Avoiding Dutch Books despite inconsistent credences

Abstract

It is often loosely said that Ramsey (in: Braithwaite (ed) The foundations of mathematics and other logical essays, Routledge and Kegan Paul, Abingdon, pp 156–198, 1931) and de Finetti (in: Kyburg, Smokler (eds) Studies in subjective probability, Kreiger Publishing, Huntington, 1937) proved that if your credences are inconsistent, then you will be willing to accept a Dutch Book, a wager portfolio that is sure to result in a loss. Of course, their theorems are true, but the claim about acceptance of Dutch Books assumes a particular method of calculating expected utilities given the inconsistent credences. I will argue that there are better ways of calculating expected utilities given a potentially inconsistent credence assignment, and that for a large class of credences—a class that includes many inconsistent examples—these ways are immune to Dutch Books and single-shot domination failures. The crucial move is to replace Finite Additivity with Monotonicity (if $$A\subseteq B$$ , then $$P(A)\le P(B)$$ ) and then calculate expected utilities for positive U via the formula $$\int _0^\infty P(U>y)\, dy$$ . This shows that Dutch Book arguments for probabilism, the thesis that one’s credences should be consistent, do not establish their conclusion. Finally, I will consider a modified argument based on multi-step domination failure that does better, but nonetheless is not as compelling as the Dutch Book arguments appeared to be.