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.