Wednesday, August 28, 2019

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.


Heath White said...

Is there a requirement that the book be decided on before the first bet? I always pictured it as a Dutch bookie sitting next to the coin-flippers, offering them losing bets opportunistically.

Anyway, this all became clearer to me when I figured out that ordinary binary logic has a notion of consistency, and once you start using the range between 0..1 you still need an analogous notion. The Dutch book is a way to make that concept of "probabilistic consistency" concrete, not a practical danger of losing money.

Alexander R Pruss said...

I was thinking that the book covers the eventualities. Here's the first bet to be offered. (Since there is no data there, that's just fixed.) Then, if the coin toss is heads, here's the second bet to be offered. If the coin toss is tails, instead, here's the bet to be offered.

On reflection, though, the proof in my post doesn't work if the Dutch bookie can read Carl's mind and offer a bet dependent on Carl's credence. I don't know right now if it can be fixed.

Alexander R Pruss said...

OK, I've convinced myself that that if the bookie can read Carl's mind, they still can't Dutch Book him. The reason is this: One will always do better against a particular sequence of offers if one's credences are strictly closer to the truth. The bookie can't Dutch Book Alice. But no matter how the two random events come out, double-heads-coin and heads-toss, there is a chance that Carl's credences are either the same as Alice's or closer to the truth. So any algorithm--even one based on reading Carl's mind--for generating sequences of bets that would be a Dutch Book for Carl no matter how his random credence generator comes out would be an algorithm generating sequences of bets that would be a Dutch Book for Alice.

Again, none of this really matters much. The post is really just a parenthetical note.