Scientific betting markets

As expected, Scott Aronson's bet about the newly published proof that P does not equal NP spurred a heated discussion about the bet and scientific betting in general. Aronson's summary of the discussion runs as follows:

I was hoping for some enlightening discussion about the ethics of scientific betting in general, not more comments on the real or imagined motives behind my bet. However, the actual comments I woke up to have convinced me that the critics of my bet were right. In principle, the bet seemed like a good way to answer the flood of queries about Deolalikar’s paper, and then get back to enjoying my trip. In practice, however, the way people respond to the bet depends entirely on what they think about my motives. If you don’t care about my motives, or consider them benign, then the bet probably provided a useful piece of information or (if you already knew the information) a useful corrective to hype. If, on the other hand, you think I’m arrogant, a media whore, etc., then the bet served no purpose except to confirm your beliefs. Since people (at least taking the commenters on this blog as a representative sample, admittedly a questionable assumption!) generally wish to ascribe the worst motives to me, it follows that the bet was counterproductive.

I have no reason to question Aronson's motives and, as I have said, to me his opinion has been more informative than anything else anyone has said about Delalikar's paper. But that's just me, and Aronson can be right: given the average response of his readers, his bet was indeed counterproductive. This doesn't mean, however, that a scientific betting market would also be counterproductive. As the number of market participants grows, you cannot gain notoriety by joining it, so the motive of an average participant becomes purely to try and make money. The only way to make money on an opinion is to bet on it and be right. I see no reason why such market would not work (or work just as well as other markets work, at any rate).

One of the commenters, a computer scientist Gil Kalai, spells out his reasons to believe prediction markets wouldn't work in science:

In the economics setting of “rational expectations” there is an unknown state of the world on which agents have private information. The market prizes aggregate the private information in a subtle way and reveal the true state of the world.

The principal reason that this is not relevant to scientific questions is that a basic implicit assumption in all these models is that if all the private information of agents about the state of the world will be made public, this will suffice to determine the correct state of the world. (Say with extremely high probability). The behavior of agents in the market (or in other strategic situations) give a way for the private information to aggregate as if it was common knowledge.

But, of course, this is not relevant here. Even if we know the sincere private opinion of all scientists on a question like is factoring in P (and even their opinions about opinions of others, and whatever additional information that we want, and we can even give them time to reconsider their opinions based on pinions of others etc etc…) making this private information public is not sufficient (and usually does not help) to determine the “state of the world”.

Kalai is vague about what he means by "the true state of the world." In the context of Delalikar's proof, is the true state of the world whether or not the proof is correct, or the actual reasons why the proof is correct or not? If the latter, then it's true, a betting market can't help much. But if the former, then a betting market is actually more informative than "making all private information of participants public," for three reasons. First, a betting market allows for expressing beliefs that can't be supported by a rational argument ("I just have a strong hunch"). Second, betting markets convey information more efficiently than public revelation of private knowledge. The amount of all private knowledge relevant to a question as complex as "P versus NP" is so huge that no individual can possibly process all of it. Market prices compress all that information into signals that are much easier to process. An third, a scientific prediction market would offer information about open scientific questions in the form that would be accessible to everyone, not only scientists. As I've said before, to me as a theoretical computer science ignoramus, Aronson's bet is actually much more informative that a detailed refutation of Delalikar's proof, because I have no qualifications to judge the technical details. I have no idea why a proof that shows 3SAT is not in P should fail for problems like 2SAT and XOR-SAT. In fact, I barely know what 3SAT, 2SAT and XOR-SAT even mean. But I do know the exact meaning of $200,000.