Infinity vs decision theory
The first question is, how do we know that utility is a useful way to describe our preferences? The main reason is because of the Von-Neumann Morgenstern (VNM) theorem. The VNM theorem says that as long as our preferences follow a few basic axioms, then it is possible to describe our preferences as maximizing our expected utility according to a particular utility function. These are the four basic axioms:1
1. Completeness. For any two lotteries,2 we either prefer one over the other, or we prefer them equally.These axioms seem fairly basic, but some experts question their validity. It is possible to construct choices and give them to test subjects, showing that real people's preferences do not strictly obey the axioms. But does this mean that the VNM theorem is unsound, or does it mean that people miscalculate their own preferences? The answer lies outside the scope of this post.
2. Transitivity. If we prefer lottery A to lottery B, and lottery B to lottery C, then we prefer lottery A to lottery C. If A and B are equally preferable, and B and C are equally preferable, then so are A and C.
3. Continuity. If lottery A is preferred to B is preferred to C, then it is possible to combine lotteries A and C to make something that is equal to B. The constructed lottery will have a certain probability of producing A, and otherwise produces C.
4. Independence of alternatives. Given lottery A, we can construct lottery A' by attaching a certain probability to produce an alternative outcome. This axiom states that if A is preferred to B, then A' is preferred to B'.
The St. Petersburg paradox and Pascal's Wager present the possibility of a lottery with infinite utility. Unfortunately, this lottery does not obey the VNM axioms:
Completeness: It's not clear if two lotteries of infinite value are comparable to each other. The best we can do is define ∞ = ∞.
Continuity: ∞ is preferred to 1 is preferred to 0. However, there is no way to combine ∞ and 0 to get a lottery with value 1. As long as there is a nonzero probability of generating ∞, the lottery is still preferable to the value 1.
Independence of alternatives: Given lottery A, we can construct lottery A' by attaching a small probability of producing infinite value. But then, regardless of the preference between A and B, A' = B' = ∞.A new number system
Could we perhaps construct a new version of the VNM theorem which uses a different system of numbers, or a different set of axioms? Perhaps... I'm not ruling it out.
For example, we could allow infinitesimal probabilities "ϵ", such that ϵ*∞ = 1. But then you'd need ϵ to be distinct from 2*ϵ and 3*ϵ, and ∞ would be distinct from 2*∞ and 3*∞. To resolve problems with the independence of alternatives, we could define ∞ to be distinct from ∞+1 and ∞+2, etc.
Basically, I'm proposing a number system where the utility is in the form of a*∞+b, and probabilities are in the form of c*ϵ+d. But now we have to define division (what is 1/(∞+1)?). Another problem: if we construct a St. Petersburg lottery,3 all we know is that its utility is infinite. But in my number system, there are many different infinite numbers, so the value of the St. Petersburg lottery is poorly defined.
There's a reason math teachers tell you that infinity is not a number! If you try to build a number system around it, you run into all sorts of problems.4
Skipping the VNM theorem
There's another way to deal with infinite utility, and that is to forget about the VNM theorem entirely. Let's just assume that people's preferences are based on maximizing expected utility, and that one possible value of utility is infinity. This is a valid route to take. Sure, it may not describe real people's preferences, but maybe real people are just wrong! Maybe it describes how real people should act, even if they don't act that way.
If that's the route you take, then consider the following lottery: I choose not to believe in God, but when I turn 100, I will roll a dice and if I get a 6, I will choose to believe in God. There is a small probability that I will live to 100, and small probability that I will get a 6, and a small probability that I will successfully believe in God, and a small probability that God exists and will give me an infinite reward for believing. Therefore, this lottery has infinite value, and is equally preferable to the lottery where I simply believe in God.5
Yes, this is logically consistent. But would you bite the bullet?
Utility is bounded
The St. Petersburg lottery is constructed as follows: For every positive integer N, you have a probability 2^-N of getting an outcome with utility 2^N. This lottery has an expected utility that is greater than any finite value.6 For all practical purposes, the lottery behaves like it has infinite utility, leading to all the problems discussed previously.
Allow me to boil the St. Petersburg lottery down to its essence. If, for any value M, there exists a lottery whose value is greater than M, then it is possible to construct a lottery with infinite value. If utility is unbounded above, then infinite utility is possible.
Alternatively, if utility is unbounded below, then negatively infinite utility is possible. If utility is unbounded both above and below, then it is possible to construct a lottery whose value does not converge, not even on infinity.
This leaves us with several options:
(a) We need stricter construction rules forbidding the construction of St. Petersburg-like lotteries.7I advocate option (d). The other options are not impossible, but simply undesirable. Option (a) amounts to reconstructing probability theory. Option (d) is the only other option where the VNM axioms could be true. This is why, in my discussion of Pascal's Wager, I claimed that the probability of infinite utility is precisely zero. The VNM axioms require that utility is bounded.
(b) Utility is unbounded above and unbounded below. Some lotteries do not have defined utility. The completeness axiom of the VNM theorem fails.
(c) Utility is either bounded above or bounded below, but not both. Infinite utility exists. The VNM theorem fails, as discussed previously.
(d) Utility is bounded above and bounded below.
How can a maximum utility exist? Couldn't we pick a lottery that returns the maximum utility, and then repeat, doubling the maximum? The answer is that having two experiences in succession does not result in the sum of the utilities of those two experiences. And that's that.
In conclusion, Pascal's Wager doesn't just have philosophical problems, it also has mathematical problems.
1. For more technical descriptions of the VNM axioms, refer to Wikipedia.
2. A Lottery is a set of outcomes, each with their own probability. In decision theory, we talk about choosing between different lotteries, rather than between different outcomes, because the outcomes aren't completely determined by our choices.
3. See section "Utility is bounded" for a description of the St. Petersburg lottery.
4. I found an article about non-standard number systems that include infinitesimals. The number system I describe is most similar to Tall's superreals, but I think it still doesn't have the properties we need to reconcile the St. Petersburg paradox with the VNM theorem. But I can't rule out that some other number system might work.
5. This argument is described in the Stanford Encyclopedia of Philosophy, credited to Duff 1986 and Hájek 2003.
6. After the time of writing, I learned that it is not necessarily true that the St. Petersburg construction leads to an infinite utility. Due to some technicalities, when the VNM theorem is applied to lotteries with infinite outcomes, it is allowed that the utility of the lottery is not the expected utility calculated from its component outcomes. This may offer a way out of the paradox, but in my opinion it actually worsens the situation. It means that we need an "extended" VNM theorem with even stronger axioms.
7. This is similar to the resolution of Russell's paradox. Russell's paradox is about the construction of an impossible set. The resolution to the paradox was a whole new set theory, with rules forbidding the construction of such a set.