On the assumption that the known probability is the implied payoff (in reverse of betting, where the known payoff is assumed to be the implied probability of the bet) you can check for a dutch-book by summing the probabilities. Above one and it books the gambler (who will probably not buy it), below one and it books the bookmaker.
This is because Dutch books have to be profitable over every possible outcome. There is a procedure called dutching which is very similar, except it doesn’t guarantee a profit; it just forms a Dutch book over a restricted set of bets. This is no longer exhaustive, so there are outcomes in dutching where all of your bets fail and you lose money.
I am not sure what changes if the payoff and the probability of paying off are not equivalent.
Yes. You can do the same for a diachronic Dutch book which takes the table of probabilities that describes an agents beliefs before the agent learns E and after the agent learns E. For all H in table two p(H) must = p(H|E) in table 1. If p(H) does not = p(H|E) the the agent these tables describe is Dutch bookable assuming she will wager at those probabilities.
On the assumption that the known probability is the implied payoff (in reverse of betting, where the known payoff is assumed to be the implied probability of the bet) you can check for a dutch-book by summing the probabilities. Above one and it books the gambler (who will probably not buy it), below one and it books the bookmaker.
This is because Dutch books have to be profitable over every possible outcome. There is a procedure called dutching which is very similar, except it doesn’t guarantee a profit; it just forms a Dutch book over a restricted set of bets. This is no longer exhaustive, so there are outcomes in dutching where all of your bets fail and you lose money.
I am not sure what changes if the payoff and the probability of paying off are not equivalent.
Yes. You can do the same for a diachronic Dutch book which takes the table of probabilities that describes an agents beliefs before the agent learns E and after the agent learns E. For all H in table two p(H) must = p(H|E) in table 1. If p(H) does not = p(H|E) the the agent these tables describe is Dutch bookable assuming she will wager at those probabilities.