An Actuarial Analysis of Pascal’s Wager

[Analytics]

Is there anything we can gain from Dan's recent admitting to living Kierkegaard's life of the brutish aesthete?

Pascal isn't appealing to the spiritual man, but the natural man to be convinced only with worldly facts. So factoring in spiritual "infinities" may not be very convincing.

it seems to me he must mean one conceivably great materialistic life, plus a second, plus a third, and so on. He's using suppositions the brute can relate to. Taking Dan as an example of raw hedonistic desires, it really comes down to avoiding death. The central message of the gospel is extending the number of pleasurable days the physical body enjoys. That either results in a finite number per NPV, or it results in 1 life + 1 life + 1 life ad infinitum, which = -1/2 lives.

If he means that each life is infinitely good, then that is horrible to his argument, because conceivably, God could bestow upon us a infinitely great pleasure for one second and then snuff us out. Pascal, per the proclivities of the Afore, is bound to a limitless continuance of the physical body. Infinity is bad, because we can argue (as PG showed, I think) that infinite pleasure by one second is the same amount of pleasure as infinity by three entire lives and even out to infinity it still equals infinity. And so you would be forced to accept that if God wished to bestow infinity upon us over one second, then that is just as good as doing so over a thousand or infinite lifetimes. And that seems to undercut what Dan wants.

And Pascal would have been off base suggesting that two is better than one and three is better than two. So it seems that discounting must happen to make Pascal rational.

My impression of DCP’s forthcoming book is that it’s going to be about “why to believe,” and the first chapter is going to be about Pascal’s Wager. My guess is that the basic message will be, “What Pascal’s wager tells us is that in order to win against atheists, the bar apologists need to cross is extremely low. All we need to do is establish that somehow my beliefs might somehow be true. Even if the probability that we are right is only 0.000001%, that’s enough to evoke Pascal’s Wager to prove that belief is the rational choice.

[Analytics]

I see a potential wrinkle in your calculations. All religious are not equal. The Mormon religion, for example, promotes the notion that it is the only religion that speaks for God. That all other religions have some truths, but only they have all the truths. So the reward for religious belief in an incorrect religion (per Mormonism) is something “less than”. There’s an infinite cost attached to not picking Mormonism as your religion of choice, according to Mormonism. And Mormonism even puts a price on it, that you will be separated from God and from your family members for eternity. I’m not sure any other religion is quite so supercilious.

How does the variable of picking the right religious belief get taken into account in your risk assessment based on Pascal’s Wager?

What I’d point out here is that Pascal’s Wager is a model. In its pure basic form, there is only one choice--either to believe or not to believe. My actuarial analysis is accepting this basic premise that this is in fact the only choice.

Of course this model in its pure form is irrelevant to making choices about religion; in the real world, there are multiple choices. If we generalize the problem so that there are n religions to choose from, then the probability of religion x being true could be notated as p(rx). It follows that p(r1) + p(r2) + p(r3) + ...+ p(rn) = 1.

With this setup, if the payoff for believing in a religion is “infinity”, then the payoff for correctly believing in religion 1 is p(r1) x infinity = infinity. The payoff for believing in religion 2 is p(r2) x infinity = infinity, etc.

The first takeaway from this is that every religious claim that has both a non-zero chance of being true and has an infinite reward are the only ones we could rationally choose. If we assume there are multiple religions to choose from that all have infinite payoffs and non-zero chances of being true, these calculations give no rational way of choosing between them. If the reward is literally infinity, these calculations give no reason to bet on the 99% religion rather than the 1% chance religion (i.e. 0.99 x infinity = 0.01 x infinity = infinity).

The easiest way to move this forward is what Physics Guy suggested above. Rather than saying the reward for believing is literally infinite, we could say the reward approaches infinity as something else (e.g. time) approaches infinity. Once you do that, it becomes a matter of not only which religion is more likely to be true, but also which one has the biggest payoff.

This is a formalized version of the Mormon who thinks, “I’m following Mormonism because I like the godhood upside, and my downside is limited because I’m confident that Jesus still recognizes that I accept Him.” And this way of thinking also captures the evangelical who thinks, “I think Mormonism is more likely to be true. However, I choose evangelical christianity because Mormonism being true has limited downside (a lesser kingdom of glory), and there is a non-zero chance that Mormons are going to be tortured in hell for all eternity, and I don’t want to risk it. Therefore evangelical christianity is the less risky bet."

[Analytics]

I certainly would, if with a little bit of my trillion in hand I could make an investment that would pay out at least a buck a year forever. Interest can work both ways, after all. And if investments compound here on Earth, why would they not also grow in value through eternity?

Exactly. This is my point. The time value of money goes in both directions. With this, you’ve conceded the reasonability of how I tweaked the premise.

It seems arbitrary to assume that Pascal's reward for belief should have a value that accrues at death and then depreciates ever after, or that it should be an eternal fixed rent which would be outperformed in the eternal long haul by any decent investment.

Let's hold on here. I never assumed the value of a year in heaven somehow depreciates over time, nor did I assume that a decent investment would somehow “out perform” eternal glory.

Rather, I am assuming the following:

1- Heaven exists over time
2- The value of a limited time period in heaven is limited (e.g. $1 million per year, or $1 trillion per year. Whatever you want)
3- There is a time preference in heaven like there is on earth (if I could experience one year in heaven, I’d rather have that one year sooner rather than later (e.g. I’d rather enjoy that year in heaven next year than in a trillion years, or I’d be willing to trade 2 years in heaven starting in a 100 years for 1 year of heaven starting now). This assumption is what establishes a positive interest rate.

As a primer in the Theory of Interest, I’ll assume that the interest rate i is 5%. (This makes the notation easier and more understandable in this format, but the rate could be anything positive).

$100 dollars now is worth $100. Over the course of a year, $100 dollars now will grow to $100 * (1.05) = $105. In n years, $100 dollars will grow to $100 * (1.05)^n. That is the accumulation function, and as you said, it works both ways.

The accumulation function in the opposite direction is called the present value function. If I need $100 in a year, then all I need right now is $100 / (1.05). That’s because in one year, that will grow to [$100 / (1.05)] x (1.05) = $100. And if I need $100 n-years from now, all I need now is $100 / (1.05) ^ n.

So I’m not assuming that the value of heaven depreciates over time, and I’m not saying a financial investment somehow beats an eternal reward. I’m saying that if a year of heaven is worth a million dollars at the start of the year and I want to purchase a year of heaven every year for n years, I’ll eventually $1,000,000 x n, but I won’t need that much now. Rather, all I need now is $1,000,000 + (1,000,000)/(1.05) + (1,000,000)/(1.05)^2 + ... + (1,000,000)/(1.05)^n. And as n approaches infinity, the grand total of this sum approaches ($1,000,000)/(0.05) =$20,000,000.

Looking at it another way, $20,000,000 would generate $1,000,000 of interest in one year. With $20,000,000 invested, I could leave the principal invested and buy my annual tickets to heaven with the interest.

We can change the interest rate to any positive amount you want, and the math works the same. But if you drop it to zero, then yes, the present value goes to infinity, but you would have to concede that trading an infinite number of future dollars (payable $1 per year) for a mere trillion now would be an infinitely terrible trade.

[Physics Guy]

What I mean by saying that eternal life is not a fixed rent is that if we are being this sophisticated about value and time then it is not at all clear that “a year of heaven” has a fixed future value.

Time is money. On Earth, in fact, time is the ultimate currency. Is a year of life now worth more or less than a year in the future? That future year is one more year of letting all your previous investments compound. So I don’t think it goes down in value because of opportunity costs. Its value is automatically indexed.

I think that Pascal would want to value eternal life in that way. It’s not a matter of receiving another 365 cookies each year. It’s one more year of doing everything possible, under optimal conditions, including whatever you could do to make the future even better. And then another year after that, ever after.

[Analytics]

What I mean by saying that eternal life is not a fixed rent is that if we are being this sophisticated about value and time then it is not at all clear that “a year of heaven” has a fixed future value.

Time is money. On Earth, in fact, time is the ultimate currency. Is a year of life now worth more or less than a year in the future? That future year is one more year of letting all your previous investments compound. So I don’t think it goes down in value because of opportunity costs. Its value is automatically indexed.

I think that Pascal would want to value eternal life in that way. It’s not a matter of receiving another 365 cookies each year. It’s one more year of doing everything possible, under optimal conditions, including whatever you could do to make the future even better. And then another year after that, ever after.

Part of the issue here is that at best, the value of living in heaven is speculative. We can make up anything we want to about it. But it’s worth clarifying what we’re arguing. Are you saying that a year in heaven is of infinite value? Or are you saying that the value of a year in heaven is exponentially better than the year before it?

If it is the former, I’d ask for your answer to Analytics’s wager 2.0: Would Pascal trade a trillion dollars now in exchange for one second less of eternal bliss after he dies? If he would, that means he thinks a second of eternal bliss is worth less than a trillion dollars. That means time matters, and the time-value of future blessings matter.

On the other hand, if every year in heaven is worth at least 5% more than the year before, and that exponential improvement of eternal bliss goes on for eternity, then sure, my actuarial nuance doesn’t make a difference. But at that point, the force of Pascal’s Wager no longer comes from “infinity” alone — it comes from a very strong and highly specific assumption about unbounded, compounding growth of utility in the afterlife, an assumption Pascal himself never articulated.

[Analytics]

This idea of spending each day in heaven to make the next one even better highlights another implicit assumption in my original model. It adds another layer of complexity, but addressing it actually strengthens my point.

Specifically, I’m talking about the relationship between the economic utility of something and how much it costs in dollars.

An implicit assumption I initially made was that if a year in heaven were worth, say, a million dollars in economic terms, the utility we derive from that would be equal for all years. That’s actually a very conservative assumption. Many people worry about monotony when contemplating eternity; would the trillionth day strolling down a street of gold be just as transcendently satisfying as the first? I assumed it would be.

But now that we’re talking about using each day in heaven to invest in making the next day even better, we need to think more carefully about the relationship between cost and satisfaction. For example, imagine how wonderful it would be to walk down a street of 10-karat gold. Now imagine how much better it would be to walk down a street of 18-karat gold with diamond-embedded lane markers. That second street might cost 100 times as much as the first--but would it provide 100 times the satisfaction?

In economics, it’s standard to assume that the marginal utility of goods and services diminishes as you consume more of them. More is still better, but twice as much is usually not twice as good.

So even if we assume that the “wealth” of each day in heaven is reinvested to make the next day better, we still need to ask how much additional happiness those marginal improvements generate. Once diminishing marginal utility is taken into account, the present value of heaven could still be finite, even under a model where every day is better than the one before.

[Gadianton]

Imagine Dan is offered the choice to add one perfect day to his finite life -- a day of anything he wishes for. Jesus, Adam Smith, And CSL all at his feet learning together -- anything. He can chose one perfect day, or a week of having a mild cold.

What does he pick? I'm pretty sure he pics the perfect day.

But now, what if he's offered to his finite life either a single perfect day, or twenty years of a mild cold?

Now that's playing hardball. He reasons, a mild cold isn't that bad, it's an excuse to watch TV all day and lounge. Twenty years of it? If it's either that or eternal blackness, then he'll take the cold. He considers the eternal goldfish; with a memory of three seconds, it won't get bored in eternity. Focusing a day at a time puts himself in the mental space of the goldfish.

[Physics Guy]

Or are you saying that the value of a year in heaven is exponentially better than the year before it?

That seems to be how actuaries value time on Earth, does it not? Any present value compounds positively into the future, and any future value is correspondingly discounted in the present. If I invest one dollar today at 5% annual interest, that investment now will bring in (1.05)^(N-1)/20 in interest, in my Nth future year. That's an exponentially growing value of each future year. So precisely this idea, that the value of subsequent years increases exponentially, seems to be the basis of all your actuarial calculations. Maybe actuaries don't normally look at it this way, and instead take passing years for granted and ask about money value, but the fact that the value of time itself automatically keeps pace with any kind of growth over time seems to be a basic actuarial assumption.

You are taking the arbitrarily long succession of afterlife years for granted, as something that everyone gets automatically, and then valuing beatitude as a fixed rent that accrues over time. And you are then pointing out that fixed rents are worth less than compounding investments, which are the competition against which actuaries measure any other kind of value. This means that you are assuming that Pascal's admittedly under-specified "infinite reward" should be treated as an infinitely less valuable kind of reward, a fixed rent rather than a compounding investment. I mean, even on Earth a CEO's compensation package may well include the annual vesting of company shares, which (at least in good times) means an exponentially increasing dollar value each year, even though it's a fixed rate in shares. Pensions are sometimes indexed to inflation, so that the currency value of each future year's income increases exponentially over time, with the goal of holding its present value constant. Why shouldn't the saints enjoy a sweet deal like that? Why should we arbitrarily assume that an infinite heavenly reward is an infinitely worse kind of reward than the non-discounting kinds of rewards that already exist in mortality?

It''s true that Pascal didn't explicitly mention any of these points, but he also didn't explicitly mention that the reward for belief in God should be valued as a fixed rent in currency. In the absence of any such specification by Pascal, I think the most reasonable assumption is that he meant what Christians in his time all thought their reward would be: eternal life. If the infinite extension of heavenly years is not a basis that everyone gets, but is itself the promised reward, then actuarial assumptions that time means compounding interest should naturally tell us that the reward of eternal life itself is not like a fixed rent but like a vesting in shares or an indexed pension, whose present value does not discount.

What Pascal did mention is all that he really needed to mention: he stipulated the premise that the reward has infinite value. He doesn't specify "present value" in the language of modern actuarial science, but since his argument is about a decision to believe that must be made in mortal life, it is clear from context that he must mean present value. If you had been able to show that infinite present values are impossible, even if paid over infinite time, then this might have been a demonstration that Pascal's argument was unsound, in that its premise was impossible. There clearly do exist non-discounting rewards even on Earth, though, and a payment in time itself must be one of them. So Pascal's premise of an infinite present value does seem to be possible in principle. It just can't be a fixed rent.

And in that case, finding a formulation of Pascal's reward that might sound infinite to a layperson, and then showing that this formulation is actually finite, only demonstrates that this formulation is invalid as a restatement of Pascal’s premise. It does not hurt Pascal's argument.

[Gadianton]

Pascal's distilled point seems to be: God can pour his blessings out from the windows of heaven faster than you can lower the chance that he exists such as to pick up the bucket in the first place. The logic behind every fraud.

Infinity is allowed. Infinitesimal is not allowed. If infinity is allowed as a reward, to be fair, zero should be allowed as a probability. If not, then just ask if Bernie Madoff offers an infinite reward? The math stays the same. But he has a credibility problem. The only way to fix it, is to infuse the credibility into the proposition, by making God in the abstract the only one who can make the offer. Ultimately, the ontological argument will be required to save the wager.

[Doctor CamNC4Me]

Pascal’s Wager is the wrong wager. Belief isn’t a switch you flip for insurance. You either find something convincing or you don’t. Pretending to believe for safety misses the point.

It also assumes there’s only one possible god. With lots of competing religions, picking one could be just as wrong as picking none.It turns religion into risk management instead of a search for truth. The real question is what is true, not what feels safest.

So. The point. The real wager is honesty and truth-seeking versus comforting yourself with ‘just in case.’