Wolfgang Schwarz

Ah, sh, push it!

I finally found the decision theory puzzle that I posted recently in a series of papers by Reed Richter from the mid 1980s. I'm not convinced by Richter's treatment though, and I'm still somewhat puzzled.

Here is Richter's version:

Button: You and another person, X, are put in separate rooms where each of you faces a button. If you both push the button within the next 10 minutes, you will (both) receive 10 Euros. If neither of you pushes the button, you (both) lose 1000 Euros. If one of you pushes and the other one doesn't, you (both) get 100 Euros.

What would you do? Most people, I guess, would push the button. After all, if you don't push it, there is a high risk of losing 1000 Euros. For how how can you be certain that X won't do the same? On the other hand, if you push the button, the worst possible outcome is a gain of 10 Euros.

In my previous version of the puzzle, I added the following:

Duplicates: X is an atom-for-atom copy of yourself, located in a situation that is a perfect duplicate of your own, and the universe is deterministic.

This means that you can be certain that X will do exactly what you do. If you decide not to push, you can be certain that you will lose 1000 Euros. It would be crazy not to push the button.

Evidential decision theory agrees: the expected payoff is 10 Euros for pushing, and -1000 Euros for not pushing. Things are less straightforward on causal decision theory, since your decision is merely evidence for X's decision; you don't actually control what X will do. Suppose you are fairly certain that both you and X will push the button. Then what would happen if you didn't push? You wouldn't thereby change X's behaviour. X is, we may assume, completely causally isolated from you. Assuming that X will push, what would happen if, counterfactually, you didn't push? Well, X would push and you wouldn't. You would get 100 Euros. Hence if you're fairly certain that X will push, causal decision theory tells you not to push. Of course, as soon as you carry out that plan, you can be certain that X won't push either and you will lose 1000 Euros.

In causal decision theory, the button case is an 'unstable' decision problem in which every option is guaranteed to appear wrong as soon as it is chosen: if you have chosen to push, it would have been better not to push; if you have chosen not to push, it would have been (much!) better to push. On some accounts, there is no rational option in such a case: everything you do is equally rational or irrational.

But that seems wrong. In the 'Duplicates' button case, the two options aren't on a par: you really ought to push the button. Or so it seems.

Evidential decision theory isn't safe from the puzzle. Here is another version.

Ideal Partner: X may or may not be a copy of yourself; at any rate, he is ideally rational. Moreover, he is certain that you are ideally rational as well.

What would you do? It still seems irrational not to push the button. Indeed, suppose it is not irrational not to push the button. Then for all you know, X may well not push it. (All you know about X, recall, is that he will do what's rational.) But if there's even a slight risk of X not pushing, it will be irrational for you not to push. Hence if it not irrational not to push, then it is irrational not to push. From which it follows that it is irrational not to push. Or so it seems.

But suppose it is irrational not to push. Then you can be certain that X will push, for you know that X will always do what's rational. And if it is certain that X will push, then not pushing is certain to give you a higher profit than pushing. This is true both in causal and in evidential decision theory: if it is irrational not to push, you can maximize your expected payoff by not pushing. That is, there are cases where doing the irrational thing is (both causally and evidentially) guaranteed to give you higher payoff than doing the rational thing!

Richter thinks that something like this is true, but to me it sounds more like a reductio of the assumption that pushing is the uniquely rational option.

However, didn't we have a reductio of the opposite assumption as well (three paragraphs back)? And isn't it intuitive anyway that one ought to push?

Let's try to find a way out.

Call a decision 'practically rational' iff it maximizes expected payoff, and let's assume X is ideally rational in this sense; he always maximizes expected payoff. It is out of the question then that sometimes an irrational decision can maximize expected payoff. The reductio against pushing being the unique rational action still goes through. We have to explain away 1) the alleged reductio against the opposite assumption, and 2) the intuition that pushing is the unique rational action.

Beginning with (1), I've assumed that if pushing is not the unique rational action, then you ought to be uncertain what X will do, given that all you know about X is that he is ideally rational. But this is not the 'ought' of practical rationality; it is the 'ought' of reasonable priors. Practical rationality doesn't stop you from being certain that X will push the button, even though not pushing it would not be irrational. And if this belief is strong enough, the practically rational decision is not to push the button. This is where the reductio went wrong: it isn't true that if there is no unique rational action, then the unique practically rational option is to push. If you have strange beliefs, you may well not push.

This also explains why it would be crazy not to push: if you don't push, you are either practically irrational or you have strange beliefs -- you are overly confident that your partner will push. This confidence is especially crazy in the Duplicate scenario where you are right now contemplating not to push, and fully aware that your partner will do whatever you will end up doing. (We should therefore not follow Jeffrey, Harper and others and say that in unstable problems, anything you can do is equally (ir)rational.)

As I said, I'm still puzzled by this, and I don't see through all the ramifications. But it seems to me that the best response will go somewhere along these lines.

Here is one of the residual puzzles. What if by 'the rational decision' we mean the decision that would maximize payoff given reasonable priors? Then there will obviously be cases where an irrational decision will have the highest expected payoff, if the agent has unreasonable priors. Will the Ideal Partner scenario be of this kind? No. Instead, it seems to fall apart. The problem is to say what the reasonable priors should be: let z be the credence an ideally rational person (in the new sense) assigns to the proposition that a random other ideally rational person will push the button in the Ideal Partner scenario. Then our ideal agent X will assume that this partner Y also gives credence z to the proposition that he, X, will push. (For X is certain that Y is ideally rational.) And X can calculate that Y will push if z < 0.925. Hence if z < 0.925, then X should be absolutely certain that Y will push; if z > 0.925, he should be certain that Y will not push. Either way we get a contradiction, since X's credence in Y pushing was supposed to be z, not 0 or 1. Finally, if z = 0.925, then X can't tell whether Y will push or not, since either is equally licensed by Y's reasonable priors. But then how can X give credence 0.925 to the proposition that Y will push? That seems unreasonable. The upshot is that any attempt to make 'reasonable prior' precise apparently leads to contradiction. (Hurray for subjective Bayesianism?)