Suppose I’m trying to infer probabilities about some set of events by looking at betting markets. My idea was to visualise the possible probability assignments as a high-dimensional space, and then for each bet being offered remove the part of that space for which the bet has positive expected value. The region remaining after doing this for all bets on offer should contain the probability assignment representing the “market’s beliefs”.

My question is about the situation where there is no remaining region. In this situation for every probability assignment there’s some bet with a positive expectation. Is it a theorem that there is always an arbitrage in this case? In other words, can one switch the quantifiers from “for all probability assignments there exists a positive expectation bet” to “there exists a bet such that for all probability assignments the bet has positive expectation”?

Right. But also we would want to use a prior that favoured biases which were near fair, since we know that Wolf at least thought they were a normal pair of dice.