An Actuarial Analysis of Pascal’s Wager

[I Have Questions]

For the purpose of clarity, here is Pascal’s Wager as posthumously compiled from his notes…

If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is. ...
... "God is, or He is not." But to which side shall we incline? Reason can decide nothing here. There is infinite chaos that separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.
Do not, then, reprove for error those who have made a choice; for you know nothing about it. "No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all."
Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.
"That is very fine. Yes, I must wager; but I may perhaps wager too much." Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to change your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite.

The root cause of all the dilemma created by Pascal’s wager is the enforced choice between the two definites - God definitely exists, God definitely does not exist. The sensible choice is to remain agnostic - the safest bet, as it were. I wonder how that plays out mathematically…

I also think the following is flawed "But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is." The assumption that in choosing to believe that God exists you lose nothing if He doesn't, suggests that is no "cost" to making that choice. Which isn't true in Peterson's case for example. Or in MG's. There is a significant cost to making that choice within Mormonism - financial, social, and moral.

[Analytics]

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.

That’s a fair point. Let’s expand Pascal’s wager so that there are three possibilities:

1. Evangelical Christianity is true (i.e. to get the infinite reward you need to accept Jesus as your personal savior).
2. Mormonism is true (i.e. to get the infinite reward, you not only need to accept Jesus, but you also pay 10% of your income to Ensign Peak advisors, go to the temple, and get your caffeine fix from dirty soda rather than coffee).
3. Atheism is true. There is no infinite reward.

If you applied Pascal’s reasoning to this universe, there is only one correct bet--bet your soul on Mormonism. It would be totally illogical to believe mere Christianity, because you’d get screwed if Mormonism were true. Evangelical’s sense this and in order to rationalize their choice in religion, they need to believe that Mormons aren’t really Christians and they won’t make it to heaven.

[Morley]

In my mind, the biggest problem with Pascal's Wager is the idea that one can actively choose to believe something that there's no evidence for.

If, for example, I've been promised that if I can convince myself that I have a full sack of flour in my pantry, I will be given immeasurable wealth. Learning this, I seek to find and catalog all the flour in my pantry--but looking, I find that I have none. I look again, opening all the cupboards and drawers, but I still can find no flour. I pull up the floorboards, call in my neighbors to help, and even hire a private detective--but still, there's no flour. So, how do I convince myself that there is, indeed, a full sack of flour there? I'm gong to try my best.

From my reading of what some believers argue, comes the idea that all I need to do is to redefine the concept of "a sack of flour." Using some version of Sorties Paradox, I decide that neither the concept of "a sack' nor the definition of 'flour' can really be defined. For instance, if I sweep up a heap of dust from a pantry where bread was made, it's certainly reasonable to think that some part of that dust must be flour. However, is my resulting heap of dust--that I hope contains some flour (and who's to say it doesn't)--a sack? Surely it must be, because if a heap can't be defined, then certainly a sack can't be either. At what point does a heap of dust become a sack of flour? At any point I want it to be. Who is to say otherwise?

However, absent some pretty severe mental gymnastics, have I honestly succeeded in convincing myself that I have a found a full sack of flour? Meh. I don't think so.

[Limnor]

Pascal’s Wager seems to argue that it is better to pretend that you believe something is true rather than actually believe it is true. It’s like buying insurance. An odd foundation for faith if you ask me.

[malkie]

In my mind, the biggest problem with Pascal's Wager is the idea that one can actively choose to believe something that there's no evidence for.

If, for example, I've been promised that if I can convince myself that I have a full sack of flour in my pantry, I will be given immeasurable wealth. Learning this, I seek to find and catalog all the flour in my pantry--but looking, I find that I have none. I look again, opening all the cupboards and drawers, but I still can find no flour. I pull up the floorboards, call in my neighbors to help, and even hire a private detective--but still, there's no flour. So, how do I convince myself that there is, indeed, a full sack of flour there? I'm gong to try my best.

From my reading of what some believers argue, comes the idea that all I need to do is to redefine the concept of "a sack of flour." Using some version of Sorties Paradox, I decide that neither the concept of "a sack' nor the definition of 'flour' can really be defined. For instance, if I sweep up a heap of dust from a pantry where bread was made, it's certainly reasonable to think that some part of that dust must be flour. However, is my resulting heap of dust--that I hope contains some flour (and who's to say it doesn't)--a sack? Surely it must be, because if a heap can't be defined, then certainly a sack can't be either. At what point does a heap of dust become a sack of flour? At any point I want it to be. Who is to say otherwise?

However, absent some pretty severe mental gymnastics, have I honestly succeeded in convincing myself that I have a found a full sack of flour? Meh. I don't think so.

Your first mistake was to look. You need to doubt your doubts, and just believe. Then, having made your choice, never allow yourself to question it. Easy-peasy!

If the immeasurable wealth fails to appear, then:

• you're to blame - it was an unreasonable assumption that such a thing was possible, and/or
• you're to blame - some little doubt must have crept in somewhere and spoiled everything

[Analytics]

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.

The primary drivers of the time-value of money aren’t limited to the fear of death--corporations and governments charge and collect interest all the time. The real reasons are:

1. You could use $7,000 now to capitalize a business that would generate enough profit over the next five years to pay back $10,000, plus an extra margin for the risk. This is the big business reason.
2. Personal psychology; there is a preference for spending now. This is impatience more than fear of death.
3. Fear of default--if I loan you money for 20 years, will you pay me back?

Your point about psychology cutting both ways is interesting. Some people spend freely, others hoard and live poor only to die rich because they are paralyzed by unknown opportunity costs. Nevertheless, market-based interest rates are positive.

Whether these principles apply in the hypothetical realm of the afterlife is an assumption we can choose, but if we are using real-world statistical formulas to calculate a real-world expected value of belief so we can decide if the real-world cost is worth it, it is fitting to use real-world discounting of the costs and benefits.

An example of the time-value of post-mortal blessings and punishments is the idea of literally selling your soul to the devil. The reason people sell their souls is because they believe 40 years of owning Space X and Tesla now is better than an eternity of hell in the future. That intuition is only plausible is if the future is discounted.

The human resistance to delayed gratification makes the point in a clean way. People value a donut now more than a healthy BMI tomorrow.

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

Let me clarify my point. Pascal’s wager is a model that has some really strong assumptions:

1. There is a non-zero probability that God exists
2. There is a binary choice: either believe or disbelieve. Those are the only choices.
3. The payoff for correctly choosing belief is the difference between an eternity in heaven or an eternity in hell

Within the scope of this model, Pascal's key insight is that belief is the more rational choice, even if the probability that God exists is very low.

Most responses to this (e.g. the Homer Simpson meme) are an attack on second premise.

The actuarial insight I’m making is a different approach. In my model, I’m fully accepting all of Pascal’s assumptions, but adding one more:

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

Adding this one assumption changes the result.

[Fence Sitter]

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.

Can I choose a “God” who is, in every observable and practical respect, indistinguishable from no God at all?

If it’s impossible to choose the wrong one, then what, if anything, actually distinguishes belief from non-belief?

[malkie]

In my mind, the biggest problem with Pascal's Wager is the idea that one can actively choose to believe something that there's no evidence for.

If, for example, I've been promised that if I can convince myself that I have a full sack of flour in my pantry, I will be given immeasurable wealth. Learning this, I seek to find and catalog all the flour in my pantry--but looking, I find that I have none. I look again, opening all the cupboards and drawers, but I still can find no flour. I pull up the floorboards, call in my neighbors to help, and even hire a private detective--but still, there's no flour. So, how do I convince myself that there is, indeed, a full sack of flour there? I'm gong to try my best.

From my reading of what some believers argue, comes the idea that all I need to do is to redefine the concept of "a sack of flour." Using some version of Sorties Paradox, I decide that neither the concept of "a sack' nor the definition of 'flour' can really be defined. For instance, if I sweep up a heap of dust from a pantry where bread was made, it's certainly reasonable to think that some part of that dust must be flour. However, is my resulting heap of dust--that I hope contains some flour (and who's to say it doesn't)--a sack? Surely it must be, because if a heap can't be defined, then certainly a sack can't be either. At what point does a heap of dust become a sack of flour? At any point I want it to be. Who is to say otherwise?

However, absent some pretty severe mental gymnastics, have I honestly succeeded in convincing myself that I have a found a full sack of flour? Meh. I don't think so.

Your first mistake was to look. You need to doubt your doubts, and just believe. Then, having made your choice, never allow yourself to question it. Easy-peasy!

If the immeasurable wealth fails to appear, then:

• you're to blame - it was an unreasonable assumption that such a thing was possible, and/or
• you're to blame - some little doubt must have crept in somewhere and spoiled everything

I missed one:

• you're to blame - it's not what god wants and knows is best for you

Why are you to blame for this? Because you should be sufficiently in tune with god to only want nothing other than what god wants

[Marcus]

...At what point does a heap of dust become a sack of flour? At any point I want it to be...

Mopologetics in a nutshell. Perfection.

...However, absent some pretty severe mental gymnastics, have I honestly succeeded in convincing myself that I have a found a full sack of flour? Meh. I don't think so...

Unless you triangulate a crooked line with a parallel straightrule, in real time, ongoing forever, and then pretend it's straight. Although it occurs to me that that's what you meant by severe mental gymnastics. Never mind.

[Gadianton]

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.