Binding and small probabilities

Suppose we neglect events with probability less than ϵ for some small ϵ > 0. Let’s suppose two independent random things have happened. First, an independent event E may or may not have happened, and P(E) = 2ϵ. Second, a fair die was rolled. You don’t have any information on whether E happened or the die was rolled. The following complex deal (The Deal) is offered to you.

If you accept The Deal, it will be revealed to you whether E happened. Then the following will happen:

1. If E happened, you get a choice between:

   1. you pay a dollar, or

   2. one following happens:

      1. you get a dollar if the die showed 1, 2, 3 or 4, but

      2. you get a year of torture if the die showed 5 or 6.

2. If E did not happen, you pay ϵ cents.

The event of E happening and the die showing 5 or 6 has probability 2ϵ ⋅ (2/6) = (2/3)ϵ which we have supposed is negligible. So, it seems that 1.b.ii can be completely neglected. On the other hand, the event of E happening and the die showing 1, 2, 3 or 4 has probability 2ϵ ⋅ (4/6) = (4/3)ϵ, which is not negligible.

What should you do? The difficulty here is that a full probabilistic evaluation of what you will do depends on what your choice in case 1.a will be. One way to handle such cases is through binding: you think of your choice as being a choice between strategies and then you stick to your strategy no matter what. This seems to be a good way to handle various paradoxes like Satan’s Apple.

What are the relevant strategies here? Well, in terms of pure strategies (we can consider mixed strategies, but in this case I think they won’t change anything), they are:

1. Reject The Deal.

2. Accept The Deal, and if E happened, pay the dollar.

3. Accept The Deal, and if E happened, don’t pay the dollar.

If you don’t neglect small probabilities, then clearly (A) is the right strategy to choose (and stick to).

Also, clearly, (B) is never the right strategy: whatever happens, you pay.

Now, suppose you do neglect small probabilities, and let’s evaluate (A) and (C). The payoff for (A) is zero. The payoff for (C) is, in dollars:

• (4/3)ϵ − (1−2ϵ)(0.01)ϵ > 0.

For the torture option drops out, as it has the negligible probability (2/3)ϵ.

So, if you neglect small probabilities, and take binding to a strategy to be the right approach to such puzzles, you should accept The Deal and bind yourself to not pay the dollar. But now notice how psychologically impossible the binding is. If in fact E happened—and the probability of E is 2ϵ, which is not negligible—then you have to choose between paying a dollar and a wager that has a 2/3 chance of yielding a dollar and a 1/3 chance of a year of torture. How could you possibly accept a 1/3 chance of a year of torture in exchange for about $1.67? Real brainwashing would be required, not just a mere resolution to stick to a strategy.

So what? Why can’t the proponent of the binding solution simply agree that (C) is the abstractly best strategy, but since we can’t practically bind ourselves to it, we are stuck with (A)? But there is something counterintuitive about thinking that (C) is the abstractly best strategy when it requires brainwashing that is this extreme.