Coherent Odds and Subjective Probability

A well-known Dutch book, due to de Finetti, shows how violations of the additivity law of probability theory (Pr(A∪B)=Pr(A)+Pr(B)−Pr(A∩B)) open the door to a sequence of bets leading to a sure loss. In this paper we show that in a market environment, when bookies act strategically, it may well be optimal for them to post incoherent (nonadditive) odds. This is true even when the bookie's own preferences are expected utility, provided she plays against at least two nonexpected utility bettors.