One objection that Henry Kyburg raises in several places to the Dutch Book argument for the notion of subjective probability is that people can avoid Dutch Books by exercise of purely deductive reasoning, and therefore they provide no constraint on betting odds or the like. As he puts it in his 1978 paper, “Subjective Probability: Criticisms, Reflections, and Problems”:
No rational person, whatever his degrees of belief, would accept a sequence of bets under which he would be bound to lose no matter what happens. No rational person will in fact have a book made against him. If we consider a sequence of bets, then quite independently of the odds at which the person is willing to bet, he will decline any bet that converts the sequence into a Dutch Book.
I think there’s something right about the general point, but this particular passage I quoted seems just plain wrong. I’ll give an example in which it seems perfectly reasonable to get oneself into such a Dutch Book.
Let’s say that back in January I was very impressed by John McCain’s cross-partisan popularity, and his apparent front-runner status as the Republican nominee for president, so I spent $40 on a bet that pays $100 if he’s elected president. After a few months, seeing his poll numbers plummet, let’s say I became more bullish on Giuliani, and spent $40 on a bet that pays $100 if he’s elected instead. But now that Republicans seem to be backing away from him too, and that Hillary Clinton may be pulling ahead in the Democratic primary, say I now think she’s the most likely candidate to win. If Kyburg is right, then no matter what my degree of belief, I wouldn’t spend more than $20 on a bet that pays $100 if she wins, because I will have converted my set of bets into a Dutch Book against myself (assuming as I do that no more than one of them can be elected). However, it seems eminently rational for me to buy a bet on Clinton for some larger amount of money, because I regard my previous bets as sunk costs, and just want to focus on making money in the future.
Something like this is possible on the Bayesian picture whenever I change my degrees of belief at all – I might have already made bets that I now consider regrettable, but that shouldn’t stop me from making future bets (unless it perhaps does something to convince me that my overall bet-placing skills are bad).
To be fair, I’m sure that Kyburg intends his claim only in the case where the agent is sequentially accepting bets in a setting where her beliefs aren’t changing, where the basic Dutch Book theorem is meant to apply. He’s certainly right that there are ways to avoid Dutch Books while still having betting odds that violate the probability axioms, unless one is somehow required to accept any sum of bets for and against any proposition at one’s published odds.
But somehow Kyburg seems to be suggesting that deductive rationality alone is sufficient to prevent Dutch Books, even with this extra flexibility. However, I’m not sure that this will necessarily happen – one can judge a certain loss as better than some combination of chances of loss and gain. And he even provides a footnote to a remark of Teddy Seidenfeld that I think makes basically this point!
It is interesting to note, as pointed out to me by Teddy Seidenfeld, that the Dutch Book against the irrational agent can only be constructed by an irrational (whether unscrupulous or not) opponent. Suppose that the Agent offers odds of 2:1 on heads and odds of 2:1 on tails on the toss of a coin. If the opponent is rational, according to the theory under examination, there will be a number p that represents his degree of belief in the occurrence of heads. If p is less than a half, the opponent will maximize his expectation by staking his entire stake on tails in accordance with the first odds posted by the Agent. But then the Agent need not lose. Similarly, if p is greater than a half. But if p is exactly a half, then the rational opponent should be indifferent between dividing his stake (to make the Dutch Book) and putting his entire stake on one outcome: the expectation in any case will be the same.
If Kyburg’s earlier claim that agents will never get themselves into Dutch Books is correct, then this argument by Seidenfeld can’t be – the same reasoning that keeps agent out of Dutch Books should make bookies buy them (unless it’s more bad to have a sure loss than it is good to have the corresponding sure gain). I suspect that each of the two arguments will apply in some cases but not others. At certain points, the bookie will feel safer buying the Dutch Book, while at others, she will favor maximizing expectation. Similarly, the agent will sometimes feel safer allowing a Dutch Book to be completed against her, rather than exposing herself to the risk of a much greater loss.
I think Kyburg is right that there are problems with any existing formulation of the Dutch Book argument, but I think he’s wrong in the facts of this particular criticism, and also wrong about subjective probability as a whole. Seidenfeld’s argument is really quite thought-provoking, and probably deserves further attention.