Clever Bookies and Coherent Beliefs

Abstract

Since the 1930s, Dutch Book arguments have been used to support the view that one's degrees of belief should conform to the probability calculus. These arguments show that if an agent's degrees of belief violate the probability calculus, then a clever bookie, knowing nothing beyond what the agent's degrees of belief are, can offer the agent a set of bets with the following two properties: (1) each of the bets in the set will be fair, given the agent's degrees of belief; and (2) the set of bets taken together guarantees that the agent will end up losing money overall. Such a set of bets is called a Dutch Book. Clearly, there is something unattractive about a belief state which leaves one open to this sort of exploitation.1 Closely related arguments have also been given in support of further conditions on rational degrees of belief, conditions which go well beyond probabilistic consistency. Some of these arguments support popular principles, which describe the way an agent's degrees of belief should change when she is confronted with new evidence. (The probability calculus itself provides no guidance in such matters, so Conditionalization principles fill an important gap in probabilistic accounts of rationality.)2 Sim-

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