A Bayesian Interpretation of The Dutch Book Argument

The “Dutch book argument” is surely the most famous argument for thinking that graduated credal states (i.e. degrees of belief, credences, levels of confidence, or judgments about probability) ought to conform to the axioms of probability. But there is little consensus on how the argument ought to be understood and whether it succeeds. I think that its aims should be recognized to be somewhat modest, but nevertheless, with such modest aims, the Dutch book argument is successful.

Proponents of different flavors of the Dutch book argument agree about several things.

First, they agree that the Dutch book argument relies on the thought that a rational individual will use graduated credal states to maximize expected utility. Any bets that, evaluated with respect to the graduated credal state, one expects to make one better will be taken. For instance, if you have assign a .5 probability that it will rain, the assumption is that you should be willing to buy a ticket for up to 1 util that pays 2 utils if it rains and nothing if it doesn't.

Second, the Dutch book argument relies on the Dutch book theorem. According to this theorem, for any graduated credal state that fails to conform to the axioms of probability, there will exist a set of bets each of which has a positive expected utility (as evaluated by that graduated credal state) but which collectively guarantees a loss.

The Dutch book argument connects the failure of a graduated credal state to conform to the axioms of probability with a certain kind of practical failing. If one has graduated credal states that don't conform to the axioms, and one aims to maximize one's expected utility, one can get into foreseeable trouble.

Since the focus of Dutch book argument is on the practical upshot of accepting graduated credal states that violate the axioms of probability, the Dutch book argument is sometimes criticized on the grounds that it is too reliant on practical considerations. Aren't the norms of probability a matter of epistemic, rather than practical rationality? But proponents of the argument seldom (if ever) claim that the reason why we should follow the norms is in order to avoid being Dutch-book-susceptible.

One popular strategy for defending the Dutch book argument from the charge of practicality suggests that Dutch book theorem shows that graduated credal states that don't obey the axioms can lead us to value the same set of tickets at different prices, depending on how they are offered. Value shouldn't be presentation-sensitive (at least in this way). Therefore, the graduated credal states themselves are inferred to be somehow inconsistent.

The thought that the inconsistency of valuation implies some kind of inconsistency in the graduated credal states themselves is tempting, but hard to make precise. Instead, I propose that we think of the Dutch book argument in a Bayesian fashion. Let a “Bayesian argument” be an argument that proceeds as follows.

P1       Pr(P | Q) >> Pr(P)

P2       Q

C         P

Bayesian arguments are of course non-deductive, and ultimately the plausibility of the conclusion depends on the prior probability of P. But when the first premise holds in a given context, the conclusion in some sense is confirmed by the second premise.

So, here is how I propose that we understand Dutch Book arguments:

P1     Intuitively, it is more likely that graduated credal states must obey rule X (P) if violating rule X

         leads to Dutch-book susceptibility(Q), for any rule X.

P2     Violating the axioms of probability leads us to Dutch-Book susceptibility.

C       Graduated credal states must obey the axioms of probability.

The Dutch book argument can be a good argument without the conclusion following deductively from the premises. So long as our prior probabilities make P1 true, P2 should count as evidence for C. There needn't be any particular reason why our prior probabilities make P1 true. They may simply be reasonable priors to have.

It is extremely plausible that P1 is true. A priori, I wouldn't expect that the correct rules of judgment would allow for Dutch-book susceptibility. Since conformity to the axioms does preclude Dutch-book susceptibility, it receives some support from that result. This argument doesn't get us any closer to explaining why graduated credal states ought to conform to the axioms of probability. But it does suggest that they ought to.