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.
Nice!