Tuesday, September 6, 2011

Weird puzzle about infinite utility

Found this on another blog. The blogger writes:

Imagine a universe containing infinitely many immortal people, partitioned into two "spheres". In one sphere, all the inhabitants live a blissful existence, whereas the members of the other sphere suffer unbearable agony. Now compare the following two variations:

1) Everyone starts off in the blissful sphere. But each day, one more person gets permanently transferred across to the agony sphere, where they reside for the rest of eternity.

2) Everyone starts off in the agony sphere. But each day, one more person gets permanently transferred across to the blissful sphere, where they reside for the rest of eternity.

Which scenario is better? The answer, paradoxically, appears to be "both".
At any moment in time, there will be infinitely many people in the original sphere, and only a finite number who have been transferred across. So option 1 is better.

However, each particular person will spend only a finite amount of time in the first sphere, whereas they will spend an eternity in their post-transfer home. So option 2 is better.

[A clarification is in order. As stated, it remains possible for some people to remain forever in their original sphere. (Suppose we assign each person a natural number. Each day we can transfer across the next even-numbered person. Then the infinitely many odd-numbered people never get transferred at all!) So let us stipulate that no-one is "skipped" in this way, and that every individual will indeed get transferred at some point.]

How are we to evaluate the options without falling into paradox?

(I owe this problem to recent discussion with ANU grad students. I think they in turn got it from Alan Hajek)

Here's my solution which I posted on the comments section:

It's an interesting problem that I think has a solution. First of all, as to the question which is "better," the term is ambiguous. "Better" can be used to mean which world has more total utility (from a "god's eye view" or from sub specie aeternitatis). In that case the two worlds are identical. But it could also mean that which world is more rational for one to choose to live in. In that case I think it's the second world. Here's my reasoning.

No matter who you turn out to be in the second world, you will eventually be transferred into the sphere where you will experience an eternity of positive utility. Your experiences of negative utility, no matter how large up to that point just before transfer, will be finite whereas your experiences in the world with positive utility will be infinite so it's rational to choose the second world.

If you had chosen the first world, this would be reversed. No matter who you are, you will eventually be sent to a sphere with infinite negative utility for you. So no matter how much positive utility you experienced prior to being sent there i while still in the first sphere, you will experience an infinite amount of total negative utility.

So choose option two. The paradox as I see it comes from the fact that the two ambiguous uses of "better" can come apart in such scenarios. But that's not surprising for me. There is a similar situation I talked about on my own blog before.

Consider this scenario:

An immortal who has alternating good and bad days (sum of his utility on those days are either positive or negative). His bad days all come on odd numbered days. So days 1, 3, 5, 7,... are all bad and the rest are good days for him where he has +1 utility.

But let's also say that on his bad days, they are a diminishing sequence, say on his first bad day, his summed negative utility is -1 and on his second bad day, -1/2, 3rd bad day, -1/3, 4th day, -1/4 and so on. So we will have a sequence (-1 + -1/2 + -1/3, + -1/4...). As the negative days approach infinity, his negative utility approaches infinity as that series is a divergent series increasing without bound. Thus there will be the same amount of positive vs negative utility in the end for him (both infinite) but only a fool cannot see that this is a very good world to live in. For *any* continuous finite span of time beginning with the first day and is longer than two days, the sum of his utilities for those days will be positive. The longer than span, the more positive it will be. The further he is along his life span, the more positive benefit he will have accrued.