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.