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I'm not sure if this is appropriate for Math Overflow. I was curious about whether any attempts have been made to set up betting markets for getting mathematicians to put wagers on what they think about the possible outcomes of various unsolved problems. I've heard that betting markets generally have a good track record of aggregating the views of people and pointing to reasonably correct results, as long as there's enough participation and people have something real at stake (see prediction market). Also, when asked to actually make bets, people usually tend to examine the claims more critically.

Although betting with real money is tricky, this can be circumvented using other "currencies" such as (to take a random example) Math Overflow points.

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    $\begingroup$ And who will pay Fermat his 1000 gold coins plus interest for his bet on the FLT? ;-) $\endgroup$
    – Johannes Hahn
    Commented May 17, 2010 at 22:40
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    $\begingroup$ But would circumventing it with other currencies preserve the integrity of the bets? I'm not sure if people care about MO points enough for this to work =). Also, my short experience with Intrade taught me that the majority of the pressure on a stock comes from the lightning-quick responses to very public information, rather than from particularly wise bettors or insiders. One obvious example was the bet on who Paterson would appoint to replace Clinton in 2008 (it wasn't Kennedy). $\endgroup$
    – Sean Rostami
    Commented May 17, 2010 at 23:45
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    $\begingroup$ I know of no such attempts. I do want to point out, though, that mathematicians do have something real at stake when making predictions: status. It's as real as money, and in some ways more so. Anyway, see Robin Hanson's blog for lots of provocative discussion of prediction markets and related problems: overcomingbias.com/tag/prediction-markets $\endgroup$
    – JSE
    Commented May 18, 2010 at 0:28
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    $\begingroup$ True story: Wiles gave 3 lectures in Cambridge in 1993 and in the 3rd one he announced a proof of FLT. After the first one my friend Andy Plater (RIP Andy) tried to place a bet on FLT being proven in the near future (in the UK, where you can find a betting shop on any high street), but he was told that all bets were off concerning FLT at that point. $\endgroup$
    – Kevin Buzzard
    Commented May 18, 2010 at 6:41
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    $\begingroup$ An interesting fact is the efficiency theorems about prediction markets don't hold for mathematical bets. (Whether they still work in practice is another matter entirely.) The reason is that Bayesian agents are too strong a computational model of mathematicians; the axioms of Bayesian probability theory require giving a probability of 1 to logical truths, and a probability of 0 to logical falsehoods. This means that ideal decision-theoretic agents are logically omniscient: they already know all mathematical truths, and so there is no disagreement among them, and hence, no market! $\endgroup$
    – Neel Krishnaswami
    Commented Jul 2, 2010 at 16:26

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"I've heard that betting markets generally have a good track record of aggregating the views of people and pointing to reasonably correct results, as long as there's enough participation and people have something real at stake"

In my opinion, there is no reason to believe that prediction markets will be useful to aggregate information which is not there. For most mathematical and scientific questions people have no real information that aggregating together will say something useful.

It is not clear also how good are financial markets in predicting their own future behavior. (But it is a complicated question how to formally as it.)

It would be useful to look skeptically also into these claims about the good track record of prediction markets that the asker heard about.

Probably the question is more suitable to a "meta discussion".

Further discussion can be found in this post over Shtetl optimized following this one.

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The key feature of prediction markets is betting on something which may or may not happen within a day, a year, maximum 10 years (this one is the longest public bet with real money I heard of - see a few more here). I remember right before I left for the US in 1993, I saw 1:10 odds for "putsch within 10 years". This was two years after the first one, and I still can't believe I did not take these odds - the next attempt came within 3 months...

But seriously, since in math very little happens quickly, the money lost/gained will not influence anyone to bet. The best thing you can do is poll the experts, kind of like what CS people did in this famous article (dealing with P=?NP).

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I distinctly remember that during the early 1990's, there was some kind of prediction market where one of the items was whether Fermat's Last Theorem would be proved before some date (the year 2000 maybe?), as judged by a panel of mathematicians or something. Unfortunately I do not remember the details; I think that there were periodic postings to sci.math giving the current "going rate," and of course as confidence in the correctness of Wiles and Wiles-Taylor grew, the market gradually dried up. I haven't been able to confirm my memory with a Google Groups search, though.

Intrade.com does have a few science-related markets but I don't know if they have ever had mathematics-related markets.

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  • $\begingroup$ There was precedent for bets connected with FLT. In Arthur Porges' short story "The Devil and Simon Flagg", a mathematician bet his soul that the Devil couldn't prove Fermat's last theorem in twenty-four hours. $\endgroup$
    – Gerry Myerson
    Commented Apr 7, 2019 at 3:20
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Cryptocurrency smart contracts provide a framework by which people can bet on the solution to mathematics or computational problems. A cryptocurrency smart contract is a piece of software used on a cryptocurrency which can be used to transfer funds if the conditions of the contract are satisfied. For example, one can make a contract that says “Gil Kalai will give Andrew Booker 20 coins if Andrew can find integers $x,y,z$ where $x^{3}+y^{3}+z^{3}=33$” (no Gil and Andrew never made this contract). Cryptocurrency smart contracts are also able to issue their own tokens and these tokens could have a value on the free market in the same way that national currencies, Bitcoin, or baseball cards have value. For example, a smart contract can be used to issue tradable tokens to the entities that find solutions to the equation $x^{3}+y^{3}+z^{3}=n$.

Since cryptocurrency smart contracts are able to issue their own tokens, these tokens can be used as a reward for proving theorems or solving computational problems in mathematics and these tokens can provide an alternate source of funding for mathematical research especially for set theory.

Smart contracts can also be used to bet on things that happen on the blockchain, but the winner of the bet can only be determined based on information on the blockchain. However, if there is a reward for solving a problem or doing a mathematical computation on the blockchain, then the solver of that mathematics problem is more likely to post the solution on the blockchain and therefore the bet can successfully be carried out on the blockchain.

Of course, cryptocurrency smart contracts have their limitations since they are slow, inefficient, and since it is a pain to completely formalize a proof of a result and post that proof on a blockchain. In the near future, I therefore see cryptocurrency smart contracts being used to bet on computational problems in mathematics rather than on theorems that require proofs since it will be a pain to completely formalize a proof and post it on the blockchain.

Cryptocurrency smart contracts are very new since the main cryptocurrency for smart contracts, Ethereum, has only been out since 2015. See this post for relevant discussion along with some issues that may arise when using smart contracts to reward computational mathematics. I have not heard of anyone else discuss using cryptocurrency smart contracts for funding and rewarding the solutions to mathematics problems.

One possible issue with using cryptocurrency smart contracts for betting on results is that it typically takes several years for a new mathematical result to be obtained and posted on the blockchain and nobody wants their coins or tokens stuck in a smart contract for all that time. A remedy for this problem would be for the people placing their bets to bet not on a single problem but instead to be on multiple problems so that the smart contract can be closed as soon as one of those problems has been solved.

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  • $\begingroup$ The downvote for this post is completely unjustified. $\endgroup$
    – Joseph Van Name
    Commented Apr 8, 2019 at 12:47
  • $\begingroup$ Stop going to my profile and looking for things to downvote. That is inappropriate. $\endgroup$
    – Joseph Van Name
    Commented Apr 8, 2019 at 19:50

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