Is there a nice way to bet on large-evidence, small-probability differences?
Normally we bet on substantial probability differences, like I say 10% you say 50% or similar. Betting makes sense there—you incentivize having correct probabilities, at least within a few percent or whatever. Is there some way to incentivize reporting the right log-odds, to within a logit or whatever?
One sort of answer might be showing that/how you can always extract a mid-range probability disagreement on latent variables, under some assumptions on the structure of the latent variable models underlying the large-logit small-probability disagreement.
Most high-liquidity securities are this—small movements with large amounts of money riding on them.
I don’t think it generalizes, though—different domains or even specific disagreements over probability of an outcome will have different mechanisms for matching counterparties with low-enough transaction costs that a wager is feasible.
I guess that is the generality: every wager is about two parties agreeing that they have a disagreement which will be resolved in the future, and further agreeing that the less-correct party will pay the more-correct party. Sometimes this can be a distributed/anonymous agreement, like a stock market (or like an actual market, where you “bet” that today’s purchases can be sold later for more). Sometimes it’s an individual wager. And, as always, transaction costs (loss of value due to the structuring and administration of the wager) make the vast majority of possible wagers infeasible.
For log-odds, there’s a complexity in that it’s not obvious how to make them add up to 1, so the wager will be (as wagers need to be) zero-sum. Every dollar won by a better predictor is a dollar lost by a worse one (or ones). The easiest way to convert a log-odds prediction to a linear zero-sum wager is to just convert to actual probability.
Is there a nice way to bet on large-evidence, small-probability differences?
Normally we bet on substantial probability differences, like I say 10% you say 50% or similar. Betting makes sense there—you incentivize having correct probabilities, at least within a few percent or whatever. Is there some way to incentivize reporting the right log-odds, to within a logit or whatever?
One sort of answer might be showing that/how you can always extract a mid-range probability disagreement on latent variables, under some assumptions on the structure of the latent variable models underlying the large-logit small-probability disagreement.
Most high-liquidity securities are this—small movements with large amounts of money riding on them.
I don’t think it generalizes, though—different domains or even specific disagreements over probability of an outcome will have different mechanisms for matching counterparties with low-enough transaction costs that a wager is feasible.
I guess that is the generality: every wager is about two parties agreeing that they have a disagreement which will be resolved in the future, and further agreeing that the less-correct party will pay the more-correct party. Sometimes this can be a distributed/anonymous agreement, like a stock market (or like an actual market, where you “bet” that today’s purchases can be sold later for more). Sometimes it’s an individual wager. And, as always, transaction costs (loss of value due to the structuring and administration of the wager) make the vast majority of possible wagers infeasible.
For log-odds, there’s a complexity in that it’s not obvious how to make them add up to 1, so the wager will be (as wagers need to be) zero-sum. Every dollar won by a better predictor is a dollar lost by a worse one (or ones). The easiest way to convert a log-odds prediction to a linear zero-sum wager is to just convert to actual probability.