Some Dutch Book Arguments

I was recently teaching David Christensen’s 1991 paper “Clever Bookies and Coherent Beliefs”, and I thought there were many good points there that I should have really noticed when I read the paper years ago. My dissertation would have been better if I’d seen all this years ago, but better late than never. Christensen gives a nice method for determining just what particular Dutch Book arguments do and don’t show. I agree with him mostly about how to apply that method to particular cases, as we’ll see below.

First a couple of arguments Christensen gives, with his evalations.

Spousal Consistency
Assume that Bill Clinton believes some proposition p to degree x, and Hillary believes it to degree y, with x > y. Let a and b be numbers such that x > a > b > y. Then a clever bookie can sell Bill a bet that pays $1 if p for $a. By Bill’s lights this is a good trade. And she can buy a bet that pays $1 if p for $b from Hillary. By Hillary’s lights, this is a good trade. But now the bookie has made a sure profit of $a-b. So it (allegedly!) follows that Bill and Hillary should never disagree about the probability of any proposition.

This is, I think, obviously a bad argument, though it takes some work to say why. There’s an amusing bit of philosophical history here. In 1991, the same year that Christensen published his paper that included this as a parody of a Dutch Book argument, Donald Gillies published two papers which rested on the idea that an argument like this is sound. I’ve never seen one argument regarded as a parody by one philosopher, and sound by another, in the same year! I’m glad Philosophical Review was on the side of parody 🙂

Here’s Christensen’s diagnosis. If we took DB arguments to be bits of practical reasoning, this argument would be as good as any other DB argument. And hence all DB arguments would be unsound, for this is clearly unsound. (That was my first, and last, thought when first looking at these arguments.) So we should interpret DB arguments as not practical considerations, but as ways of revealing some kind of underlying tension. (David Lewis says something similar in the notes on “Why Conditionalise?”.) In this case the tension is not one that is normatively criticisable, since it is perfectly reasonable to have spouses who have different degrees of belief.

And that’s the general pattern we should follow. A DB argument reveals an underlying incompatibility between two or more evaluations. It doesn’t tell us whether this tension is epistemically acceptable or not. If one person evaluates the same thing two different ways, that’s not acceptable. If two people evaluate the same thing two different ways, that is acceptable.

Here’s another illustration.

Calcification
Assume that Bill Clinton believes some proposition p to degree x, and tomorrow he will believe it to degree y, with x > y. Let a and b be numbers such that x > a > b > y. Then a clever bookie can today sell Bill a bet that pays $1 if p for $a. By Bill’s lights this is a good trade. And she can buy a bet that pays $1 if p for $b from Bill tomorrow. By his lights then, this is a good trade. But now the bookie has made a sure profit of $a-b. So it (allegedly!) follows that Bill should never change his mind about the probability of any proposition.

Again the DB argument fails. This one also reveals a tension, but in this case it is a perfectly acceptable tension between Bill’s beliefs today and Bill’s beliefs tomorrow.

Lewis’s comments in “Why Conditionalise?” suggest a different response. He says a DB argument is only good if the bookie is only given access to the same information as the victim has. This is too strong a constraint, for it would rule out too many good arguments.

Not Very Introspective
John believes p to degree 1/2, and ~p to degree 3/4. So there is a familiar Dutch Book to be made against him. But to make it you have to know his credences, and John does not. John’s margin of error on assessments of his own credences is 1/4. So he knows that his credence in p is in [1/4, 3/4], and his credence in ~p is in [1/2, 1] but he knows no more than that.

According to Lewis, there is no DB argument against this kind of agent, for the bookmaker needs information the agent doesn’t have. According to Christensen, there is such an argument, and indeed it is successful. I think Christensen is probably right, and this supports his analysis of Calcification. It is prima facie successful, and it does reveal a tension. But the tension it reveals is epistemically acceptable, indeed in many cases epistemically praiseworthy.

The final two cases (both mine) are a little more difficult.

Water
Suzy’s chemistry textbook has a bad misprint. It says throughout that water is H3O, not H2O. Suzy has heard many people talk about water being H2O, and she think they are probably correct. But she gives some credence to her textbook being correct, for it is correct on many other things. So her credence that water is H2O is 0.9. Now there’s a Dutch Book of a kind to be made against her. She will sell you for 95 cents a bet that pays $1 if water is H2O. Since it is metaphysically necessary that this bet wins, Suzy faces sure loss.

I think a Christensen-style analysis of this argument should be that it does reveal a tension between Suzy’s beliefs and the nature of modal space. But given Suzy’s position, this is a perfectly understandable tension, so it does not reveal any irrationality on Suzy’s part. The final case is even more problematic.

Fermat and Brian
It is 1991, and Fermat’s Last Theorem has not been proven. One of Brian’s math professors suggests, only half-jokingly, that the best explanation for the failure of any proof is that there is a counterexample out there somewhere. Brian thinks this is unlikely, but serious enough that his credence in Fermat’s Last Theorem is adjusted to 0.9. Now there’s a Dutch Book of a kind to be made against him. He will sell you for 95 cents a bet that pays $1 if Fermat’s Last Theorem is true. Since it is mathematically necessary that this bet wins, Brian faces sure loss.

I think a Christensen-style analysis of this argument should be that it does reveal a tension between Brian’s beliefs and the nature of modal space. But given Brian’s position, this is a perfectly understandable tension, so it does not reveal any irrationality on Brian’s part. The question is whether this is consistent with probabilism on any understanding of that term. I’m inclined to think it isn’t really, so Christsen’s analysis of DB arguments leaves a gap in the argument for probabilism. But that’s a contentious application of his approach, not something equivalent to the general approach which as I said seems like a much better way to think about DB arguments than any alternative.