Being subject to a Dutch Book

I’ve periodically wondered why doing poorly when faced with a Dutch Book is supposed to be a sign of irrationality, but it’s not a sign of irrationality that rational people do poorly when faced with someone who hits all and only rational people on the head with a baseball bat.

This occurred to me today:

1. One cannot get a rational person to act against their own interest except by force, luck or superior information.

2. Putting a Dutch Book over someone with inconsistent credences does not require force, luck or superior information.

This seems to get at some of the intuition as to why being subject to a Dutch Book is supposed to be a sign of irrationality.

But I don’t know how much confidence we should have in (1). The exception clause already admits three exceptions. This sounds ad hoc. Would we be very surprised if more exceptions had to be added?

Still, there is some plausibility to (1), at least for self-interested rationality.

Lebesgue sums previsions don't always lead to Dutch Books for inconsistent credences

Suppose E is the Lebesgue-sum prevision. Namely, if W is a wager on a finite space Ω with a credence (perhaps inconsistent P) and U_W is the utility function corresponding to W, then EW = ∑_yP({ω : U_W(ω)=y}).

Suppose your decision procedure for repeated wagers is to accept a wager if and only if the wager’s value is non-negative (independently of whatever other wagers you might have accepted). Suppose, further, that Ω has exactly two points and the credence of each point is non-negative and of at least one it is positive.

Proposition: Then, no finite sequence of wagers forms a Dutch Book.

Proof: Consider the sequence of utility functions U₁, ..., U_n that corresponds to a Dutch Book sequence of wagers W₁, ..., W_n. Then U₁ + ... + U_n < 0 everywhere on Ω and yet EW_i ≥ 0 for all i. Let a and b be the two points of Ω. Reordering the wagers if necessary (the order doesn’t matter on this decision procedure), we can assume that the wagers W₁, ..., W_m are such that U_i(a)≠U_i(b) for i ≤ m, and that W_m + 1, ..., W_n are such that U_i(a)=U_i(b) for i > m. Then EW_i = U_i(a)=U_i(b) for i > m. Hence, U_i is positive everywhere on Ω for i > m. So, if W₁, ..., W_n form a Dutch Book, so do W₁, ..., W_m. Now, EW_i = αU_i(a)+βU_i(b) where α and β are the probabilities of a and b respectively. It follows that EW₁ + ... + EW_n = α(U₁(a)+...+U_m(a)) + β(U₁(b)+...+U_m(b)). Since this is a Dutch Book, it follows that the two sums on the right hand side are both negative. Since α and β are non-negative and at least one is positive, it follows that EW₁ + ... + EW_n < 0, and hence this isn’t a Dutch Book.

Dutch Books and update rationality

It is often said that if you depart from correct Bayesian update, you are subject to a diachronic Dutch Book—a sequence of bets you will have to rationally agree to that is sure to make you lose—and this is supposed to indicate a lack of rationality. That may be, but I want to point out that the lack of rationality is not constituted by being subject to a Dutch Book: being subject to a Dutch Book is merely a symptom. I expect most people working this stuff know this, but perhaps it’s worth giving an explicit argument for.

Here is why. Alice, Bob and Carl are observing a coin that is either double-headed (D) or fair (F). Their prior probabilities for the two hypotheses are 1/2, and they have the reasonable and consistent priors: they assign probability 3/4 to heads showing up, and so on. The coin is flipped and the result is observed. If the coin lands tails, all three correctly update their probability for D to 0. If the coin lands lands heads, Alice, Bob and Carl each follow a different rule for updating their credence for D. Alice updates to 2/3 in accordance with Bayes’ theorem. Bob updates to 3/4 as that intuitively seems right to him. Carl, on the other hand, initiates a process in his brain which randomly updates to a uniformly chosen credence between 1/2 and 1.

Alice is not subject to a Dutch Book.

Bob is.

But Carl, once again, is not. [Proof: For any betting book, there is a non-zero chance that Carl would be rationally permitted to respond to that book in a way that it would be rationally permitted for Alice to respond. For Carl and Alice differ in their credences only in post-toss bets dependent on D in the special case that the first toss is heads, but the direction in which they differ in their credences is random: Carl has a non-zero chance of having a lower credence than Alice in D at this point and a non-zero chance of having a higher one. If at Alice’s credence of 2/3 the bet is rationally permitted to take, then either (a) for all credences lower than 2/3 it is rationally permitted to take, or (b) for all credences higher than 2/3 it is permitted to take, since the expected outcomes are linear functions of the credence. But there is a non-zero chance that Carl’s credence is lower than Alice’s and a non-zero chance that Carl’s credence is higher than Alice. Thus, there is a non-zero chance that Carl can permissibly take the bet, if Alice can permissibly take the bet. And the same argument applies if Alice can permissibly refuse the bet.]

However, Carl is not more rational than Bob, despite not being subject to a Dutch Book due to his unpredictability. Hence, not being subject to a Dutch Book is only a symptom of irrationality, not constitutive of it.

Truth and probabilistic consistency

Suppose Alice has an inconsistent probabilistic assignment P^A. Then, famously, there is a series of bets on single propositions (call these binary bets) that is a Dutch Book against Alice: i.e., Alice by her lights will accept each bet, and is guaranteed to lose money.

But now suppose Bob has a probabilistic assignment P^B—perhaps a consistent one—that is strictly further from the truth than Alice’s inconsistent one in the sense that

1. for any p, if p is false, then P^B(p)≥P^A(p),

2. for any p, if p is true, then P^B(p)≤P^A(p), and

3. at least one of the inequalities is strict.

Then Alice will do at least as well as Bob on every portfolio of offers of binary bets, and on some portfolios she will do strictly better than Bob. In particular, even if Bob’s probabilistic assignment is consistent, and there is a binary bet Dutch Book against Alice, Bob will fare no better than Alice with respect to that book.

Thus, if we start with a consistent assignment and then by some process move towards truth, we will do better (against binary bet portfolios) even if we lose consistency.

So why is Alice’s probabilistic assignment supposed to be rationally bad in a way that Bob’s isn’t? Well, the difference is this. A bookie can fleece Alice simply on the basis of knowing Alice’s probability assignment. But simply knowing Bob’s probability assignment won’t be enough to know which portfolio will fleece him.

However, the more I think about this, the more I lose the intuition that all this shows there is something particularly rationally problematic about Alice’s assignments just because they are inconsistent. Why should game-theoretic performance against a competitor who knows one’s credences be particularly indicative of rationality or the lack thereof? When nature offers us betting portfolios (to pursue this trail or that trail after a wounded deer in the woods, say), these portfolios are normally independent of our credences. Of course, in business and war, we have to worry about mind-reading competitors. But much of our life, we don’t.

Suppose I find myself with inconsistent credences. What should I do? Should I force them to be consistent? If I am dealing with mind-reading competitors who have no more information about the external world than I do, then I should go for consistency. But going for consistency will force me to modify some of my probabilities, and for all I know, these probabilities may get modified away from truth. And that might be more harmful.

There may be interesting trade-offs. Maybe some intellectual strategies work better against mind-reading competitors and others work better with the portfolios set by nature. We should not take doing well with respect to one selection of portfolio to be particularly informative about the nature of rationality.

Diachronic Dutch Books

You have a Dutch Book (DB) against you at t provided that, given your credences at t, you would assent to each of a set of bets such that you're guaranteed to lose on balance if you assent to them all.

This morning, I was thinking about cases where people are offering diachronic DB argument.

Suppose you rationally change your mind about p, adjusting your credence between today and tomorrow, say from 1/4 to 3/4, in the light of new evidence, all duly according to Bayes. My initial thought was that there is then a diachronic DB against you in the following sense: there is a pair of bets such that if one is offered today and another tomorrow, you will accept both and be guaranteed to lose overall. (For instance, today, you will accept the deal that you will pay three dollars if p and get a dollar if not p, and tomorrow you will accept the deal that you will pay three dollars if not p and get a dollar if p. But then you're going to lose two dollars whether or not p is true.)

But that was careless of me. A Dutch Book would do better to be defined as a set of bets that you're individually rational in accepting and that are sure to lose you money given the information you have. But you don't have a guarantee that you will change your mind about p from 1/4 to 3/4 in this case. (This is at the heart of the diachronic DB argument that has been given for the Reflection Principle.)

Is there anything to be learned from my case above, other than to be more careful in thinking about DBs? Maybe. Consider your situation during the second bet, the one tomorrow. You accept that bet. In accepting the bet, you bring it about that by your present lights you are bringing it about that you have played a game that you are sure to have overall lost. So one lesson of this is that it is not irrational to bring it about that you have played a game that you are sure to have lost. Moreover, this case suggests that there is a crucial temporal or causal directionality to DB-based arguments. DB arguments have been used to argue that you should now adopt any credences you know for sure you will rationally have (with some provisos). But one had better not use DBs to argue that you should now adopt any credences you know for sure you rationally had: that way lies stasis.