The assumption you should relax is that of an objective probability. If you treat probabilities as purely subjective, and that saying that P(X)=1/3 means that my decision procedure thinks the world with not X is twice as important as the world with X, then we can make a trade.
Lets say I say P(X)=1/3 and you say P(X)=2/3, and I bet you a dollar that not X. Then I pay you a dollar in the world that I do not care about as much, and you pay me a dollar in the world that you do not care about as much. Everyone wins.
This model of probability is kind of out there, but I am seriously considering that it might be the best model. Wei Dai argues for it here.
I know Wei’s model and like it a lot, but it doesn’t solve this problem. With subjective probabilities, the exchange of information between players in a market becomes very complicated, like Aumann agreement but everyone has an incentive to mislead everyone else. How do you update when the other guy announces that they’re willing to make such-and-such bet? That depends on why they announce it, and what they anticipate your reaction to be. When you’re playing poker and the other guy raises, how do you update your subjective probabilities about their cards? Hmm, depends on their strategy. And what does their strategy depend on? Probably Nash equilibrium considerations. That’s why I’d prefer to see a solution stated in game-theoretic terms, rather than subjective probabilities.
ETA: see JGWeissman’s and badger’s comments, they’re what I wanted to hear. The answer is that we relax the assumption of zero-sum, and set up a complex system of payouts to market participants based on how much information they give to the central participant. It turns out that can be done just right, so the Nash equilibrium for everyone is to tell their true beliefs to the central participant and get a fair price in return.
Game theory in these setting is built on subjective probabilities! The standard solution concept in incomplete-information games is even known as Bayes-Nash equilibrium.
The LMSR is stronger strategically than Nash equilibrium, assuming everyone participates only once. In that case, it’s a dominant strategy to be honest, rather than just a best response. If people participate multiple times, the Bayes-Nash equilibrium is harder to characterize. See Gao et al (2013)] for the best current description, which roughly says you shouldn’t reveal any information until the very last moment. The paper has an overview of the LMSR for anyone interested.
The assumption you should relax is that of an objective probability. If you treat probabilities as purely subjective, and that saying that P(X)=1/3 means that my decision procedure thinks the world with not X is twice as important as the world with X, then we can make a trade.
Lets say I say P(X)=1/3 and you say P(X)=2/3, and I bet you a dollar that not X. Then I pay you a dollar in the world that I do not care about as much, and you pay me a dollar in the world that you do not care about as much. Everyone wins.
This model of probability is kind of out there, but I am seriously considering that it might be the best model. Wei Dai argues for it here.
I know Wei’s model and like it a lot, but it doesn’t solve this problem. With subjective probabilities, the exchange of information between players in a market becomes very complicated, like Aumann agreement but everyone has an incentive to mislead everyone else. How do you update when the other guy announces that they’re willing to make such-and-such bet? That depends on why they announce it, and what they anticipate your reaction to be. When you’re playing poker and the other guy raises, how do you update your subjective probabilities about their cards? Hmm, depends on their strategy. And what does their strategy depend on? Probably Nash equilibrium considerations. That’s why I’d prefer to see a solution stated in game-theoretic terms, rather than subjective probabilities.
ETA: see JGWeissman’s and badger’s comments, they’re what I wanted to hear. The answer is that we relax the assumption of zero-sum, and set up a complex system of payouts to market participants based on how much information they give to the central participant. It turns out that can be done just right, so the Nash equilibrium for everyone is to tell their true beliefs to the central participant and get a fair price in return.
Game theory in these setting is built on subjective probabilities! The standard solution concept in incomplete-information games is even known as Bayes-Nash equilibrium.
The LMSR is stronger strategically than Nash equilibrium, assuming everyone participates only once. In that case, it’s a dominant strategy to be honest, rather than just a best response. If people participate multiple times, the Bayes-Nash equilibrium is harder to characterize. See Gao et al (2013)] for the best current description, which roughly says you shouldn’t reveal any information until the very last moment. The paper has an overview of the LMSR for anyone interested.
Thanks for the link to Gao et al. It looks like the general problem is still unsolved, would be interesting to figure it out...
Maybe I should try to turn this comment into a full discussion post.