmetric multi-player games due to him, or was it somewhere in the literature before?

  1. Do philosophers take this idea seriously? I have not seen any professional literature which uses this. I am not asking whether philosophers should take the idea seriously, because I think they should. I am asking whether they do and if anyone can point me to specific examples of this that can be found in the literature.

3 Answers 3



Is the mathematically precise definition of superrationality in symmetric multi-player games due to him, or was it somewhere in the literature before?

Not really. The idea had emerged several times in philosophy, economics and mathematics before Hofstadter wrote about it. Notably, Martin Gardener wrote about a puzzle involving the notion in Scientific American in the 70s, in the same column that Hofstadter ultimately took over and wrote about superrationality in.

A good place to start would be SEP's entry on Common Knowledge.

Do philosophers take this idea seriously?


Here's a quote from Hofstadter's Metamagical Themas (1985) where he provides a definition of superrationality:

You need to depend not just on their being rational, but on their depending on everyone else to be rational, and on their depending on everyone to depend on everyone to be rational - and so on. A group of reasoners in this relationship to each other I call superrational. Superrational thinkers, by recursive definition, include in their calculations the fact that they are in a group of superrational thinkers. (Chapter 30)

Prior to this, the idea appeared a number of times, dating back to Hume. See the linked SEP article for a discussion.

I will present two mentions from the economics literature.

In 2005, Thomas Schelling and Robert Aumann shared the Nobel prize in economics for "having enhanced our understanding of conflict and cooperation through game-theory analysis" (see the press release).

Schelling in particular can be credited with preempting Hofstadter's definition of superrationality in coordination games:

  • Thomas Schelling: There's a nice quote from The Strategy of Conflict (1960) that the linked SEP article uses:

    When a man loses his wife in a department store without any prior understanding on where to meet if they get separated, the chances are good that they will find each other. It is likely that each will think of some obvious place to meet, so obvious that each will be sure that it is "obvious" to both of them. One does not simply predict where the other will go, which is wherever the first predicts the second to predict the first to go, and so ad infinitum. Not "What would I do if I were she?" but "What would I do if I were she wondering what she would do if she were wondering what I would do if I were she … ?" (p. 54)

    Schelling ran a variety of experiments based around games like the one sketched above, and ultimately developed the idea of a focal point equilibrium:

    Most situations - perhaps every situation for people who are practiced in this kind of game - provide some clue for coordinating behavior, some focal point for each person's expectation of what the other expects him to expect to be expected to do. (p. 57).

  • Robert Aumann is the first person credited with a rigorous notion of common knowledge. He presents the graph reachability notion of common knowledge on partitions on information space, which is now the defacto defition for modern logicians working on Epistemic Logic. Here's the abstract of his paper Agreeing to Disagree (1976) (full text):

    Two people, 1 and 2, are said to have common knowledge of an event E if both know it, 1 knows that 2 knows it, 2 knows that 1 knows it, 1 knows that 2 knows that 1 knows it, and so on.

    Theorem: If two people have the same priors, and their posterios for an event A are common knowledge, then these posteriors are equal

  • Sorry, these are not superrationality. Superrationality requires at least the analysis of the Luring lottery, to show that the superrational answer for N players is to flip an N-sided dice to and send a postcard if it comes up "1". All the other things are philosophical blah blah blah with no precise counterpart, and no essential modification of economical game-theoretic reasoning required.
    – Ron Maimon
    Aug 2, 2012 at 8:18
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    I quoted Hofstadter's definition of superrationality. I don't really see how to distinguish Hofstadter's definition as not "philosophical blah blah". It looks very similar to Schelling and Lewis' definitions. The Luring Lottery is quite similar to Schelling's games involving focal points, with the exception that Hofstadter's proposed focal point could not be observed empirically... although admittedly a contest in Scientific American is not a reliable instrument for doing behavioral economics.
    – Matt W-D
    Aug 2, 2012 at 19:53
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    Ok, I looked at the focal point, and it is completely unrelated, so I should downvote, but I won't because you are sincere in confusing the two concepts. The Schelling fellow does not predict cooperation in one-shot prisoner's dilemma, and his theory is to explain how to coordinate without communication, which is only vaguely related to the Hofstadter idea. Hofstadter's thing is mathematically precise-- I can tell you the superrational strategy in any symmetric game you dream up.
    – Ron Maimon
    Aug 10, 2012 at 1:13
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    That sounds like your own interpretation. I wouldn't call it plagiarism. It neglects the ideas Hofstadter had about rationality, coordination, and aggregate payoff. It speaks to the precision of Hofstadter's concept if you must resort to interpretation in order to provide a definition. Finding an adequate definition in Hofstadter is a matter of scholarship - if you can't find a direct quote then this just boils down to a matter of "he-said, she-said". Perhaps Hofstadter's vagary is part of the reason he's neglected in the mainstream game theory literature.
    – Matt W-D
    Aug 20, 2012 at 14:15
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    -1:It is not interpretation! This is exactly what Hoftstadter writes, it is exactly what he means, and there is absolutely no confusion here. Hofstandter never talks about "aggregate payoff", the concept never comes up! It is indeed true that the superrational strategy maximizes aggregate payoff in symmetric prisoner's dilemma (since the aggregate payoff divided by the number of players is the average payoff in a symmetric game), but this is not the definition. Hofstadter isn't vague. The reason he is neglected is because he is right, and philosophers prefer political clowns who are wrong.
    – Ron Maimon
    Sep 2, 2012 at 13:57

If your action forces other players to behave the same way, you are not playing a real game. you are playing a one-player game, a decision problem. The game theoretic fallacy Hofstadter makes is not new. An extensive discussion of the "symmetry fallacy" can be found in Ken Binmore's Game Theory and the Social Contract, Vol. 1: Playing Fair in Chapter 3.

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    It is not forcing anything, rather it is saying that your decision is correlated with the other player because you are both finding the unique answer to a well defined problem. Assuming it is well defined and there is a unique answer is what forces the answers to be the same, not any causal mechanism. This is not a fallacy, no matter what Ken Binmore says, it is a point of view that was just missed by philosophers and economists both, because it is essentially religious. I don't downvote new users, and I also cannot really participate on this site anymore, unfortunately.
    – Ron Maimon
    Dec 13, 2012 at 0:22
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    Of course it has been missed--- you just missed it! The "unique answer" is only to defect if you assume the answer is uncorrelated between the two players, that it is a "random variable", when it is a result of a procedure of deciding what to do, and is far from random. If you assume that the result of "figuring out what to do" is unique and perfectly correlated, it is just as obviously to cooperate. You are simply wrong, and this is why it is important to explain Hofstadters position, because people are wrong, and continue to say wrong things even after being patiently corrected.
    – Ron Maimon
    Dec 22, 2012 at 8:57
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    That is plain nonsense. Both defecting is the only correlated equilibrium of the PD. The term correlation is only defined for random variables, so I'm not responsible for your abuse of language. Why don't you provide a formal argument insteadof just repeating dogma? Dec 22, 2012 at 9:16
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    The argument is trivially formal: when "rational" means "superrational", so that there is a unique self-consistent answer to all (symmetric) games, then for the prisoner's dilemma, the unique superrational strategy is the one that maximizes (either) player's utility assuming all players play it. That's CC. This is not a standard equilibrium, it is the new concept of "Hofstadter equilibrium". The term "random variable" is fine, you are right, sorry. But you are missing the central point, that the superrationality is equally self consistent, and much better deserves the name "rationality".
    – Ron Maimon
    Dec 25, 2012 at 4:56
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    Your statement that it is a "fallacy" is defective. It is not a fallacy, it is manifestly fine, and it is manifestly formal. The thing that is amazing is that there is a whole community of people who blindly collectively reject this. This is simply making a formal error, there are no two ways about it. Please read Hofstadter's description of superrationality, it is cogent and clear, and if you like, you can also read my description of how to extend it to asymmetric games. Unfortunately, behaving according to the extension to asymmetric games prevents me from participating on this website.
    – Ron Maimon
    Dec 25, 2012 at 4:59

Actually, Hofstadter's notion of superrationality is closely related to evidential decision theory, where the action you take is one such that, conditional on you having taken that action, your expected utility is highest. If you believe your action in the Prisoner's Dilemma is highly correlated with that of your opponent, then your expected utility is higher conditional on you cooperating than conditional on your defecting, so evidential decision theory would say that you should cooperate. In contrast, under causal decision theory, you should still defect even if you believe the actions are correlated, because there is no causal effect of your action on the other's.

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