# Is Pascal’s Wager Sound?

OK, this isn’t actually technically a game, but since a lot of people think of it as such (given the common depiction through payoff matrices, with probabilities of different scenarios), we’re going to cover it anyway.

The basic gist of Pascal’s Wager is that by hedging one’s bets and choosing the path of religion, one can expect a greater payoff than by being not religious. Its proponents offer two possible states of the world: either God exists, or God does not. From a person’s standpoint, one can choose to be religious, or not:

 Choice God Exists God Does Not Exist Religious $\infty - C$ $- C$ Not Religious $G -\infty$ $G$ Fig. 1: Standard Pascal’s Wager

One gets infinite payoff from being religious if God exists, as then one goes to heaven. If one is not, and God exists, one goes to hell, and gets a payoff of negative infinity. By being religious, one incurs a finite cost C, as religion isn’t so much fun apparently; if one is not religious, one gets positive payoff G since one gets to party all the time.

Now suppose God exists with some probability P>0, however small that might be. Then the expected payoff from being religious is, according to the argument, P(∞ – C) – (1 – P)C = ∞. The payoff from being irreligious is P(G – ∞) + (1 – P)G = -∞. Thus one is better off being religious.

A lot of people really hate this argument, and so do their utmost to bring it down. Yet this argument is not as bad as they think it is. Let’s go through some of the criticisms.

The first criticism is that Pascal automatically assumes that if God exists, then Catholicism is true. But there are many religions out there that posit the existence of heaven and hell, yet are mutually incompatible. For example, Catholics (at least conservative ones) would condemn Muslims for not accepting Jesus as their Lord and Savior; Muslims would condemn Catholics as polytheists for this very acceptance. Since one can do this calculus for both religions, the arguments negate each other in paradox, as we end up granting both infinite payoffs and negative infinite payoffs to the same groups!

The second criticism is the so-called Atheist’s Wager. It could be instead that God wants us to live good lives, and be rational. Since one can live a better life by being irreligious, this is what God would prefer of us. Hence, according to the Atheist’s Wager, the payoff matrix should look as follows:

 Choice God Exists God Does Not Exist Religious $- \infty - C$ $- C$ Not Religious $G + \infty$ $G$ Fig. 2: Atheist’s Wager

Thus, says the atheist, it is a dominant strategy to be irreligious.

The problem with both these arguments is actually a problem with the initial formulation of Pascal’s Wager. However, while we can tweak Pascal’s Wager to make the problem go away, this flaw is quite possibly fatal to the above two criticisms. So now that I’ve built up the drama, here’s the problem: infinity is not a well-defined number on the real line (which we use for expected utilities). Instead, we must use a limit of some number B as it increases toward infinity. Thus, Pascal’s Wager should look like:

 Choice God Exists God Does Not Exist Religious $lim_{B\rightarrow\infty} B - C$ $- C$ Not Religious $lim_{B\rightarrow\infty}G - B$ $G$ Fig. 3: Revised Pascal’s Wager

Addressing the first criticism, we can then compare the possibilities that each religion is true by seeing which is the most probable among them. Thus, taking Catholicism and Islam, with probabilities PC and PI, and difficulties CC and CI, respectively, to establish Catholicism (without loss of generality) as the better way to go of the two, we just need to check whether

PCB – PIB – CC > PIB – PCB – CI,

which, as B gets large, is equivalent to just checking whether PC > PI. As a Jew, I would probably argue that the evidence/support for Judaism is the greatest of all the religions that have a system of heaven/hell, even if the support for any of these religions is slight.

Similarly, we can check whether, given the evidence in front of us, the probability that the Atheist’s Wager (AW) is true is greater than that of Pascal’s Wager (PW):

 Choice God Exists & PW God Exists & AW God Does Not Exist Religious $lim_{B\rightarrow\infty} B - C$ $lim_{B\rightarrow\infty} - B - C$ $- C$ Not Religious $lim_{B\rightarrow\infty}G - B$ $lim_{B\rightarrow\infty}G + B$ $G$ Fig. 4: Revised Pascal’s Wager vs. Atheist’s Wager

Comparing the respective probabilities of the Atheist’s Wager and Pascal’s Wager, PA and PP, we check whether

PAB – PPB + G > PPB – PAB – C

Given that almost all purported religious claims in the past about heaven and hell have been based on being religious, and none have supported the Atheist’s Wager (the closest you get is some who claim that all people go to heaven, whether religious or not), I would think that PP > PA (though I admit the possibility that I am wrong). Thus, the Atheist’s Wager loses out to Pascal’s Wager by comparing expected utilities.

Thus, there is a very strong case to make that if one is merely comparing expected utilities, then  no matter how small the probability is that God exists (as long as it is not zero), Pascal’s Wager is actually sound.

Of course, one might not be comparing expected utilities. After all, if there is only an extremely miniscule possibility of a hugely negative payoff, no matter how bad it is, it might not be a bad thing to completely ignore that possibility. But that is in the realm of decision theory, not game theory. Thus I’ll leave it to your intuitions: if you’re risk averse (as pretty much everyone is – that’s why we all buy insurance), and carry this reasoning even against remote possibilities, then by all means, Pascal’s Wager seems to work. But if you’re willing to take the risk, since it doesn’t matter if you think there’s only, say, a one-in-a-trillion chance that you’ll go to hell if you’re not religious, then go out and boogie.

### 2 Comments on “Is Pascal’s Wager Sound?”

1. Actually, there is a far worse flaw in the wager: The assumption that the deity in question would be fooled/tolerate this type of behaviour. At least the omniscient-god religions would need to seriously question this assumption, bearing in mind that the consequences could be either no payout or a severe penalty. (Of course, you could factor in a p_{God is fooled} or similar, but without this value being unknown and likely very small, this would not be much help.)

To quote Terry Pratchet (http://www.lspace.org/books/pqf/hogfather.html )

This is very similar to the suggestion put forward by the Quirmian philosopher Ventre, who said, “Possibly the gods exist, and possibly they do not. So why not believe in them in any case? If it’s all true you’ll go to a lovely place when you die, and if it isn’t then you’ve lost nothing, right?” When he died he woke up in a circle of gods holding nasty-looking sticks and one of them said, “We’re going to show you what we think of Mr Clever Dick in these parts…”

— (Terry Pratchett, Hogfather)

2. edwinkite says:

I hope neither of you ever fall victim to Pascal’s Mugging … http://www.nickbostrom.com/papers/pascal.pdf