aspera

I had a discussion recently with some Less Wrongers about a decision problem involving infinities, which appears to have a paradoxical solution. We have been warned by Jaynes and others to be careful about taking the proper limits when infinities are involved in a problem, and I thought this would be a good example to show that we can get answers that make sense out of problems that seem not to. The problem is the "Infinite Spheres of Utility." To quote a description from Philosophy et cetera, > Imagine a universe containing infinitely many immortal people, partitioned into two "spheres". In one sphere [sphere A], all the inhabitants live a blissful existence, whereas the members of the other sphere [sphere B] 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. > > At first consideration, the problem appears to cause a paradox: > 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. > > Given these reasonable but hard-to-reconcile viewpoints, how do we make a decision? Deciding how to decide We first need to decide which kind of decision analysis we want to use to choose a starting sphere. For example, we might simply have an arbitrary preference for putting everyone in the bad sphere. Paradox over.