An Actuarial Analysis of Pascal’s Wager

[Analytics]

Post-mortal punishments and rewards have a time value in the same way that real-world money has a time value

Pascal should have understood that this is the very reason why punishments and rewards are so greatly exaggerated. Because they are discounted so heavily, the promises must be heaped and heaped to move the needle to do what the wicked priest commands. I don't recall who invented time preference, during Pascal's time interest was probably considered usury. Reading IHAQs quote from Pascal, he was on the verge of a mental breakthrough. One life in change on the floor was surely enough, but two or three lives? An infinitude? If it wasn't for his own confirmation bias, he could have racked up some major breakthroughs.

Yes, he should have understood this. While I’m standing on the shoulders of 350 years of subsequent mathematicians and have the advantage (and bias) of doing this type of calculation on a daily basis, Pascal would have understood my point immediately. Whether or not it was called ‘usury’ doesn’t change the fact that positive interest rates imply time preference, even if the concept wasn’t formalized in modern terms. And you don’t need to understand the nuances of calculus to understand that if there were a forever interest rate of 5%, then $20 now is equivalent to $1 per year, every year, for an infinite number of years. Pascal would have seen my point.

The best counterargument I can think of is that while yes, human beings have a preference for good things now and bad things later, this preference isn’t based on logic or reasoning, but rather is some sort of cognitive bias: objectively, this argument goes, $1 per year for an infinite number of years is worth an infinite amount of dollars--what is illogical is to do discounting.

But that raises the question, if the rational eternal being thought $1 per year for an infinite number of years and $1,000,000 per year for an infinite number of years both have the same infinite value, then a rational decision maker would have no basis to say $1,000,000 per year is better than $1 per year.

[Physics Guy]

But that raises the question, if the rational eternal being thought $1 per year for an infinite number of years and $1,000,000 per year for an infinite number of years both have the same infinite value, then a rational decision maker would have no basis to say $1,000,000 per year is better than $1 per year.

Paradoxes like this are the reductio ad absurdum that prove that we cannot consider infinity as an ordinary number. It just doesn't work that way. There are too many things that are all in some sense infinite, yet are not at all the same as each other. Infinity is not just one thing.

What you have to do is consider all your numbers to be finite, and then define "the limit" in which one or more of them become infinite. If more than one thing is becoming infinite, then you also have to say how the different things become infinite, for example by specifying a ratio among them which remains constant as they all become huger. So for example $1,000,000 per day is always a million times more than $1 per day, even as the number of days goes to infinity. One could also have some quantity A being proportional to N, while B is proportional to N^2, and then in the limit where N becomes infinite, A and B both become infinite, but B becomes infinitely larger than A.

In physics we deal with limits like this all the time—like, constantly, practically in every second sentence. We probably have things becoming zero more often than becoming infinite, but the logic is the same in both cases, since saying that epsilon is approaching zero (from above) is the same as saying that 1/epsilon is approaching infinity. We seldom, if ever at all, really mean that something is literally infinite. Physics has a lot of large numbers, though, and we are content with good approximations. So if there is one big number, such that dividing any of our other numbers by it will leave something so tiny that we don't care about it, we often say that the big number is infinite, just as a brief way of speaking.

Back to Pascal's Wager: I'm afraid I still don't see that actuarial science adds much to the discussion. The Wager has the premise that there is an infinite reward for believing in God. Saying, "Nuh-uh, the reward can't be infinite" is just denying the premise. It should always have been clear that this was an option. It doesn't seem to add much to the discussion just to point out that rewards on Earth are not infinite. The Wager doesn't assume that infinite rewards are common things; it postulates one only in the special case of a believer's afterlife. The applicability of earthly actuarial practice to an eternal afterlife seems dubious. So we don't need to invoke actuarial principles to reject Pascal's premise, and I don't see that they even help us to reject the premise.

And, anyway, the literal infinity of the reward is not actually essential to Pascal's argument. It can very well just be a physics-like figure of speech to mean "so large that if you divide something by it, the result is not worth considering". I mean, I have a hard time seeing how even the most dedicated atheist can rationally set the chance of God's existence to zero. It may seem unlikely, but there is no rational way to rule it out completely. So to make Pascal's argument effective, the reward for believing in God only has to be greater, by some impressive factor, than one divided by the possibly small but non-zero chance that God exists. That's all it takes to make belief the clear winning strategy. So calling the reward infinite can just be a way of saying that it is bound be plenty big enough even if the chance that God exists is extremely low.

[Analytics]

Paradoxes like this are the reductio ad absurdum that prove that we cannot consider infinity as an ordinary number. It just doesn't work that way. There are too many things that are all in some sense infinite, yet are not at all the same as each other. Infinity is not just one thing.

What you have to do is consider all your numbers to be finite, and then define "the limit" in which one or more of them become infinite. If more than one thing is becoming infinite, then you also have to say how the different things become infinite, for example by specifying a ratio among them which remains constant as they all become huger. So for example $1,000,000 per day is always a million times more than $1 per day, even as the number of days goes to infinity. One could also have some quantity A being proportional to N, while B is proportional to N^2, and then in the limit where N becomes infinite, A and B both become infinite, but B becomes infinitely larger than A.

I agree that if we model rewards as a function of time and take the value of the rewards as t approaches infinity, then a million dollars per time period is always a million times greater than one dollar per time period. But this order of counting implies time matters. You are the one arguing that Pascal’s premise is an infinite reward, not a reward that approaches infinity as t approaches infinity. Once you drop time-indexed growth and talk about infinity as a completed total, Cantor’s work shows that you no longer have a natural way to order ‘how much bigger’ one infinite total is than another without reintroducing the very time structure you set aside.

If you insist that the correct way of doing it is to introduce t into the formula, then I agree. That is my point.

Here is Analytics’ wager. A demon comes down to you that is 100% trustworthy. He tells you the following: “Physics Guy, I have a proposition for you. I will give you one trillion dollars. Along with one trillion dollars, I’ll guarantee you a healthy, naturally long life. And along with the money, I’ll grant you with the highest level of wisdom so that you can deploy the money in a way that will best achieve your goals, whatever they may be.

But there’s one catch.

In exchange for one trillion dollars, wisdom, and a long, healthy life, you’ll have to give me one dollar every year. For. All. Eternity. Wah ha ha ha ha!!!"

What would you do? Would you agree to an infinite financial punishment after death in exchange for a mere $1 trillion now?

Back to Pascal's Wager: I'm afraid I still don't see that actuarial science adds much to the discussion. The Wager has the premise that there is an infinite reward for believing in God. Saying, "Nuh-uh, the reward can't be infinite" is just denying the premise.

My point is a little more subtle than saying nuh-uh. What I’m doing is clarifying what the premise actually is. Is the premise that correctly believing results in an infinite reward? Or is the premise that correctly believing results in a reward that is very high over every time period and that lasts for an infinite amount of time?

If it is the latter, then discounting for time-value makes sense and the present-value of the reward isn’t infinite. If Pascal clarified it is the former and said every time period of the next life is infinitely happy in its own right, I’d push back on that with 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.

[Analytics]

I'm afraid I still don't see that actuarial science adds much to the discussion. The Wager has the premise that there is an infinite reward for believing in God....

As a more concise version of my insight, actuaries deal with streams of cash flows over time. For example, we forecast insurance premiums, taxes, annuity payments, social security benefits, long-term care costs, etc. Any well-defined stream of cash flows has multiple values. We can look at the present value of the cash flows. Or the accumulated value. Or the statistical expected value. Or the statistical expected value discounted for interest (i.e. the “actuarial present value”).

Pascal’s wager deals with the premise of an “infinite reward for believing in God.” My actuarial background makes me approach this premise with more nuance. Is the infinite reward the sum of the finite rewards each period that last for an infinite amount of time? If it is, actuarial mathematics shows that even though the sum of these undiscounted rewards is infinite, the present value--the quantity relevant for a rational choice made now--is not.

Pascal’s wager could be salvaged by clarifying that the “infinite reward” means that the present value of the reward is infinite, but that is a much stronger premise that isn’t necessarily supported by the Christian view of the afterlife.

[malkie]

I'm afraid I still don't see that actuarial science adds much to the discussion. The Wager has the premise that there is an infinite reward for believing in God....

As a more concise version of my insight, actuaries deal with streams of cash flows over time. For example, we forecast insurance premiums, taxes, annuity payments, social security benefits, long-term care costs, etc. Any well-defined stream of cash flows has multiple values. We can look at the present value of the cash flows. Or the accumulated value. Or the statistical expected value. Or the statistical expected value discounted for interest (i.e. the “actuarial present value”).

Pascal’s wager deals with the premise of an “infinite reward for believing in God.” My actuarial background makes me approach this premise with more nuance. Is the infinite reward the sum of the finite rewards each period that last for an infinite amount of time? If it is, actuarial mathematics shows that even though the sum of these undiscounted rewards is infinite, the present value--the quantity relevant for a rational choice made now--is not.

Pascal’s wager could be salvaged by clarifying that the “infinite reward” means that the present value of the reward is infinite, but that is a much stronger premise that isn’t necessarily supported by the Christian view of the afterlife.

Would I be correct in thinking that your profession is "happier" with relatively stable values of quantities such as life expectancy, and that disruptive events - like a pandemic - put a serious dent in the calculations of future values?

If so, is there some potential discontinuity that would distort rational choices for belief in God?

I'm not sure if I'm precise enough in what I'm asking, or if I really know what I'm asking, so I'll be happy if any answer you might give also helps to improve the question.

[I Have Questions]

As a more concise version of my insight, actuaries deal with streams of cash flows over time. For example, we forecast insurance premiums, taxes, annuity payments, social security benefits, long-term care costs, etc. Any well-defined stream of cash flows has multiple values. We can look at the present value of the cash flows. Or the accumulated value. Or the statistical expected value. Or the statistical expected value discounted for interest (i.e. the “actuarial present value”).

Pascal’s wager deals with the premise of an “infinite reward for believing in God.” My actuarial background makes me approach this premise with more nuance. Is the infinite reward the sum of the finite rewards each period that last for an infinite amount of time? If it is, actuarial mathematics shows that even though the sum of these undiscounted rewards is infinite, the present value--the quantity relevant for a rational choice made now--is not.

Pascal’s wager could be salvaged by clarifying that the “infinite reward” means that the present value of the reward is infinite, but that is a much stronger premise that isn’t necessarily supported by the Christian view of the afterlife.

Would I be correct in thinking that your profession is "happier" with relatively stable values of quantities such as life expectancy, and that disruptive events - like a pandemic - put a serious dent in the calculations of future values?

If so, is there some potential discontinuity that would distort rational choices for belief in God?

I'm not sure if I'm precise enough in what I'm asking, or if I really know what I'm asking, so I'll be happy if any answer you might give also helps to improve the question.

I think you are talking about volatility and how that impacts the assessment of risk. But I would see that you are missing a component in the equation - time space over which you assess the risk. For example, assessing the risk of investing the stock market would consider potential volatility in whatever the sector was and global events that were foreseeable. But the risk rating would be different for you if you were going to leave your money in for 12 months, or 12 years. The longer you leave it in, the less individual volatile impacts on share prices will impact your return. Does that influence your question to analytics?

[Analytics]

Would I be correct in thinking that your profession is "happier" with relatively stable values of quantities such as life expectancy, and that disruptive events - like a pandemic - put a serious dent in the calculations of future values?

If so, is there some potential discontinuity that would distort rational choices for belief in God?

I'm not sure if I'm precise enough in what I'm asking, or if I really know what I'm asking, so I'll be happy if any answer you might give also helps to improve the question.

There is a hackneyed joke that the blind CEO of an insurance company is driving the company down a winding road. The VP of Sales is reaching over with his foot and flooring the gas pedal, while the VP of Underwriting is simultaneously reaching over with his foot and slamming on the brake. Meanwhile, the actuary is telling the CEO which way to steer by looking out the rear window.

So yes, I’d say actuaries are happier if the future looks like the past.

There are multiple ways we try to build in conservatism against the future turning out differently than expected. For example, we might assume a future interest rate of 4% even though we can currently invest new money at 5.5%. We also hold surplus on top of reserves to cover things that could go sideways in ways we didn’t explicitly model.

As for rational choices about belief in God, I think Pascal’s basic intuition makes sense. Stripped down and stylized, Pascal is reasoning along the lines of:

• Payoff for believing if religion is true: infinite
• Payoff for believing if religion is false: $0
• Probability religion is true: non-zero
• Cost of believing: finite

Under those assumptions, the expected value of believing is infinite, so belief dominates disbelief. And importantly, that conclusion is robust: it doesn’t really matter whether the probability is 1%, 0.5%, or 0.00001%.

My point is that even if there is an infinite payoff for believing that is paid over an infinite length of time, the present value of that payoff could still be finite. Once that happens, the robustness disappears. Assumptions about probability and cost suddenly matter a great deal, and the conclusion is no longer invariant to small changes in those assumptions.

That’s where actuarial thinking actually changes the structure of the problem, rather than merely objecting to Pascal’s premise.

[Gadianton]

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.

[I Have Questions]

As for rational choices about belief in God, I think Pascal’s basic intuition makes sense. Stripped down and stylized, Pascal is reasoning along the lines of:

• Payoff for believing if religion is true: infinite
• Payoff for believing if religion is false: $0
• Probability religion is true: non-zero
• Cost of believing: finite

Under those assumptions, the expected value of believing is infinite, so belief dominates disbelief. And importantly, that conclusion is robust: it doesn’t really matter whether the probability is 1%, 0.5%, or 0.00001%.

My point is that even if there is an infinite payoff for believing that is paid over an infinite length of time, the present value of that payoff could still be finite. Once that happens, the robustness disappears. Assumptions about probability and cost suddenly matter a great deal, and the conclusion is no longer invariant to small changes in those assumptions.

That’s where actuarial thinking actually changes the structure of the problem, rather than merely objecting to Pascal’s premise.

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?

[Physics Guy]

In exchange for one trillion dollars, wisdom, and a long, healthy life, you’ll have to give me one dollar every year. For. All. Eternity. Wah ha ha ha ha!!!"

What would you do? Would you agree to an infinite financial punishment after death in exchange for a mere $1 trillion now?

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? 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.

Mainstream Christianity offers few details about eternal bliss. It's supposed to be "eternal life", and it is taken for granted that this is a good thing, but that's about it. Jesus had several financial parables, however, and they turn on realistic details like wage setting and interest. The parable of the talents is perhaps the most relevant; its main plot point is the fact that burying cash in the ground is much worse than investing it to yield increase. Sown seeds yielding many more grains than were sown, or growing into large trees, are also prominent Jesus memes, while the rich man who decides to retire after building new barns for his hoard is condemned. Christian thinking, going right back to Jesus, has always considered growth a basic feature of life. It's been a prominent theme, not just a one-off remark.

So although you're right that Pascal didn't specify that his huge reward should be a present value, I don't think it's fair to say that a Christian's "treasure in heaven" should by default be a wasting asset or a fixed rent with finite present value. The implicit default would rather seem to be that eternal life should be growth, like those talents and trees.

Pascal didn't mention any of this, of course. So maybe the actuarial analysis does add more interest to the discussion that I realised.