So, dutch books are collections of bets (the synchronic kind) or sequences of bets (the diachronic kind) that result in a sure loss. As an example, consider paying .51 for a bet on A with unit payoff, and paying .51 on a bet against A with unit payoff. If you make both bets (if you accept the book that consists in both bets), you will gain 1, and pay 1.02, resulting in a sure loss of .02. The idea is that a rational agent should never accept such a bet.
On the betting interpretation of our degrees of belief, the presence of such Dutch Books present a reason for our degrees of belief (alternatively, credences) to be probabilities. If our credences in a proposition are defined as the prices we would be willing to pay for a (fair) bet on the truth of that proposition, or if our credences are measured by those prices, then a theorem due to De Finetti shows that if our credences are not (at least finitely-additive) probabilities, then a dutch book can be constructed against us. A converse theorem due to (I believe) Skyrms shows that if our credences are probabilities, then no such dutch book can be constructed against us. That is, if our credences are the prices for the bets we would view as fair given our credences, and were we to bet on those prices then dutch books can be constructed against us if our credences are not probabilities and we can avoid dutch books if they are probabilities. The betting interpretation of credences thus provides an easy way for establishing our credences as probabilities, allowing us to demonstrate various results in formal epistemology about our credences, and motivating the subjective, personalist interpretation of probability.
However, the betting interpretation itself runs into familiar problems. First, there are agents who may have fundamental reasons not to make bets of any kind. A religious person whose doctrine forbids gambling may refuse to make any bets whatsoever. The betting interpretation will thus deliver the result that the person assigns a credence of 0 to every proposition, despite the obvious possibility that the religious person may well have positive credences for many propositions. Second, there is no obvious reason that a rational person should be willing to bet on any and all propositions for which they have credences. Something is clearly right about the betting interpretation, but it can’t be that our credences are, or are measured by, our actual betting behavior (or actual dispositions to bet, or whatever).
In light of these difficulties, Alan Hajek and Lina Eriksson propose taking the concept of degree of belief to be a primitive concept that is not in need of, or perhaps cannot be, analyzed into some more basic notion. While their discussion has a certain appeal, I think a better approach is simply to move from analyzing our credences as our actual betting behavior to merely what we would view as a fair bet. A fair bet is defined as one where each side of the bet (for A, or against A) is equivalent in terms of expected value. We view a bet as a fair bet simply if we would be indifferent to either side of the bet. If we do this, then the objections raised above to the betting interpretation of degrees of belief fall away. After all, even if a person is religious, or risk averse, or for whatever other reason does not want to engage in various bets, there is no reason to suppose they may not still view bets as fair or not. They could view a bet as fair, without any corresponding behavior or disposition to bet on their part.
If we move to what we would view as a fair bet, rather than defining an agent as actually disposed to making bets on what they view as fair, then it might appear as if we lose our dutch book justification for having our credences be probabilities. After all, if we give up the traditional betting interpretation’s assumption that we would take any bet we view as fair, then even if a dutch book could be constructed against us there is no longer the result that we would end up with a sure loss (precisely because we would no longer automatically make this bet).
However, there still might be a dutch-book related argument for requiring our credences to be probabilities. Henry Kyburg argued that dutch books would never present a problem for a rational agent, because the rational agent could deduce from the dutch book alone that it resulted in a sure loss, and so would never accept that bet. For the purposes of this post, the rational agent could deduce that the bet was not fair, because it resulted in a sure loss. However, the nature of a dutch book is that every individual bet is one that the agent would view as fair. As a result, if a dutch book could be constructed against an agent, then the rational agent would think both that the dutch book was fair (because it was a combination of individual bets, each of which was fair) and that the dutch book was not fair (because the agent could deduce that accepting one side resulted in a sure loss, while accepting the other side resulted in a sure gain).
Given de Finetti’s theorem, then, if our credences are not probabilities then the agent would be in a position to believe a contradiction (that the bet was fair and was not fair). Given Skyrms’, if our credences are probabilities, then the agent would be safe from this contradiction. Given the paucity of reasons for our credences to not be probabilities – there may be reasons not to reveal our credences as probabilities, and to represent them non-probabilistically, but no reasons I know of for them not to be probabilities – these considerations should be sufficient to establish that our credences should be probabilities even without the sure-loss result of traditional dutch book arguments. So we can still have that our credences should be probabilities, even given the failures of the traditional betting interpretation of probabilities so long as our credences are defined as/measured by what we would view as fair bets.