When I had lunch with Julian, Amy, and Tim on Saturday, the following intriguing question came up: If you were a Muslim, and you died and went to the Muslim heaven, how would you space out your enjoyment of the 72 virgins? Suppose that you actually find virginity desirable, and suppose that the virgins’ maidenheads are not magically restored periodically. If the afterlife has infinite duration, then no matter how long you wait to deflower your 72nd virgin, you’ll still be looking at an infinitely long virgin-less future thereafter. (More abstractly, the question is how to allocate your consumption of a finite non-durable good over an infinite period of time.)
To begin with, assume the existence of a discount rate, so that later periods are valued less than earlier periods (when viewed from the present). Even so, you would not choose to take them all in one night if virgins have diminishing marginal utility. You would save some virgins for the future, because the gain in value from regaining your appetite exceeds the loss in value from time-discounting. Still, under reasonable assumptions, eventually you’d use up your virgins. Your appetite for virgins presumably increases with the length of time since your most recent virgin; but unless your appetite continuously rises at a rate greater than your discount rate, you’ll eventually find it worthwhile to take the last virgin.
But does it make sense to have a discount rate when you’re dead? Discounting of the future presumably reflects our uncertainty about whether it will arrive at all. If you know your future will last forever, arguably you should weigh all periods equally. So what happens if we assume no discounting? Then we have a paradox: you might choose never to take the 72nd virgin. This will happen so long as your appetite always increases with the length of time since your most recent virgin. Even if the increase in your appetite is diminishingly small, the absence of discounting means you’ll wait any amount of time for the tiniest increase in satisfaction. To escape this conclusion, we must assume your appetite for virgins “tops out” after a certain amount of time – say, T years. In that case, there would be infinitely many optimal allocations of your virgins. One such solution would be to take one immediately, wait T years, take another, wait T years, etc., until you take the 72nd virgin after 71T years. But an equally good solution would involve waiting T + 1 years between virgins, or T + 2, etc. In fact, the intervals need not have equal length, and you could wait as long as you wanted before taking the first virgin.
Fewer solutions work if your appetite, after topping out at T years, begins to decline. In that case, you must space your virgins at intervals of exactly T to maximize your utility. How long you should wait before taking the first virgin depends on how great your appetite for virgins is when you arrive in heaven.
All of the above assumes that only the actual act of taking a virgin gives satisfaction, though the amount of satisfaction may depend on how long it’s been since your last virgin. Things change dramatically if recollection or anticipation is a significant source of one’s satisfaction. A person who enjoys looking back fondly on his experiences would rationally choose to take the virgins earlier – possibly in one fantastic orgy – so as to increase the duration of the pleasant memories. Of course, there will always be infinite time after the last virgin, so it could be countered that taking the virgins earlier will not increase the time spent recollecting. But I think it makes sense in this case to make comparisons of infinities, since one infinity begins earlier.
The real paradox of choice arises, I think, in the case where anticipation is highly important. If the joy of looking forward to taking a virgin were the primary source of your satisfaction from doing so, then your optimal plan would require always having one virgin ahead of you. Every period, you would have the choice of taking the virgin now or taking her tomorrow, and taking her tomorrow would always generate a greater sum of instantaneous utility and anticipatory utility (if we maintain the assumption of no time-discounting). But in that case, you would never actually take the last virgin – in which case your anticipation would be unjustified.