Monday, July 6, 2009

Pascal's wager and infinity

(Cross-posted to prosblogion).

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.

Crucial to Christianity being favored over Islam was the fact that BC/I was bigger than BI/C: that Islam is more accepting of salvation for Christians than Christianity is of salvation for Muslims. If BC/I and BI/C were the same, then we'd have a tie between the infinities in EBC and EBI, and we'd have to decide based on comparisons between finite numbers like a, b and c (and finite summands in the other columns that I omitted for simplicity)--how much trouble it is to be a Christian versus being a Muslim, etc. However, in real life, I think the probabilities of Christianity and Islam aren't going to be the same (recall that above I assumed both were 0.1), because there are better apologetic arguments for Christianity and against Islam, and so even if BC/I and BI/C are the same, one will get the result that one should become Christian.

It is an interesting result that Pascal's wager considerations favor more exclusivist religions over more inclusivist ones--the inclusivist ones lower the risk of believing something else, while the exclusivist ones increase it.

It's easy to extend the table to include deities who send everybody to hell unless they are atheists, etc. But the probabilities of such deities are very low. There is significant evidence of the truth of Christianity and some evidence of the truth of Islam in the apologetic arguments for the two religions, but the evidence for such deities is very, very low. We can add another column to the table, but as long as the probability of it is small (e.g., 0.001), it won't matter much.


Vlastimil Vohánka said...

Thanks for this intro to the non-standard Wager.

Martin Cooke said...

This is all perfectly rigorous.
But what you want is something true, reliable or plausible. Nonstandard Analysis is based upon the first-order possibility of infinite numbers satisfying Peano's axioms for the natural numbers, for example. Go to second-order and you get falsities (e.g. there is no such infinite number by definition).

Mr Veale said...

A very interesting introduction. And yes, I can see that infinite values need not render the wager irrational.

But when Pascal talked about infinities, I think he meant something akin to "incommensurable".

And the wager becomes a little too abstract to be of practical use once we start comparing infinities.

Of course, that's not to deny the considerable philosophical interest generated by the wager using infinities.
And this was a helpful post for the non-mathematician.

Many thanks.

Alexander R Pruss said...


I am not sure what you mean by "there is no infinite number by definition".

Once one accepts ZFC, one gets the existence of nonstandard reals as a theorem.

I suppose there is a sense in which nonstandard arithmetic is higher order, viz., that it involves ultrafilter constructions that are not numbers but set theoretic constructions from numbers. But in the very same sense, real numbers aren't first order. They are sets of rational numbers (i.e., Dedekind cuts), and rational numbers are equivalence classes (and hence sets of) ordered pairs of integers, and integers are themselves formed out of naturals.

Mr Veale said...

"It's easy to extend the table to include deities who send everybody to hell unless they are atheists, etc."

I hate to go on about this - but shouldn't it be zero?
It should be fairly easy to deduce a contradiction between the concept of a God worthy of worship, and a theology in which that God sends everyone to Hell.

But I take it that the infinities can remain in an evidentially concerned wager.
But it would be conflicting religious claims that send us to the evidence - not the possibility of an atheist universe. Is that correct?
If so, that's very interesting.

Thank you for this thread. It's been a while since I looked at the wager.

Alexander R Pruss said...

Those would be deities that are not God. :-)

Mr Veale said...

Or parodies of God. We could call them "Zods" for ease of reference(-;

Mr Veale said...

In any case, it's the property of being most "worthy of worship" that makes something God and not Zod. That would include something like "greatest being imaginable".
And any reason for thinking that a Zod is possible -
(say our Zod is a spiritual, irrational being powerful enough to torture you forever for worsipping God) -
would be reason for thinking that God is possible.
And you want to bet on the greatest being (who can give most reward). Not cheap substitutes.
After all Pascal believed in Satan and demons, and presumably believed that at least some idolaters worshipped demons; but they did not show up in the wager.


Martin Cooke said...

Hi Alex,

There is no such infinite number by definition; i.e. the natural numbers are all finite, by definition. So what you say is perhaps a reason not to accept ZFC.

The surreal numbers simply postulate algebraically such numbers as 1/w (where w is the ordinal number of the natural numbers in their natural ordering), but there is again the question of how applicable they are. A similar question arises with the standard transfinite numbers, such as w.

Martin Cooke said...

...a problem with applying 1/w to measurable stuff is sliding, i.e. that w things of length 1/w make up an aggregate of length 1, and yet adding another 1/w to the start of that w-sequence and shifting the other bits along to make room for it also makes up 1. The question is whether or not that is acceptable behaviour in the measurable stuff. Similarly, rearranging the stuff can change its measured value (as in Hilbert's Hotel).

I think that similar problems arise with NSA, but there is certainly the problem that NSA is based upon the first-order logical possibility of actually impossible measures (infinite natural numbers). For measuring stuff you might need such infinitesimals as the so-called irreal ones that are found in lines full of 1/0 points (where 1/0 is an 'actual' cardinal number). The problem with them is that their analysis remains to be done.

metametatics said...

Whether the technical features of something can be realized in ZFC does not determine its truth. After all, the technical features of assigning to the existence of God a probability of zero can be realized in ZFC as well. I don't think btw one even needs Choice to accomplish this.

If one is going to allow infinite surreals in one's model, one might as well also allow infinitesimals. The infinitesimals of signifance would be those that are greater than zero but less than any ordinary real greater than zero. This too has a mathematics.

Then, one might as well say that the probability of an infinite payoff is mapped to such an infinitesimal.

It does not seem ontologically possible for a finite being to possess an infinite good for in possessing an infinite good (such as infinite joy), one would in that respect not be finite.

If one wants to say that the payoff is not truly infinite but only incommensurate with other finite payoffs, i.e. transfinite, then it is not clear why life in heaven would be the only case of such incommensurability. The payoff from sex with certain women for some men may be incommensurate with the payoff from eating ice cream, i.e. they would prefer one night with certain women to any amount of ice cream of any kind for the rest of their lives, even if their lives were immortal. So here, even though the latter be of infinite duration, the former is incommensurably greater (for some people).

In claiming that the payoff is transfinite one can say that any frequency of the payoff is transfinite or that an instance of the payoff is itself finite but that the frequency (eternity of heaven) makes it transfinite. In the former case we have an ontological problem similar to the one I mentioned. In either case, it does not guarantee incommensurability with payoffs in this finite life on earth.

If God exists, it seems more plausible that she would grant people what they prefer in accordance with preference utilitarianism. Personally, I would prefer an eternity of carnal pleasure and intellectual curiousity as is tasted on earth to an eternity of worship and contemplation.

The probability of a god that would consign someone to not being able to enjoy those things he happens to find pleasurable and would at the same time be a god worthy of worship would seem to be either zero or infinitesimal.

The broader problem with Pascal's wager is that if God exists and the infinite payoff is to be achieved, it would seem it cannot be achieved when motivated by utility maximization and that the selfless love of God for her own sake that would give the infinite payoff is as likely to be achieved by utility maximization as it is by utility minimization.

Mr Veale said...

Dr Pruss

Along with the Hedonistic Paradox, I wonder if Nozick's experience machines couldn't help us evaluate rewards?
They might also help us envision what infinite loss might be like for humans.

I've been having a little trouble with googlemail. Someone seems to have accessed my account.
If you can continue the conversation I can be reached at

Graham Veale