bigideas

It seems to me there are at least two troublesome things about the situation we’re presented with.

First, and this is the one we got into most last night, you have the “free will or determinism” thing. If the Predictor is really 99.9% accurate, it must surely have something like a time machine or an ability to see into the future. Hence, the Predictor already knows, with 99.9% certainty, what you’ll choose before putting the money in the boxes.

We can unroll the “time travel” malarkey by changing the scenario to this, equivalent one involving not a Predictor but a mere Administrator. You tell the Administrator a week in advance which choice you’ll make. The Administrator then rolls an enormous 1000-sided dice (or die, if you must). If it comes up 638 (say) then the Administrator does the opposite of what he usually does. You’re constrained to do what you said you’d do when the fateful day arrives.

In this case you can apply an “expected utility” argument. The expected utility of something is “what it’s worth to me”. Say I’ve got a scratch card that has a one-in-ten chance of paying out £100, otherwise it pays out nothing. How much, at most, would you buy it for? Anything less than £10, right? Because if you bought 10 of them you’d spend £100 and get £100 as a prize (from one of them). The expected utility of the scratch card is £10.

Using a similar argument, the expected utility of choosing both boxes is £1999 (using the hypothetical 99.9% probability), whereas box B alone is £999000. You’d be mad not to choose box B in the modified scenario, where you choose first.

Given that the Predictor is so accurate, it’s very tempting to think that they can, or might as well be able to, see into the future and they therefore more or less know what you’re going to do. You can think of your choice as causing — backwards in time — the contents of box B to be either £1,000,000 or nothing.

In fact, the expected utility argument concludes that Box B should be chosen even if the Predictor isn’t all that good. Any success rate better than 50.1% (according to my back-of-envelope calculation) is enough to make it worth choosing Box B in the event that you think backward causality is at work.

Secondly, however, you have to remember that the Predictor has really made a choice abotu what goes in the boxes. As you sit there looking at them, you know for sure that Box A contains £1,000 and Box B contains either £1,000,000 or nothing. Nobody’s going to change the contents of the boxes as a result of your choice.

Taking only Box B gives you either £1,000,000 or £0. Taking both boxes gives you either £1,001,000 or £1,000. Which do you prefer? Of course you should choose both boxes. Choosing Box B means you’re throwing away £1,000 whether or not the Predictor got it right.

The “paradoxical” part springs from the fact that both of these seem like rational arguments, yet they lead to completely contradictory strategies. And even if you don’t believe the Predictor can see the future, so there can’t be any backward causality, the “expected utility” argument still looks convincing. Last night, all of us went for Box B only the first time around, and most of us stuck with it after discussion as well.

Would reducing the Predictor’s observed success rate to, say, 60% make a difference to our choices?

By Ornette on Feb 23, 2007

Addendum: I just got into this again over lunch with 3 colleagues. All three went for Box B on gut reaction. One figured out the expected utility strategy and stuck to it like glue (hence chose box B every time). Of the other two, only one changed his mind to pick both boxes.

Maybe this problem is really about how willing we are to believe that our choices can affect things that have already been determined. If we feel like we can, we’ll go for box B; if we feel we can’t, we’ll go for both boxes.

I still choose both boxes, unless I know the Predictor really can see into the future.

By Ornette on Feb 23, 2007