Spelling out a Dutch Book argument

Dutch Book arguments are often used to justify various epistemic norms – in particular, that credences should obey the probability axioms and that they should evolve by condionalization. Roughly speaking, the argument is that if someone were to violate these norms, then they would be prepared to accept bets which amount to a guaranteed loss, and that seems irrational.

But it's hard to spell out how exactly the argument is meant to go. In fact, I'm not aware of any satisfactory statement. Here's my attempt.

For concreteness, I'll focus on the argument for probabilism, but the case of conditionalization is similar.

The argument begins with an uncontroversial mathematical fact, the Dutch Book Theorem:

Let a unit bet on a proposition A be a deal that pays $1 if A is true and otherwise $0. Suppose for any proposition A, an agent is prepared to buy a unit bet on A for up to $Cr(A) – the dollar value corresponding to her credence in A – and she is prepared to sell a unit bet on A for $Cr(A) or more. If her credences do not satisfy the axioms of non-negativity, normalization, and finite additivity, she will then be prepared to buy and/or sell unit bets in such a way that if she makes all these transactions she incurs a guaranteed loss.

How do we get from here to an argument that rational credences should conform to the probability axioms? A few problems immediately stand out. (See Hajek 2008.)

First, the theorem seems to say nothing about people who aren't prepared to trade bets in accordance with their expected monetary payoff. Surely epistemic rationality does not require having a utility function that is linear with respect to monetary payoff. An epistemically rational agent need not care about money at all.

Worse, even if an agent does care only about money, and her utility function is linear with respect to monetary payoff, she ought not to be prepared to buy a unit bet on any proposition A for up to the dollar value corresponding to her credence in A. For example, let A be the proposition that the agent will not buy any bets today. An agent's credence in A may well be high, yet she ought not to pay much for the corresponding bet, since doing so would render A false.

Even if the relevant propositions are unaffected by the considered bets, there can be interference effects between different bets. For example, what if our agent has earlier bought a high-stakes bet on the proposition that she will not buy any more bets today? Then she may not be prepared to buy a unit bet on any proposition whatsoever. Relatedly, the Dutch Book argument for finite additivity involves at least three bets; if a probabilistically incoherent agent cares about the net outcome of all her transactions, rather than myopically about the isolated outcome of whatever transaction she currently considers, it is not clear why she ought to make all three transactions. (This is the "package principle objection". Interestingly, it seems not to arise for the case of conditionalisation, as Skyrms 1993 shows.)

Stepping back, why is the mere possibility of making a sure loss normatively relevant? After all, as Lewis said, "there aren't so many sneaky Dutchmen around".

Finally, why is the possibility of financial loss a sign of epistemic, rather than practical irrationality?

A neat way to get around most of these problems, which I haven't seen in the literature, is to invoke some broadly Humean principles about the independence of belief and desire. In outline, the idea is that for any probabilistically incoherent agent X there is a possible agent Y who (1) has the same credences as X, (2) only cares about the monetary payoff of whatever transaction they presently consider, and (3) is offered the relevant bets that make up a Dutch Book. Y then makes a sure (and avoidable) loss, despite trying to get as much money as possible. Something has gone wrong. But the fault must lie in Y's beliefs, for neither her utilities nor her decision process is faulty. So Y's beliefs are irrational. But Y's beliefs are identical to X's. So X's beliefs are irrational.

That's the outline. Let's fill in the details.

Let X be an arbitrary agent whose credences violate one of the probability axioms. Our aim is to show that X is epistemically irrational.

Let Y be a possible counterpart of X with the same (centred) credences. But Y has strange desires. Whenever Y is offered a monetary gamble, she only myopically cares about the net amount of money she will make through the present transaction. Specifically, if Y has the option to buy a unit bet on some proposition A for some amount $x, then the only thing she cares about is whether she will eventually (i) win $1 after having paid $x, or (ii) not win after having paid $x, or (iii) not win after not having bought the bet; the utility she assigns to these outcomes are, respectively, (i) $1-$x, (ii) -$x, (iii) $0. Similarly, mutatis mutandis, if Y has the opportunity to sell a unit bet.

I'll also stipulate that when faced with a choice, Y always chooses an option with maximal expected utility.

All this still doesn't ensure that Y is prepared to pay up to $Cr(A) for a unit bet on A because her credence in A may be affected by getting an offer to buy the bet or even by the act of buying (as when A is the proposition that she won't buy any bets today). If we want Y to accept a Dutch Book that involves several transactions, we must also ensure that, say, buying the first bet does not affect the expected utility of buying the second.

I'm not sure how best to get around these problems. Here's a brute force response.

Let's say that X's (and Y's) credence function is stable with respect to some propositions A,B,...,N iff Y regards $Cr(A) as the fair price for a unit bet on A, $Cr(B) as the fair price for a unit bet on B conditional on having bought/sold a unit bet on A, and so on. That is, a credence function is stable with respect to a list of propositions if the credence in each proposition on the list is not affected by whether a bet on that proposition or a proposition earlier in the list has been bought or sold.

If we assume that there is a list of propositions for which X's credences are stable and violate the probability axioms, we can stipulate that Y is made the relevant offers and gets caught in a Dutch Book.

So we need to assume that X's probabilistic incoherence isn't restricted to unstable parts of her credence function. To get a general argument for probabilism, we'll need the following premise.

Premise 1. If any restriction of an agent's credence function to stable propositions should satisfy the probability axioms, then so should her entire credence function.

(Here and throughout, a violation of the probability axioms means that either (i) some proposition has negative probability, or (ii) the tautology does not have probability 1, or (iii) there are disjoint propositions whose disjunction has a probability that is not the sum of the probability of the disjuncts. Boolean closure is not treated as an axiom.)

The motivation for Premise 1 is that the probability axioms are supposed to be general consistency requirements on rational belief. They are meant to hold for beliefs or any kind, not just for beliefs with a specific content.

Jeffrey makes a similar move in Subjective Probability: The Real Thing (pp.4f.):

If the truth [of a proposition about distant planets] is not known in my lifetime, I cannot cash the ticket even if it is really a winner. But some probabilities are plausibly represented by prices, e.g., probabilities of the hypotheses about athletic contests and lotteries that people commonly bet on. And it is plausible to think that the general laws of probability ought to be the same for all hypotheses – about planets no less than about ball games. If that is so, we can justify laws of probability if we can prove all betting policies that violate them to be inconsistent.

Jeffrey here assumes that there's a reasonably wide set of propositions for which our credences match our betting prices. I assume something much weaker. But I'd still like to know how to do better.

(One reassuring fact to keep in mind is that probabilistic incoherence is infectious: if, for example, your credence in an exclusive disjunction A v B is not the sum of your credences in A and B, there will be lots of other propositions for which you'll violate additivity. So it requires some fine-tuning to limit incoherence to unstable fragments of a credence function.)

Moving on, recall that X was an arbitrary agent whose credences fail to satisfy the probability axioms. We want to show that X is epistemically irrational. By Premise 1, we can assume without loss of generality that there are some propositions with respect to which X's credence function is stable but fails to satisfy the probability axioms.

The next premise is that all the differences between X and Y are irrelevant to whether the credence function shared by X and Y is epistemically rational. More precisely:

Premise 2. If X is epistemically rational, then so is Y.

The basic idea is that whether someone's beliefs are epistemically rational does not depend on her goals or desires. If we want to know whether it is epistemically rational (as opposed to practically useful) for an agent to have such-and-such beliefs, we don't need to know anything about her goals or desires.

I've also assumed that Y is an expected utility maximizer, which X may not be. But again, arguably the epistemic rationality of someone's beliefs would not be undermined by finding that they are an expected utility maximizer.

Finally, Premise 2 implies that it does not affect the epistemic rationality of an agent's beliefs if they are about to be offered a series of bets. (That's the final difference between X and Y.)

Premise 2 looks fairly good to me.

Now the Dutch Book theorem tells us that there are certain transactions that Y is prepared to make that would amount to a guaranteed loss. Let's stipulate that Y is made the relevant offers and thus really does make a sure loss.

The next premise states that something has then gone wrong.

Premise 3. It is irrational of Y to make choices that together amount to a sure loss (a loss she could have avoided by making different choices).

Here the guiding idea is that it is irrational for an agent whose sole aim is to maximize monetary profit to knowingly and avoidably enter transactions that are logically guaranteed to cost her money.

Premise 3 relies on the Converse Dutch Book Theorem: that probabilistically coherent agents cannot be Dutch Booked.

I'm not entirely happy with Premise 3. The problem is that, by assumption, Y does not care about her net wealth. When offered a series of choices, she only cares about the net outcome of the present choice. It would be nicer if we could stipulate that Y cares about the net payoff of all the choices she's about to make. This would make Premise 3 quite compelling, I think. But then we'd need to explain why Y accepts the individual deals that together constitute a Dutch Book. (Skyrms offers such an explanation in his 1993 paper on conditionalization, but the argument sadly doesn't generalize to the case of finite additivity.)

The rest of the argument is simple.

Premise 4. If an agent makes irrational choices, then either she is epistemically irrational or her desires are irrational or her acts don't maximize expected utility.
Premise 5. Y's desires are not irrational.

Y's myopic desires are admittedly weird, but on a suitably weak notion of rationality, I think they should pass. Since Y maximizes expected utility, we can conclude that Y's credences are irrational. Intuitively, Y misjudges the profitability of the relevant bets.

So Y is epistemically irrational. By Premise 2, it follows that X is epistemically irrational. QED.


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