(Cross-posted to my own blog.)
Some people, I think, are still under the impression that the infinities in Pascal's wager create trouble. Thus, there is the argument that even if you don't believe now, you might come to believe later, and hence the expected payoff for not believing now is also infinite (discounting hell), just as the payoff for believing now. Or there is the argument that you might believe now and end up in hell, so the payoff for believing now is undefined: infinity minus infinity.
But there are mathematically rigorous ways of modeling these infinities, such as Non-Standard Analysis (NSA) or Conway's surreal numbers. The basic idea is that we extend the field of real numbers to a larger ordered field with all of the same arithmetical operations, where the larger field contains numbers that are bigger than any standard real number (positive infinity), numbers that are bigger than zero and smaller than any positive standard real number (positive infinitesimals), etc. One works with the larger field by exactly the same rules as one works with reals. This is all perfectly rigorous.
Let's do an example of how it works. Suppose I am choosing between Christianity, Islam and Atheism. Let C, I and A be the claims that the respective view is true. Let's simplify by supposing I have three options: BC (believe and practice Christianity), BI (believe and practice Islam) and NR (no religious belief or practice).
Now I think about the payoff matrix. It's going to be something like this, where the columns depend on what is true and the rows on what I do:
Here, X is the payoff of heaven and -Y is the payoff of hell, and X and Y are positive infinities. I assume that the Christian and Islamic heavens are equally nice, and that the Christian and Islamic hells are equally unpleasant. The lowercase letters a, b and c indicate finite positive numbers. How did I come up with the table? Well, I made it up. But not completely arbitrarily. For instance, BC/C (I will use that symbolism to indicate the value in the C column of the BC row) is 0.9X-0.1Y. I was thinking: if Christianity is true, and you believe and practice it, there is a 90% chance you'll go to heaven and a 10% chance you'll go to hell. On the other hand, BC/I is 0.7X-0.3Y, because Islam expressly accepts the possibility of salvation for Christians (at least as long as they're not ex-Muslims, I think), but presumably the likelihood is lower than for a Muslim. BI/C is 0.6X-0.4Y, because while there are well developed Christian theological views on which a Muslim can be saved, these views are probably not an integral part of the tradition, so the BI/C expected payoff is lower than the BC/I one. The C and I columns of the tables should also include some finite numbers summands, but those aren't going to matter. A lot of the numbers can be tweaked in various ways, and I've taken somewhat more "liberal" (in the etymological sense) numbers--thus, some might say that the payoff of NR/C is 0.1X-0.9Y, etc.
What should one do, now? Well, it all depends on the epistemic probabilities of C, I and A. Let's suppose that they are: 0.1, 0.1 and 0.8, and calculate the payoffs of the three actions.
The expected payoff of BC is EBC = 0.1 (0.9X - 0.1Y) + 0.1 (0.7X - 0.3Y) + 0.8 (-a) = 0.16X - 0.04Y - 0.8a.
The expected payoff of BI is EBI = 0.15X - 0.05Y - 0.8b.
The expected payoff of NR is ENR = 0.08X - 0.12Y + 0.8c.
Now, let's compare these. EBC - EBI = 0.01X + 0.01Y + 0.8(b-a). Since X and Y are positive infinities, and b and a are finite, EBC - EBI > 0. So, EBC > EBI. EBI - ENR = 0.07X + 0.07Y - 0.8(b+c). Again, then EBI - ENR > 0 and so EBI > ENR. Just to be sure, we can also check EBC - ENR = 0.08X + 0.08Y - 0.8(a+c) > 0 so EBC > ENR.
Therefore, our rank ordering is: EBC > EBI > ENR. It's most prudent to become Christian, less prudent to become a Muslim and less prudent yet to have no religion. There are infinities all over the place in the calculations, but we can rigorously compare them.