Brownian motion and regret

Let B_t be a one-dimensional Brownian motion, i.e., Wiener process, with B₀ = 0. Let’s say that time 0 you are offered, for free, a game where your payoff at time 1 will be B₁. Since the expected value of a Brownian motion at any future time equals its current value, this game has zero value, so you are indifferent and go for it.

But here is a fun fact. With probability one, at infinitely many times t between 0 and 1 we will have B_t < 0 (this follows from Th. 27.24 here). At any such time, your expectation of your payout at B₁ to be negative. Thus, at infinitely many times you will regret your decision to play the game.

Of course, by symmetry, with probability one, at infinitely many times between 0 and 1 we will have B_t > 0. Thus if you refuse to play, then at infinitely many times you will regret your decision not to play the game.

So we have a case where regret is basically inevitable.

That said, the story only works if causal finitism is false. So if one is convinced (I am not) that regret should always be avoidable, we have some evidence for causal finitism.

Avoiding regrets

I’ve recently been troubled by cases where you are sure to regret your decision, but the decision still seems reasonable. Some of these cases involve reasonable-seeming violations of expected utility maximization, but there is also the Cable Guy paradox, though admittedly I think I can probably exclude the Cable Guy paradox with causal finitism.

I shared Cable Guy with Clare Pruss, and she said that the principle of avoiding future regrets is false, and should be modified to a principle of avoiding final future regrets, because there are ordinary cases where you expect to regret something temporarily. For instance, you volunteer to do something onerous, and you expect that while volunteering, you will be regreting your choice, but you will be glad afterwards.

In all the cases that I’ve been interested in, while you are sure that there will be regret at some point in the future, you are not sure that there will be regret at the end (half the time the Cable Guy comes at the time you bet on him coming, after all).

Independence axiom

Here is an argument for the von Neumann – Morgenstern axiom of independence.

Consider these axioms for a preference structure ≾ on lotteries.

1. If L ≺ M and K ≺ N, then pL + (1−p)K ≾ pM + (1−p)N.

2. If M dominates L, then pL + (1−p)N ≾ pM + (1−p)N.

3. If A ≾ pM + (1−p)N′ for all N′ dominating N, then A ≾ pM + (1−p)N.

4. If L ≺ M, there is an M′ that dominates L but is such that M′ ≺ M.

5. If M dominates L, then L ≺ M.

6. Transitivity and completeness.

7. 0 ⋅ L + 1 ⋅ N ∼ 1 ⋅ N + 0 ⋅ L ∼ N.

Now suppose that L ≺ M and 0 < p < 1. Let M′ dominate L but be such that M′ ≺ M by (4). Let N′ dominate N. Then pL + (1−p)N ≾ pM′ + (1−p)N by (2) and pM′ + (1−p)N ≾ pM + (1−p)N′ by (1). This is true for all N′ that dominates N, so pL + (1−p)N ≾ pM + (1−p)N by (3).

Now suppose that L ≾ M. Let M′ dominate M. Then L ≺ M′ by (5). By the above pL + (1−p)N ≾ pM′ + (1−p)N. This is true for all M′ dominating M, so by (3) we have pL + (1−p)N ≾ pM + (1−p)N. Hence we have independence for 0 < p < 1. And by (7) we get it for p = 0 and p = 1.

Enough mathematics. Now some philosophy. Can we say something in favor of the axioms? I think so. Axioms (5)–(7) are pretty standard fare. Axioms (3) and (4) are something like continuity axioms for the space of values. (I think axiom (4) actually follows from the other axioms.)

Axioms (1) and (2) are basically variants on independence. That’s where most of the philosophical work happens.

Axiom (2) is pretty plausible: it is a weak domination principle.

That leaves Axiom (1). I am thinking of it as a no-regret posit. For suppose the antecedent of (1) is true but the consequent is false, so by completeness pM + (1−p)N ≺ pL + (1−p)K. Suppose you chose pL + (1−p)K over pM + (1−p)N. Now imagine that the lottery is run in a step-wise fashion. First a coin that has probability p of heads is tossed to decide if the first (heads) or second (tails) option in the two complex lotteries is materialized, and then later M, N, L, K are resolved. If the coin is heads, then you now know you’re going to get L. But L ≺ M, so you regret your choice: it would have been much nicer to have gone for pM + (1−p)N. If the coin is tails, then you’re going to get K. But K ≺ N, so you regret your choice, too: it would have been much nicer to have gone for pM + (1−p)N. So you regret your initial choice no matter how the coin flip goes.

Moreover, if there are regrets, there is money to be made. Your opponent can offer to switch you to pM + (1−p)N for a small fee. And you’ll do it. So you have made a choice such that you will pay to undo it. That’s not rational.

So, we have good reason to accept Axiom (1).

This is a fairly convincing argument to me. A pity that the conclusion—the independence axiom—is false.