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Dutch Book Arguments

Bayesianism is a movement in epistemology and the philosophy of science that suggests that beliefs come in degrees. Dutch book arguments are a type of argument showing that rational agents must have degrees of belief that obey various principles. These arguments were introduced by Bruno de Finetti to show that for a rational agent, her degrees of belief in various propositions must obey the three axioms of probability theory. That is, every proposition has a nonnegative probability, any proposition the agent is certain of has Probability 1, and the probability of A or B is the sum of the probabilities of A and B, if A and B are incompatible. Dutch books have also been used to establish further constraints on degrees of belief and on how agents should update them over time. The basic form of the argument is to find, for any agent who violates the proposed constraint, a set of bets that the agent finds individually favorable but that collectively would guarantee that the agent loses money—this unfavorable collection of bets is the so-called Dutch book. This entry outlines how Dutch book arguments work, shows how they can be used to support some of the basic tenets of Bayesianism and discusses some of the problems they face.

Assumptions

The background of any Dutch book argument is the claim that (beyond the standard propositional attitudes of belief, knowledge, desire, and so on) there is a propositional attitude of degree of belief, and an assumption connecting this degree of belief to the agent's evaluation of bets as favorable or unfavorable. The assumption is that for every agent and every proposition, there is some number p that counts as her fair price for a bet on that proposition. What this means is that she is willing to pay any amount less than $p for a bet that wins her $1 if A is true, while she will accept any amount greater than $p for a bet that loses her $1 if A is true (or similarly $[100p] for a bet that pays $100). These two actions are respectively called buying and selling a bet on A. We typically think of gamblers buying bets and a casino selling them, but the Dutch book argument assumes each agent is willing to do either. This fair price p is then said to be the agent's degree of belief in A.

For example, let's say your degree of belief is .6 that it will rain tomorrow in Seattle. The argument assumes that you will be willing to pay $59.99 to place a bet where you win $100 if it rains tomorrow in Seattle, but you will not be willing to pay $60.01. However, you will be willing to accept if someone else pays you $60.01 to place a bet where you have to give them $100 if it rains tomorrow in Seattle. This is clearly an idealization—in some cases there will be a range of values at which you will be unwilling to either buy or sell, while in other cases there may be a range at which you are happy to do either. This is connected to the phenomenon of risk aversion.

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