Suppose--as is traditional in these setups--you're spending an eternity in Purgatory, but sometimes you have the chance to spend some time in Heaven, and sometimes you get stuck spending some time in Hell. One day in Heaven cancels out one day in Hell, and there are no time discounts.
One day Satan appears to you and offers you the following deal:
"I'll offer you a St. Peter's card. A St. Peter's card is worth a certain number of days in Heaven, determined by a St. Petersburg process--1/2 chance of one day in Heaven, 1/4 chance of 2 days, 1/8 chance of 4 days, etc. As you know, that's infinitely good--it should be worth any bounded number of days in Hell. You can cash the card in any time--you have forever.
"But the next day, I'll offer you another deal. You give me the St. Peter's card back; and you accept a new card that sentences you to a certain number of days in Hell--let's say 1024; and you get a new St. Peter's card.
"The next day I'll offer you another deal--give me your St. Peter's card back, and I'll give you a new St. Peter's card--and another card that sentences you to 2048 days in Hell.
"The next day, I'll offer you the same deal again--except this time you get a card that sentences you to 4096 days in Hell.
"And so on. You can cash in your cards you have any time you want--but then I won't offer you any more St. Peter's cards. And when you cash in your cards, you have to take them all, and you can take them in any order.
"I know you're a sensible person; you won't wait forever for the hope of a higher and higher payoff. You'll take, say, 1024 days in Heaven (in excess of your days in Hell) as a reasonable stopping point. Sure, a new St. Peter's card will always be worth more than 1024 days in Heaven; but if you trade in for a new card every day, you'll never get to Heaven at all. How about it?"
If you're capable of choosing a strategy and sticking to it, you can indeed decide to stop as soon as you have at least 1024 days in Heaven. This may not be the best strategy--you could always pick a number higher than 1024--but as Arntzenius, Elga, and Hawthorne point out, there's no guarantee that there will be a best strategy in cases like this. You can aim for some finite number, but there'll always be a higher one.
But, in this case, it's not very likely that you'll get to Heaven at all.
There's a 1/1024 chance that your first St. Peter's card will get to your threshold of 1024 days in Heaven.
If you don't make it the first time, you get a card worth 1024 days in Hell--so the second day's St. P needs to be worth at least 2048 to allow you to stop. That's a 1/2048 chance.
The next day, you've got another 2048 days in Hell, so you need at least 4096 in Heaven to allow you to stop--a 1/4096 chance. You can see that there's only a 1/512 chance that you'll ever get to stop.
And the trouble here isn't the trouble that arises in "Trumped," where you never wind up cashing in the gains that you've been accumulating. (The same with the Everbetter wine, which gets better every day for all eternity, or Chris Bertram's case in which every day you have a higher chance of getting out of Hell.) There's a 511/512 chance that you are always so deep in the hole that you don't even want to cash in.
Now, that 1/512 of the time, your utility isn't capped at 1024; you effectively have a 1/512 chance at a St. Petersburg's worth of days in Heaven. Any fraction of infinity is still infinite, so you'd like that. But it's a bit disturbing to me that, in this scenario, every trade-in seems rational, but the chances are extremely small that you'll even get to a set finite threshold at any point.
(It might be possible to set this up so that it's even worse. Say, every day that you have a positive balance, you spend in Heaven, but every day you have a negative balance, you spend in Hell--the days coming off your balance. Then you'd spend the first day in Heaven, but I think there'd be a 255/256 chance you'd spend eternity in Hell. That wouldn't be worth the 1/512 chance at at least 1024 surplus days in Heaven. But this way of framing it probably does raise some of the same issues as "Trumped" and the Everbetter wine.)
Posted by Matt Weiner at February 4, 2004 01:06 PM