Tuesday, August 10, 2010

Does P != NP and why do some people not understand the point of making bets

Earlier this week, an HP Labs researcher Vinay Deolalikar has announced that he found the Holy Grail of computer science: a proof that P does not equal NP. ("P versus NP" is a problem about computational difficulty of solving problems. See here.) Deolalikar's paper has only been out there for about two days, so we don't know for sure if his proof is actually correct (and if not, whether it's fixable). After skimming it, computer scientist Scott Aronson had this to say:
In fact, I could think of only one mechanism to communicate my hunch about Deolalikar’s paper in a way that everyone would agree is (more than) fair to him, without having to invest the hard work to back my hunch up. And thus I hereby announce the following offer:

If Vinay Deolalikar is awarded the $1,000,000 Clay Millennium Prize for his proof of P≠NP, then I, Scott Aaronson, will personally supplement his prize by the amount of $200,000.

I’m dead serious—and I can afford it about as well as you’d think I can.
Maybe it's just me--but I think Aronson's statement is perfectly clear and incredibly informative (in fact, to me, it is more informative than any non-commital statement made by someone who has actually analyzed the paper in full detail could ever be, because I'm completely unqualified to judge the merits). However, a reaction of one of Aronson's commenters is I think symptomatic of a curious attitude of abhorrence and incomprehension that a lot of people display towards betting on one's beliefs. The commenter calls Aronson's post "nasty" and says:
I do not see why you should “win” a game that you do not participate; you “win” a lot of games in which you do participate. And if you really really want to express your honest opinion without being nasty, how about: “In my opinion, it is very unlikely that the argument will stand”.
How is betting a very large amount of money on your belief "not participating?" And how on Earth is saying "In my opinion it is very unlikely that X" the same thing as betting $200,000 on the fact that not-X?

The fact is that serious bets convey large amounts of information. The reason for this is twofold. First, there are two dimensions to every opinion: its meaning and its strength. Its meaning is simply what a given opinion says. Its strength is the likelihood that the person holding an opinion (implicitly) assigns to the opinion being true. As it is, people tend to egregiously misreport the strength of their opinions in non-commital statements. That's why commitment, monetary or otherwise, works.

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