An Actuarial Analysis of Pascal’s Wager

[Analytics]

Daniel C. Peterson has said, "Fundamental to a long-term writing project of mine — or, at least, to its opening chapter — is a famous argument that has come to be called 'Pascal’s wager.’”

In one of his papers on the topic, he approvingly quotes Ray Barfield as saying:

Pascal wants to say that if it’s even possible that there’s more to
the universe and myself than atoms in the void, and if it might
be the case that this possible more is God, then it approaches
certainty that I should include this in my life planning as I
answer the question, What should I do?

When referring to Pascal’s proposition, Peterson points out that Pascal isn’t arguing that God exists--rather, he is arguing that regardless of the evidence, it is in our selfish best interest to believe.

As an actuary I specialize in pricing decisions in the face of uncertainty, and there is a fundamental flaw in his argument. The flaw is in this assertion "If God exists, the potential gains for acting on that correct assumption are infinite (e.g., salvation and eternal life in heaven).” (Peterson 2016).

The idea is that since living a life in the Celestial Kingdom would be way better than life in a lower kingdom, and since this way better life would last an infinite amount of time, the upside is infinite.

The flaw in this is the time-value of blessings. An actuary (or an economist) would approach this as follows. Let’s assume that the value of living in the Celestial Kingdom for a finite amount of time is finite. For example, say I were in the Telestial Kingdom and that the Celestial Kingdom is so much better than the Telestial that I’d spend $1,000,000 for a one-year vacation to the Celestial Kingdom, reflecting how spectacular life in the CK would be. That’s a lot of money for a one-year vacation, but it is still finite.

Here is the problem. While I might spend $1,000,000 now for one-year in the Celestial Kingdom starting now, I would not spend $1,000,000 now for one-year in the Celestial Kingdom starting one-million years from now.

The discounting rate for blessings might be 5%. With this assumption, I’d spend $1,000,000 now for a vacation in the CK starting now, but only $952,381 for one starting in a year, and only $907,029 for one starting the year after that. It turns out that if you sum up an infinite number of years at a modest 5% discount rate, you get a finite value of the infinite blessings. With these assumptions, if one-year in the CK is worth $1M, then an eternity in the CK is worth $20 million.

Understanding that the value of the promised blessings are finite, we realize we need to actually do the math. If the odds of Mormonism being true are one-in-a-million, then the expected value of its promises are twenty bucks.

While I might pay $5.00 for a lottery ticket with an expected value of $3.00, I would not compromise my integrity and steel a lottery ticket with an expected value of $3.00. likewise, I wouldn’t sell my intellectual integrity and pretend to believe a religion with an after-death expected payoff of $20.00.

[Everybody Wang Chung]

[malkie]

[huckelberry]

I think that in there is only one God so you cannot choose the wrong one. Still One may be able to say insulting things about god . Sometimes religion might do that.

[drumdude]

I think in DCP’s mind, he gets brownie points for believing in Mormonism even if traditional Christianity turns out to be true. This is probably common with most Christians. There are just too many sects to make worrying about it a real issue.

[Gadianton]

It turns out that if you sum up an infinite number of years at a modest 5% discount rate, you get a finite value of the infinite blessings.

I agree, and I've made the same argument although not specifically in the context of Pascal's wager. Everybody Wang Chung's Homer argument is my go-to, but I certainly agree with the NPV argument also.

There are other finance concepts that don't work to Dan's favor. For instance, think about bonds. If the yield-to-maturity is off the charts, it's because the market deems it as a risky investment. What Pascal is really saying is that there's "loose change on the floor" and everyone is stupid if they don't scoop it up. Just like Bernie Madoff, the religionist inflates the payoff to astronomic values. But we we know, thanks to modern portfolio theory and rational expectations -- concepts Dan should espouse since he claims to believe in a free market -- that inflating the prospects of eternal life such that nobody in their right mind fails to invest is really not doing anything except for blowing up risk premium.

If it sounds too good to be true then it probably is.

[malkie]

I think that in there is only one God so you cannot choose the wrong one. Still One may be able to say insulting things about god . Sometimes religion might do that.

I guess I should have pointed out that I was just reposting Everybody Wang Chung's image because it didn't show up for me in his post. However, I'm OK defending the image, in a way, because I believe as much in multiple gods as I do in one god, whichever one is chosen.

[Physics Guy]

Here is the problem. While I might spend $1,000,000 now for one-year in the Celestial Kingdom starting now, I would not spend $1,000,000 now for one-year in the Celestial Kingdom starting one-million years from now.

Why is there a discounting rate for future rewards, though? Is it not mainly because I might die before I get my reward?

Besides that, there is also the steadily increasing opportunity cost of an investment with a longer-delayed reward. The longer I have to wait to get my reward, the higher the chances become that I will have to forego some even better reward that comes up in the meantime because I won't have the money for that ticket, having spent it already now.

Neither of these factor seems relevant to Pascal's wager, however. So I'm afraid I question this actuarial rebuttal to Pascal's argument.

A possible way to salvage the rebuttal might be to admit that a longer-lasting reward in God's heaven can retain a value directly proportional to duration, without exponential discounting, but then to argue that this hypothetically postulated non-discounting has to be reflected in the probability that we assign to the hypothesis. In our mortal experience, rewards in the future have to be discounted for all kinds of all-too-likely reasons. So if Pascal's proposition were that belief in God now brings bliss that lasts N hundred years, then we should assign Pascal's proposition a prior probability that decreases with N exponentially. This would reflect how much less probable it seems that nothing would somehow go wrong with the whole streets-of-gold thing even over ten thousand years, in comparison with however likely we find it to be that heaven would stay great for a century.

By applying that kind of reasoning, I can see how maybe one could reject Pascal's conclusion. The discounting would effectively apply on the risk side, rather than on the reward, and not in the usual actuarial way. It's more like an actuarial assessment of an investment opportunity that is contractually guaranteed to have no depreciation because all possible liabilities or missed opportunities will be compensated, and so the assessment is entirely about whether the whole offer is simply a fraud. The better the deal, the less likely it is to be true.

[I Have Questions]

Pascal’s Wager is a rigged horse race. It excludes so many reasonable possibilities from the equation as to render it totally binary and a bit silly. It assumes only two possible outcomes - eternal reward or eternal punishment. It doesn’t allow for “non existence” after death. It doesn’t allow for negative costs associated with belief in God, such as the eternal loss of loved ones who chose disbelief. Likewise, it doesn’t account multiple Gods. Surely if you are going to “price” up Pascal’s Wager you would build in the risks and options that Pascal has neglected to include?

[Physics Guy]

In a strong interpretation, where Pascal's wager is supposed to imply that anyone must be stupid not to be Catholic, the argument fails completely in several ways. I think this may be unfair to Pascal, though. He was pretty fanatical, but he did also know his game theory, having invented it. His Wager is also just a jotted note to himself, found after his death in a pile of such notes. We don't know whether he himself considered it a solid argument.

A weaker interpretation of the Wager is an important point, though. Choosing the principles and assumptions on which one will base one's life is not just a scientific assessment of what is most likely to be true, but rather a move in a game. Risk is not the only factor to consider: there is also reward. A heavy bet on a long shot can be the wisest move, if the payoff is disproportionate.

This isn't a super-subtle point, but it can be overlooked even now. I think Pascal deserves some credit for bringing it explicitly into the theory of religious belief. By no means does the Wager settle the question of what to believe, but I think it improves the question, if we don't take the Pensées version of the Wager too literally.