Suppose we have an infinite fair lottery with tickets 1,2,3,…. Now consider a wager W where you get 1/n units of utility if ticket n wins. How should you value that wager?
Any value less than zero is clearly a non-starter. How about zero? Well, that would violate the dominance principle: you would be indifferent between W and getting nothing, and yet W is guaranteed to give you something positive. What about something bigger than zero? Well, any real number y bigger than zero has the following problem as a price: You are nearly certain (i.e., within an infinitesimal of certainty) that the payoff of W will be less than y/2, and hence you’ve overpaid by at least y/2.
But what about some price that isn’t a real number? By the above argument, that price would have to be bigger than zero, but must be smaller than every positive real number. In other words, it must be infinitesimal. But any such price will violation dominance as well: you would be indifferent between W and getting that price, yet it is certain that W would give you something bigger—namely one of the real numbers 1, 1/2, 1/3, ....
So it seems that no price, real numbered or other, will do.
(This argument is adapted from one that Russell and Isaacs give in the case of the St Petersburg paradox.)
One way out will be familiar to readers of my work: Reject the possibility of infinite fair lotteries, and thereby get yet another argument for causal finitism.
But for those who don’t like controversial metaphysics solutions to decision theoretic problems, there is another way: Deny the dominance principle, price W at zero, and hold that sometimes it is rational to be indifferent between two outcomes, one of which is guaranteed to be better than the other no matter what.
This may sound crazy. But consider someone who assigns the price zero to a dart tossing game where you get a dollar if the dart hits the exact center and nothing otherwise, reasoning that the classical mathematical expected value of that game for any continuous distribution of dart tosses (such as a normal distribution around the center) is zero. I think this response to an offer to play is quite rational: “I am nearly certain to lose, so what’s the point of playing?” Now, that case doesn’t violate the same dominance principle as the lottery case—it violates a stronger dominance principle that says that if one option is guaranteed to be at least as good as the other and in some possible scenario is better, then it should be preferred. But I think the dart case may soften one up for thinking this:
- If (a) some game never has an outcome that’s negative, and (b) for any positive real, it is nearly certain that the outcome of some game will be less than that, I should value it at zero or less.
And if we do that, then we have to value W at zero. Yes, if you reject W in favor of nothing, you’ve lost something. But probably very, very little.
Here is another weakish reason to be suspicious of dominance. Dominance is too similar to conglomerability, and conglomerability should be suspicious to anyone who likes exotic probabilistic cases. (By the way, this paper connects with this.)
7 comments:
Question: You say, "One way out will be familiar to readers of my work: Reject the possibility of infinite fair lotteries, and thereby get yet another argument for causal finitism."
Are you suggesting I assign a credence of *zero* to the possibility of fair lotteries? I don't know if I can do *that,* even though you have successfully gotten my credence in their possibility exceedingly low.
But then, I still have a dilemma. Let's consider wager W* where I get wager W if wager W is possible, or else I get 0 utility. What should I evaluate W* at? Well, it should be the product of my credence that wager W is possible and the expected utility of wager W if it was possible. Thus, how does rejecting the possibility of infinite fair lotteries help if I can't shake a nonzero credence in them?
That's a good point. However that may just fit with a family of unanswerable questions about how to make decisions about crazy low probability scenarios. The big one: sure I'm confident in classical logic. But 100%? Maybe not. How to hedge on that? I have no idea.
I think one solution is to say that fully rational agents have figured out the philosophical issues and assign 1 to the necessary truths there. We're not such, and what nonideal agents should do is a tough problem all over.
Is this really a problem?
It seems clear enough what you should do. At a price of $0 or less, accept the wager. At any strictly positive price, reject it. This does not seem obviously unreasonable or inconsistent. Does it matter that there is no equivalent value?
Maybe the problem is something like this. You can imagine a wager as itself being a kind of "generalized price". You might think that a fair "generalized price" for the wager is an independent copy of it. So suppose that W1 is this wager and W2 is the independent copy. But you know that whatever you get from W1, it'll be more than W2. In other words, each possible outcome of W1 beats getting W2. (Of course, this is basically a variant on nonconglomerability.)
You can do the same sort of thing with two St Petersburg wagers. Before the event, they look similar. But given the result of one, the other always looks better. Note that St Petersburg uses standard probability. But it requires unbounded utilities, and you have to be prepared to run a supertask if you want to be sure of completing it in finite time using coin flips.
The spinners of the previous two posts give an interesting example. For a single spinner, the rotated result dominates the raw result. But both the raw results and the rotated results are probabilistically similar to the result of a second independent spinner.
I’m not sure what to make of such examples.
Ian:
Yeah, that observation about St P was in the Russell-Isaacs paper that started me thinking about this stuff.
Do we know that "the raw results and the rotated results are probabilistically similar to the result of a second independent spinner"? Maybe a lesson to be learned from these results is just that there are no uniform spinners, and hence that a rotation of a spinner is not probabilistically similar to the spinner itself?
In the post, you suggest rejecting dominance. Another option would be to reject totality. Preference between wagers could be a preorder, but not all wagers need be comparable. Then independent probabilistically similar (isomorphic) wagers need not be comparable.
You could say, for example, that a St Petersburg wager + $1 dominates the original wager (so is preferable), but that neither is comparable to an independent St Petersburg wager. This would avoid the problems discussed in Russell-Isaacs. Similarly for the example in this post – two independent versions of the wager need not be comparable.
Of course, this has its own problems – it gives no guidance in choosing between incomparable options. But maybe that is as it should be. [Note that standard theory gives no guidance for (maximally) Lebesgue non-measurable sets.] In any case, I’m not sure that that’s worse than rejecting dominance.
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