Suppose I know that from now on for an infinite number of years, I will be offered an annual game. A die will be rolled, I will be asked to guess whether the die will show six, and if I guess right, I will get a slice of delicious chocolate cake (one of my favorite foods).
Intuitively, I rationally should guess “Not a six”, and thereby get a 5/6 chance of the prize instead of the 1/6 chance if I guess “Six”.
But suppose that instead of the prizes being slices of chocolate cake, there is an infinite supply of delightful and varied P. G. Wodehouse novels (he’s one of my favorite authors), numbered 1, 2, 3, …, and each prize is the opportunity to read the next one. Moreover, the pleasure of reading book n after book n − 1 is the same regardless of whether the interval in between is longer or shorter, there being advantages and disadvantages of each interval that cancel out (at shorter intervals, one can make more literary connections between novels and remember the recurring characters a little better; but at longer intervals, one’s hunger for Wodehouse will have grown).
Now, it is clear that there is no benefit to guessing “Not a six” rather than “Six”. For whatever I guess, I am going to read every book eventually, and the pace at which I read them doesn’t seem prudentially relevant.
At this point, I wonder if I should revise my statement that in the cake case I should guess “Not a six”. I really don’t know. I can make the cake case seem just like the book case: There is an infinite supply of slices of cake, frozen near-instantly in liquid helium and numbered 1, 2, 3, …, and each time I win, I get the next slice. So it seems that whatever I do, I will eat each slice over eternity. So what difference does it make that if I guess “Not a six”, I will eat the slices at a faster pace?
On the other hand, it feels that when the pleasures are not merely equal in magnitude but qualitatively the same as in the cake case, the higher pace does matter. Imagine a non-random version where I choose between getting the prize every year and getting it every second year. Then on the every-second-year plan, the prize days are a proper subset of the prize days on the every-year plan. In the cake case, that seems to be all that matters, and so the every-year plan is better. But in the Wodehouse case, this consideration is undercut by the fact that each pleasure is different in sort, because I said the novels are varied, and I get to collect one of each regardless of which installment plan I choose.
Here is another reason to think that in the cake case, the pace matters: It clearly matters in the case of non-varied pain. It is clearly better to have a tooth pulled every two years than every year. But what about varied torture from a highly creative KGB officer? Can’t I say that on either installment plan, I get all the tortures, so neither plan is worse than the other? That feels like the wrong thing to say: the every-second-year plan still seems better even if the tortures are varied.
I am fairly confident that in the novel case—and especially if the novels continue to be varied—the pace doesn’t matter, and so in the original game version, it doesn’t matter how I gamble. I am less confident of what to say about the cake version, but the torture case pushes me to say that in the cake version, the pace does matter.
