Solving a decision theory puzzle—how to avoid spending eternity in hell

I mentioned marginal cost in my previous post, which reminded me of a puzzle I solved a number of years back. I found the problem through Alex Pruss, but I don’t remember exactly where. But here it is:

You are in purgatory. Each day you spend in purgatory has a value of 0. Each day you spend in heaven is +1 and each day you spend in hell is -1.

God appears to you and gives you a deal. Spend one day in hell and you will get two days in heaven.

That’s obviously a good deal. You take a loss of -1 but you gain +2 for a profit of +1. So the rational thing to do is take the deal.

After you spend your day in hell God comes to you again with the same offer. Using the same logic, you take the deal again.

…You see where this is going.

God offers you the deal an endless number of times and you end up spending an eternity in hell!

So what went wrong?

My solution was the following: Consecutive days in hell are NOT worth -1. The marginal disutility grows exponentially. Let’s say each subsequent day in hell is 50% worse than the previous. Then we get the following series:

The disutility quickly gets out of hand.

But now let’s say God keeps sweetening the deal. Let’s assume that each day in heaven is +1 (though I could imagine each subsequent day in heaven being 50% better than the previous. It’s heaven after all!). On night 2, God offers 4.5 days in heaven for just 1 day in hell. This costs -2.25 for +4.5, which is a profit. God offers 6.75 days in heaven for the next day in hell. And so on. Soon God will offer millions of days in heaven for a single subsequent day in hell. It’s always a good deal, so you still end up in hell forever. 

However, there is no reason to assume that just because someone can withstand a -1 disutility that they could withstand a -1,000. Let’s say your absolute limit is -500 disutility. Then it would only be possible for you to spend up to 16 days in hell in a row. With God offering double rewards each day, you would receive 2,623 days (7+ years) in heaven at the cost of only 16 days in hell. Not bad!

There is nothing irrational in turning down God's deal when at your limit, and the number of days spent in hell is not arbitrary; it’s limited by your capacity for disutility and by the marginal disutility rate.

Why Pascal's Wager Cannot Be Salvaged

To wager is to follow God’s commands not because you are fully convinced in God, but because you believe there is some chance you will go to hell if you don’t and/or go to heaven if you do. This applies to believers and non-believers alike: You might begin to follow God’s commands or remain following God’s commands not because you’re certain it’s true, but because it would be far better for you to be on the winning side come judgment day. Given how high the stakes are, wouldn’t it be crazy not to follow God? You have little if anything to lose and eternity to gain. So, why not wager?

Questions abound. Who is this God? What, exactly, are God’s commands? Do non-believers really go to hell? How has this God communicated all of this to us? 

These questions make it impossible to wager on just any report of salvation. So for a wager-style argument to get off the ground, the following conditions must be met:

1. There must be a report of a God capable of granting us heaven or hell after death. If the promise of heaven or threat of hell is from a being incapable of making good on such promises or threats, then we have no reason to wager.

2. This report must include God communicating to us the conditions of salvation. If there is no promise of heaven or threat of hell, then we have no reason to wager.

3. The conditions of salvation must be possible for us to fulfill. If we must follow God’s commands to be saved, but cannot find, understand, or perform God’s commands, then we cannot wager.

4. The conditions of salvation must specify that only those who live according to God’s commands receive heaven and/or avoid hell. If everyone immediately goes to heaven regardless of what they did in life, then we have no reason to wager.

5. This report must come from a major religion. Fulfillment of the above criteria implies a religion, and it’s unbelievable that a tiny religion would turn out to be the right one such that a thousand people end up in heaven while over a hundred billion end up in hell. So for this report to reach a bare minimum of credibility, it must be fairly widespread. (How widespread a religion should be to be considered credible is a question for another time.)

There are two reports of salvation that, at first glance, meet all five criteria: those contained in Christianity and Islam. Therefore the question isn’t should we wager on God, but should we wager on Christianity or Islam. So, again, why not wager?

Let’s take Christianity. Christianity clearly fulfills criteria 1, 2, and 5. For 4, there are questions about universalism, postmortem salvation, and whether converting guarantees salvation. Without getting into the details, the most credible form of Christian universalism is one where hell is real but temporary. So converting to Christianity would still maximize your chances of avoiding a temporary hell, which, while not infinitely good, is still good enough to enable a wager-style argument. And while postmortem salvation may be possible, it plausibly would only apply in extreme cases. So relying on a postmortem chance for your salvation would be foolish. And while converting to Christianity may not guarantee salvation, it would still maximize your chances. So all in all we can say Christianity firmly fulfills criterion 4.

The question then comes down to criterion 3: Is it possible for you to fulfill the conditions for salvation under Christianity? I won’t get into the thorny question of what it means to be a Christian or what it means to be saved on Christianity. My central point is that criterion 3 is the ultimate weak point for any wager-style argument. After giving a serious effort in evaluating the truth claims of Christianity, you either will or will not end up a Christian. If you do, then you believe not on the basis of the wager, but on the basis of the arguments and religious experiences you encountered in your investigation. If you don’t, then the challenges to Christian belief made it impossible for you to believe.

And that’s it. The non-believer can always ultimately say that a report of salvation violates criterion 3, because the report was investigated and the challenges discovered therein made it psychologically impossible to obey the conditions of salvation.

To show this in slightly more technical terms, here is an expected value equation:

E = -C + P(V)

Where C is the cost of playing, P is the probability of winning, and V is the value you receive if you win. In the case of Pascal’s Wager, that expected value calculation might look like: 

-1,000 + .01(∞) = ∞.

Where -1,000 is an arbitrary number showing how the finite cost of wagering is dwarfed by the expected value. Now consider: 

-1,000 + (1/∞)(∞) = -999.

This shows you would have to give Christianity a zero or next to zero probability of being true to nullify the wager. However, the objections to Christianity need not be so strong as to justify such a low credence in Christianity as long as they are strong enough to render wagering impossible. If you can’t bring yourself to fulfill God’s commands and satisfy criterion 3, then effectively the cost of wagering is -∞. In this we see how the pragmatic question is not independent from the epistemic question, which means the wager fails as a tiebreaker in epistemic ties.

The only sense in which the wager is successful is that it points to the fact that when there are prima facie credible reports of salvation, it’s irrational to not put in a serious effort in evaluating the truth of those reports. But given the irrationality of ignoring worldview issues, we knew that already. The wager therefore also points to the fact that the philosophy of religion, the means by which we evaluate religions, is indispensable. But we knew that already too.