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.

When being right doesn't pay

Like me, you might have naively speculated that the more truth you know, the better you’ll do in gambling scenarios. But this is mistaken, at least when taken in the strong sense that there is a guarantee of doing better (or even just as well).

For instance, suppose that Alice and Bob are betting on two coin flips. Alice has credence 1/2 for heads for each coin. Bob has credence 1 for heads for the first coin (maybe because he peeked) and credence 1/2 for the second coin. The house happens to offer Alice and Bob this bet:

• you get $10 if the first coin is heads and you pay $16 if the second coin is heads.

And as it happens both coins are heads.

Alice calculates the expected payoff at (1/2)⋅$10 − (1/2)⋅$16 = −$3 and declines. Bob calculates the expected payoff at $10 − (1/2)⋅$16 = $2 and accepts. But of course the actual payoff on double heads is −$6, so Bob is worse off than Alice for having been right about the first coin.

Can we at least say that in the long run Bob will be better off (financially, maybe not morally) for peeking than Alice? Yes, if the house offers the same bet each time and the coins are fair and Bob accepts the bet whenever he sees the first coin to be heads. But if the house also peeks at the coins and varies the offering based on the outcome, and Bob doesn’t notice this variation, then the house can fleece Bob (e.g., the house can offer the above bet whenever both coins are heads, and in all other cases offer some tiny bet worth a penny).

So in what sense can truth be guaranteed to help? Well, if you are betting on a single proposition, you will do better (or at least no worse) the closer your credence is to the actual truth value (where 0 is falsehood and 1 is truth).

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.

Betting on paradoxical sets

Suppose that a point z will be uniformly randomly chosen on the surface of a sphere S and you are asked to place bets as to which set z is in. Then, plausibly:

1. If two sets A and B are equivalent under rotations about the center of the sphere, you should accept this offer: get three dollars if z is in A and pay two dollars if z is in B.

But now consider a paradoxical decomposition of the whole sphere S, by a version of the Banach-Tarski Paradox[note 1]. In this, the sphere is partitioned into two subsets C and D, each of which can be decomposed into a finite number of subsets that can be rotated to form the whole sphere. Applying (1) to each set in the decomposition of C and its rotation, you will accept a sequence of deals that adds up to:

2. If z is in C, you get three dollars and if z is in S you pay two dollars.

Repeating this with D's decomposition, you get a sequence of deals that adds up to:

3. If z is in D, you get three dollars and if z is in S you pay two dollars.

But of course if z is in C or D, it is also in S, and if it's in S, then it's in exactly one of C or D. It follows that the deal adds up to:

4. No matter what, you get three dollars and you pay four dollars.

So, repeated application of (1) yields an unacceptable conclusion.

One might say that this is an artifact of the fact that there is no finitely additive rotation-invariant probability measure on the sphere. But I think the above formulation is a little bit more telling. I make no reference to probabilities here. All I assume is (1), which is a very intuitive rationality judgment, namely that when one has two equivalent scenarios, one should accept an unequal bet between them that is in one's favor.

What to conclude? One conclusion might be that a single application of (1) is fine, but the sequence of applications needed to yield (4) is not.

My own conclusion, however, is that it is metaphysically impossible to have a betting scenario like the above. But why not? What's wrong with it? Well, one possibility is that space is necessarily discrete, but that doesn't seem very plausible to me.

My own preference, however, is to conclude that it is impossible to have anything causally depend on whether a random point (or a particle or the like) is in one of these weird sets that are found in the paradoxical decomposition of the sphere. Why is that? I think it's because it would in effect be a violation of causal finitism, the thesis that no event can causally depend on infinitely many things. But the full story here requires significant amounts of work to complete.