Self-locating belief and diachronic Dutch Books
If beliefs are modeled by a probability distribution over centered worlds, belief update cannot work simply by conditionalisation. How then does it work? The most popular answer in philosophy goes as follows.
Let P an agent's credence function at time t1, P' the credence function at t2, and E the evidence received at t2. Since E is a centered proposition, it can be true at multiple points within a world. Suppose, however, that the agent assigns probability 0 to worlds at which E is true more than once. Then to compute P', first conditionalise P on the uncentered fragment of E -- i.e. the strongest uncentered proposition entailed by E. This rules out all worlds at which E is true nowhere. Second, move the center of each remaining world to the (unique) point at which E is true.
Call this rule "IC", for "Indirect Conditionalisation". With minor variations, this is the rule defended in Piccione and Rubinstein (1997), Halpern (2004), Elga (2007), Meacham (2008), Titelbaum (2008), Kim (2009) and Briggs (2010), and probably other places as well. Apart from details of the presentation, the variations mostly concern what to do if the new evidence is not certain to be true at most once per world.
In my paper on updating self-locating beliefs, I present a Dutch Book argument against IC. Rachael Briggs, in her 2010 article, gives a proof that IC is immune to Dutch Books. At least one of us must be wrong.
Here is a simple version of my Dutch Book against IC. Assume at t1, an agent's credence is divided between three uncentered worlds: w1, w2 and w3. For each of these, there is exactly one place where she might be, according to her beliefs. At t2, the agent will receive either evidence E or E', and she knows that this is the case. E is true at a unique point in w1 and w2, but nowhere in w3. E' is true at a unique point in w1 and w3, but nowhere in w2. (The point where E is true in w1 is different from the point where E' is true.) Initially, the agent gives credence 1/2 to w1 and credence 1/4 to each of w2 and w3. She therefore regards as fair a bet that pays $-3 in case of { w1 } and $3 otherwise. Now suppose she learns E and updates by IC. The uncentered fragment of E rules out w3, but not w1 and w2. The new probabilities are therefore 2/3 for w1 and 1/3 for w2. Similarly, if she learns E', the new probabilities are 2/3 for w1 and 1/3 for w3. Either way, she will regard as fair a bet that pays $2 in case of { w1 } and $-4 otherwise. Hence, whether { w1 } is true or not, and whatever evidence she receives, the agent is certain to make a loss.
Now for Rachael's argument why there cannot be any such Dutch Book. One thing worth pointing out is that Rachael only allows bets on uncentered propositions. In my paper, I give a more general Dutch Book argument in support of a different update rule. This argument makes use of centered bets. However, the Dutch Book just presented only involves the uncentered proposition { w1 } and its negation, so this can't be the relevant point of disagreement.
Rachael's argument relies on a proof in Teller (1973) showing that an agent with uncentered beliefs who updates by conditionalising on uncentered evidence is immune to diachronic Dutch Books. Rachael's argument then is that if there were a Dutch Book B against an agent following IC, we could convert it into a Dutch Book B' for an imaginary agent with uncentered beliefs who updates by conditionalisation. By Teller's result, this is impossible.
For my Dutch Book in place of B, Rachael's argument is actually quite simple. The variant B' is B itself. So imagine an agent with only uncentered beliefs, who gives credence 1/2 to w1 and 1/4 to w2 and w3. Like our original agent, this agent initially regards as fair the bet that pays $-3 in case of { w1 } and $3 otherwise. Moreover, just as our original agent updates by conditionalising on either E or E', the imaginary agent updates by conditionalising on either u(E) or u(E'), where u(X) is the strongest uncentered proposition entailed by X. Since u(E) is true at w1 and w2, and u(E') is true at w1 and w3, either way the imaginary agent ends up with credence 2/3 in w1. So now they regard as fair the bet that pays $2 in case of { w1 } and $-4 otherwise. We have converted the Dutch Book for the original agent into a Dutch Book for the imaginary agent. Yet by Teller's theorem, the imaginary agent can't be Dutch Booked!
In fact, the converted Dutch Book is not a genuine Dutch Book. The reason is that u(E) and u(E') are not mutually exclusive: they both contain w1. And then we can't assume that the (imaginary) agent is certain at t1 that their total evidence is going to be either u(E) or u(E'). By contrast, E and E' are mutually exclusive. The converted Dutch Book does satisfy conditions (a) and (b) on p.19 of Rachael's paper. But it does not follow, as Rachael assumes, that it constitutes a Dutch Book against the imaginary agent.
This, I think, is the mistake in Rachael's argument: uncentering a partition of centered evidence does not always result in a partition of uncentered evidence.