4 Dutch book arguments

We shall now consider some of the best-known arguments for probabilism: that is, for the three Kolmogorov axioms we saw in section 2. These are known as Dutch book arguments.

The point of them is to show that agents whose credences do not satisfy these axioms are systematically exploitable: they can end up taking actions which together amount to choosing to give up money, or something else they value, for no gain whatsoever. And, so the argument goes, since agents who are systematically exploitable are irrational, rationality requires that they do not have the sorts of non-probabilistic credences which got them into trouble.

There is also a Dutch book argument for the principle of conditionalisation, though we will not cover that this week. (We may, time permitting, come back to it in week 5.)

In order for any Dutch Book argument to work, we need some link between what an agent believes and what she does. This is because credences on their own are inert: without some principle linking them to an agent’s behaviour, we cannot conclude anything about what an agent will do merely from their credences (degrees of belief).

The standard assumption made in these arguments about the link between credences and an agent’s behaviour is that credences match up with the agent’s betting behaviour.

Write an $S bet on A for a deal that pays $S if A is true and nothing otherwise.

So her credence fixes a price, and she is willing to take either side of it. We will assume this for the time being. It is a substantial assumption, and it is probably false.

Two of the axioms can be argued for with a single bet. These are the easy cases, because nothing in them depends on how a sequence of offers is handled.

Non-negativity. Suppose \Cr(A) = -0.5. By the betting interpretation she will sell a $100 bet on A for any price above -\$50 — and a negative price means she pays. So she will happily pay someone $49 to take the bet on, and then owes them another $100 if A turns out true.

She loses $49 whatever happens, from one transaction.

Normalisation. Suppose \Cr(\Omega) = 0.9. She will sell a $100 bet on \Omega for $91. But \Omega is guaranteed, so she collects $91 and pays out $100.

And if instead \Cr(\Omega) = 1.1, she buys the same bet for $109 and collects $100, losing $9 again.

These cases are easy, because they can be done in a single bet. Finite additivity is not so easy, because it depends on how credences relate to other credences, and so multiple bets need to be taken.

Suppose P and Q are mutually exclusive, and that, as an example of a finite additivity violation, our agent has credence 0.2 in P, credence 0.2 in Q, but credence 0.8 in P \cup Q. Since P and Q are disjoint, finite additivity requires \Cr(P \cup Q) = 0.2 + 0.2 = 0.4.

She is offered three bets in turn, each for a moment and then withdrawn.

• Bet 1 She sells a $100 bet on P for $21. Her price for that bet is $20, so selling above it is a gain.
• Bet 2 She sells a $100 bet on Q for $21. Likewise.
• Bet 3 She buys a $100 bet on P \cup Q for $79. Her price for that bet is $80, so buying below it is a gain.

Each bet, taken on its own, is one she wants. So it looks as though she should take all three. But if she does, she is down $37 however things turn out.

It is worth seeing that happen. Each frame below plots her net position in the three states — P, Q, and neither — first for each bet taken on its own, and then for the bets accumulated one after another.

So far we have quietly assumed that she treats each offer as if it were the only decision she will ever face. That is called myopic choice. An agent who instead looks ahead, and takes into account what she will do later, is called sophisticated. (Much more on this in week 5.)

The temporal structure implied by the argument for additivity above is given by the following decision tree.

The squares are decision nodes — choice points for the agent — and the edges leaving them represent the particular choices open to her. She begins at decision node 1 and may, depending on what she chooses, reach decision node 3 and a final choice. The payoffs are written as three-place vectors: the first place is her gain or loss in dollars if P occurs, the second her gain or loss if Q occurs, and the third her gain or loss if neither of them does.

A sophisticated agent will obviously not take all three bets. For suppose she did. Then she would go on taking bets at node 2 as well, and so at node 1 her choice would really be between

It turns out that a sophisticated agent here makes one trade and then walks away, though we will not show this now.

It can. The Dutch book for finite additivity can be reconstructed so that it works even against an agent who uses sophisticated choice — which is what she ought to be using. If there is time, we will see how in week 5.

The Dutch book theorem is a piece of mathematics. To get an argument for probabilism out of it we need some further premises, and the obvious way of supplying them is not very good.

The flat-footed version runs: if your credences are non-probabilistic, a cunning Dutchman might come along and take your money; if they are probabilistic, he cannot; so you had better have probabilistic credences.

On this version of the argument, the reason why we should not have non-probabilistic credences is that avoiding them protects us from actually being exploited in a Dutch book situation. However, suppose that for some reason an agent knew for certain that no Dutch book situation would arise. In that case they would know that having non-probabilistic credences will not hurt them, and the reason to adopt probabilistic credences would vanish. But presumably a proponent of probabilism would not want to say that an agent like this is rational: although they will not be exploited, it is their pattern of credences itself which is supposed to be irrational.

Here is a better version:

• If an agent does not bet in line with the betting interpretation for the credences they have, then they are irrational.
• If an agent has non-probabilistic credences and bets in line with the betting interpretation, then they are vulnerable to exploitation by a Dutch book.
• If an agent is vulnerable to exploitation by a Dutch book, then they are irrational.
• \therefore If an agent has non-probabilistic credences, then they are irrational.

Suppose the agent has non-probabilistic credences. Either she bets in line with the betting interpretation or she does not. If she does not, then by (1) she is irrational. And if she does, then by (2) she is vulnerable to a Dutch book, and so by (3) she is irrational. Either way, she is irrational.

One might object to Dutch Book arguments in general on the ground that it is not necessarily irrational to lose money. After all, it does not seem that we are rationally required to value money in particular, and if we do not value it, losing it cannot be a sign of irrationality.

Dutch Book arguments, however, do not need to be about money. Anything quantitative that can, in principle, be a prize in a bet, will do the job just fine. (The bets do not need to be practical or realistic, just possible in principle.) So long as there is any quantity the agent cares about like that, and they have non-probabilistic credences, and there is a suitable link to their betting behaviour (whether that comes via the betting interpretation or some weaker principle), the agent seems to be vulnerable to a Dutch Book, and thus open to the charge of irrationality.