Here’s another puzzle about infinite decision
theory. (Although I haven’t been flooded with responses to the last one
yet, but I like making these up.)

The background is the same as last week. Leopold
is in purgatory. He occasionally gets to spend days in Heaven or Hell depending
on how various gambles with angels turns out. Recently these have all involved
St. Petersburg type gambles.

Some quick terminology will help us along. (This
is all from the previous story.) A rock is an envelope with a number inside
it, called its number. The number is chosen by a St. Petersburg style process.
A coin is flipped until it lands tails. If it lands tails the first time, the
number is 1, the second time the number is 2, the third time the number is 4,
and so on doubling ad infinitum. An x-rock is a rock where the number
generated by this process is multiplied by x. So if the coin lands tails
on the third trial when building a five-rock, the number written would be 20.
(4, for the third coin toss, by 5, the multiplier for the rock.) A green rock
entitles the bearer to spend its number of days in Heaven, a yellow rock condemns
the bearer to spend that many days in hell. We won’tbe using this here,
but in the previous story I also defined an x-rock minus y, whose
number is determined by working out what its number would be were it an x-rock,
then subtracting y. (The main issue in the previous post was whether
rationality demanded one prefer a ten-rock to a ten-rock minus ten.)

Leopold’s preferences are such that the marginal
utility of days in Heaven and Hell are equal, opposite and constant. So he’s
indifferent between the status quo and accepting a pair of gambles, one of which
gives him a days in Heaven with probability b, and the other sends
him to c days in Hell with probability d iff ab = cd.

Today two angels came by and offered him some deals.
The first offered him two green rocks if he would also take a yellow two-rock.
The second offered him a green two-rock if he would also take two yellow rocks.
What
should he do?