Tuesday, April 14, 2015

Truth and Dutch Books

Suppose I initially assigned probability 0.5 to p and 0.5 to ~p. Suppose p is in fact true, and my credence in p comes to be magically increased to 0.8 without my credence in ~p being changed. I thus have inconsistent probabilities: 0.8 for p and 0.5 for ~p. This is supposed to be bad: it lays me open to Dutch Books. For instance, I will accept the following pair of options:

  1. Pay $0.75 to win $1.00 if p
  2. Pay $0.45 to win $1.00 if ~p.
But if I do that, then I will pay $1.20 and get $1.00, for a net loss of $0.20.

Yes, that's an unhappy result. But note that I am actually better off than earlier when my credences were consistent. Earlier I would have rejected (1) since my credence in p was 0.5, but I would have accepted (2). So I would have paid $0.45 and got nothing to show for it. Thus my revision in the direction of truth made me be better off, even though it also led me to accept a Dutch Book.

This suggests that pragmatically and synchronically speaking what matters is truth, not probabilistic consistency. Better be inconsistent and closer to truth than consistent and further from truth.

Diachronically, of course, at least logical inconsistency could be dangerous, as it can lead to lots of absurd conclusions. But in practice we all have inconsistent beliefs and we manage to contain the inconsistency without much in the way of explosion.

So what's so bad about Dutch Books? It seems to be this: an opponent who knows (with certainty) your credences and doesn't know (at least with certainty) whether p is true can offer you a series of bets that you are guaranteed to lose money on. This is a big deal if you're playing an adversarial game against such an opponent. But such games are, I think, a special case, and while they do occur in war, business, sport and other competitive pursuits, we should not let competitive pursuits against fellow humans dictate the nature of rationality to us. And note a curious thing: consistency is not the only available strategy against such an opponent—hiding your credences will also help. If you revise your credence in the direction of truth but your opponent doesn't know about your revision, you will do at least as well as before, and quite possibly better.


Heath White said...

The following is an inconsistent set:

1. If people have inconsistent credences, they will take bets they are guaranteed to lose.
2. If people will take bets they are guaranteed to lose, then there is money to be made as a Dutch bookie.
3. Where there is money to be made, people will almost certainly try to make it.
4. There are no people trying to make money as Dutch bookies.
5. People do have inconsistent credences.

Only 1 and 5 are at all likely to be false, but 1 seems the much more likely culprit to me. And that does a great deal to defuse the Dutch book argument, which after all says that you shouldn't have inconsistent credences because they are bad for you.

Alexander R Pruss said...

This is a very nice argument. Took me a while to get what you were doing! Very nice indeed.