Topics Rationality

Question Suppose it's your birthday, and you get your Aunt (who has an infinite amount of money in the bank) to mail you a signed check with the dollar amount left blank. Your Aunt says you cash the check for any amount you want, provided it is finite. Assume that the check will always go through, and that each extra dollar you request gives you at least some marginal utility. It seems in this case, every possible course of action is irrational. You could enter a million dollars in the dollar amount, but wouldn't it be better to request a billion dollars? For any amount you enter in the check, it would be irrational not to ask for more. But surely you should enter some amount onto the check, as even cashing a check for $1 is better than letting it sit on your dresser. But any amount you put onto the check would be irrational, so it seems that you have no rational options. Does this mean that the concept of "infinite value" is self-contradictory? If so we have a rebuttal to Pascal's Wager.

Accepted:
September 3, 2008

Comments

I hope that some of my co-panelists who think more about decision theory will chime in, but here are a few thoughts.

Cheap first try: it seems plausible that even if every additional dollar brings some marginal utility, by the time we reach, say, a trillion trillion dollars (a septillion dollars) the utility provided by the septiliion+1th dollar is so tiny that the utility cost of worrying about it exceeds the utility it could provide. Of course, that's not really an answer to your question. What you have in mind is a scenario on which it's not just that each additional dollar adds utility, but on which the total area under the utility curve goes to infinity. But it's worth noticing that these are separate ideas. Even if each additional dollar adds value, the infinite sum might still converge to a finite number.

So we can restate the problem this way: there's an infinite well of utility available, and you can choose to have any finite amount of it, but you have to specify the quantity of utiles (where each utile adds a constant amount of utility.) In that case, it seems, no matter what amount you pick, it would have been rational to ask for more; it would always have been possible to increase your payoff by a non-trivial amount. It's a nice problem, but it's not quite clear what it shows. We can agree that no matter what amount you pick it would have been rational to ask for more. But the conclusion you've suggested is stronger: that it would be irrational not to have asked for more. That's not so obvious. Compare: in ordinary situations, it's not clear that people who "satisfice" -- decide to make do with an amount of expected utility that's less than the maximum they could have achieved -- are being irrational.

However, there are some delicate issues here, best left to those who know more than I. Suppose we grant for argument's sake that in the situation you've posed, every option is irrational. Two questions: first, does this show that the very notion of infinite value is incoherent? And second, if the answer is yes, does this show that Pascal's Wager is fatally flawed?

On the first question, the answer is not an obvious yes. After all, suppose you were given this choice: (a) pick nothing; (b) pick a finite amount of utility; (c) pick infinite utility. This decision problem seems to have a clear answer. Why not say that the problem isn't with the idea of infinite utility; the conclusion is simply that if there could be infinite utility, some decision problems would have no good answer, while others would. The ones that do are the ones whose constraints allow you to maximize.

As for Pascal, suppose that what I've just said is wrong, and that the idea of infinite utility really does make no sense. Then certain classic versions of Pascal's wager are incoherent (we'll leave aside what Pascal himself may have had in mind), but there are neighboring arguments that don't simply collapse. Suppose that you are extremely skeptical about God's existence, but allow that it has at least some positive probability ε, however small. Then if the rewards of belief are great enough, assuming God exists, then there's still an "ordinary" expected utility argument in the offing. It's easy to construct little 2x2 tables with appropriate numbers (exercise for the reader), and we don't even need to assume that God would punish non-believers. All that Pascal-style arguments need assume is that what God would have on offer is wonderful enough to swamp other considerations, even given a low value for ε.

Needless to say, there are plenty of other criticisms that Pascal's Wager has to deal with. All that's being suggested here is that the problem of establishing the coherence of infinite utilities need not be one of them.