## What impact has Douglas Hofstadter's superrationality had in terms of philosophy?

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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.

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TL;DR:

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?

Yes.

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

1-1: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 – 2012-09-02T13:57:36.013

I found a reprint of the articles here, please read them and tell me this is "interpretation". I know what a reinterpretation is, and I know what plagiarism is.

– Ron Maimon – 2012-09-02T14:09:23.313

Quote: ... All it means is that all these heavy-duty rational thinkers are going to see that they are in a symmetric situation, so that whatever reason dictates to one, it will dictate to all. From that point on, the process is very simple. Which is better for an individual if it is a universal choice: C or D? That’s all. Actually, it’s not quite all, for I’ve swept one possibility under the rug: maybe throwing a die could be better than making a deterministic choice. Like Chris Morgan, you might think the best thing to do is to choose C with probability p and D with probability 1−p ... – Ron Maimon – 2012-09-02T14:11:53.277

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 – 2012-08-02T08:18:42.373

3I 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 – 2012-08-02T19:53:29.577

The Schelling reference might be relevant. The other ones are philosophical blah blah. Hofstadter's isn't because he can solve the Luring Lottery (it is possible to do Luring Lottery empirically--- simply do a prisoner's dilemma like game with CC payoff of $10 each, DD payoff of$0, and CD payoff of $1000/$1. In this case, I think that it is conceivable to see coin-flipping behavior in humans. The concept of "focal point" is similar, although DD is also a focal point of sorts in prisoner's dilemma, so I need to read Schelling before upvoting or accepting. – Ron Maimon – 2012-08-10T01:11:25.680

1Ok, 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 – 2012-08-10T01:13:55.520

@RonMaimon: I made my reply briefer. Can you edit your original post to contain a complete quote from Hofstadter defining superrationality, with a focus on aspects of it you consider essential? I don't see where he lays out a framework for systematically analyzing any arbitrary symmetric game. – Matt W-D – 2012-08-11T21:12:25.093

I can define it for you--- "a superrational strategy in a symmetric game is a mixed strategy (meaning the players can throw dice if they want to) that maximizes the expected payoff for any one of the superrational players, under the assumption that they all play the exact same strategy." This is the precise definition, it is not stated this way in Hofstadter, but it is blindingly obvious given what he writes, so much so that it would be plagiarism to claim this is not what he meant. This definition does not match the focal point, or anything else I have seen. – Ron Maimon – 2012-08-20T07:49:16.613

2That 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 – 2012-08-20T14:15:40.043

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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.

3It 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 – 2012-12-13T00:22:10.377

@RonMaimon This point has certainly not been missed. There is a unique solution to the prisoners dilemma that is compatible with both players being rational. Rationality forces the choices of both players to be the same: Both players defect. The argument that this leads to any form of correlation is flawed though. Constant random variables are automatically uncorrelated (and even independent). – Michael Greinecker – 2012-12-17T12:28:31.703

2Of 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 – 2012-12-22T08:57:15.590

1That 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? – Michael Greinecker – 2012-12-22T09:16:46.193

2The 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 – 2012-12-25T04:56:47.383

2Your 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 – 2012-12-25T04:59:15.847

The Harsanyi-selten equilibrium selection procedure does essentially select a unique solution for every game and prescribes defection in the prisoners dilemma. Nash has aready shown that every symmetric game has a symmetric Nash equilibrium in his thesis. So symmetry doesn't really help. There is a survey called "Rationality and Knowledge in Game Theory" by Dekel and Gul, in which they explicitely discuss the prisoners dilemma in terms of epistemic fundamentals and point out what goes wrong with the kind of argument that Hofstadter makes (and yes, I read his stuff). – Michael Greinecker – 2012-12-25T14:48:23.020

There is nothing wrong with Hofstadter's argument, it is just plain correct (arguably, but it is unarguably self-consistent). However, it is detested by the anti-religious factions that dominate game-theory. Hofstadter is just as self-consistent as Harsany-Selten-Nash rationality. The only limitation in Hofstadter is the requirement of symmetry, but I know how to extend it to asymmetric situations. The result is a rational religious ethics, along with a concept of a personal God. The problem with Nash rationality is that it has a unique solution, and it then doesn't use this fact to condition. – Ron Maimon – 2013-01-03T01:53:02.573

The main issue is this--- if I am maximizing my utility using an algorithm to select the best possible outcome, should I assume in advance that the opponent will use the exact same algorithm? In standard rationality, you assume the opponent will use the same algorithm, but that this algorithm will be an equilbrium finding a-la Nash. The result is that you defect in a prisoner's dilemma (the unique Nash equilibrium), and this is just plain foolish. You can equally self-consistently say that you will assume the sameness of algorithm first, and then find the algorithm to maximize the utility. – Ron Maimon – 2013-01-03T01:55:05.273

Both positions have equal claim to being the "rational" answer, and to isolate one of the two and say "only this choice is rational" is to really put down the Hofstadter way. It is perfectly rational to do what Hofstadter does. But you need to know how to extend to asymmetric games. In this case, the correct generalization is to assume there is a unique superrational extension of the algorithm for symmetric games, which is self-consistent for all games, and reduces to Hofstadter's solution in the symmetric case. Such an algorithm is equivalent to the assumption that the players all behave... – Ron Maimon – 2013-01-03T01:57:18.630

as if an all-knowing entity, with it's own desires and wishes, was telling them what to do. This is the religious ethics of monotheistic religions, and the desires of the superrational self-consistent strategy can be productively identified with the religious concept of the "will of God". The debate between the two conceptions of rationality is then the religious debate in disguise, and the Nash rationality is the position of the atheist, while the Hofstadter superrationality is the position of the monotheist (at least in logical positive form, where you eliminate the supernatural stuff). – Ron Maimon – 2013-01-03T01:59:03.063

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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.