First, the original question seems incomplete. Presumably the alternative to accepting the deal is something better than the guaranteed hell forever, say, 50⁄50 odds of ending up in either hell or haven.
Second, the initial evaluation of utilities is based on a one-shot setup, so you effectively precommit to not accepting any new deals which screw up the original calculation, like spending an extra day in hell.
I see. Then clearly your initial evaluation of the proposed “optimal” solution (keep banking forever) is wrong, as it picks the lowest utility. As in the other examples, there is no best solution due to unboundedness, but any other choice is better than infinite banking.
First, the original question seems incomplete. Presumably the alternative to accepting the deal is something better than the guaranteed hell forever, say, 50⁄50 odds of ending up in either hell or haven.
Second, the initial evaluation of utilities is based on a one-shot setup, so you effectively precommit to not accepting any new deals which screw up the original calculation, like spending an extra day in hell.
The problem starts after you took the first deal. If you cut that part of the story, then the other choice is purgatory forever.
I must be missing something. Your original calculation assumes no further (identical) deals, otherwise you would not accept the first one.
The deal is one day at a time: 1 day hell now + 2 days heaven later, then purgatory; or take your banked days in heaven and then purgatory.
At the beginning you have 0 days in heaven in the bank.
I see. Then clearly your initial evaluation of the proposed “optimal” solution (keep banking forever) is wrong, as it picks the lowest utility. As in the other examples, there is no best solution due to unboundedness, but any other choice is better than infinite banking.
I was attempting to complete the problem statement that you thought was incomplete—not to say that it was a good idea to take that path.
I thought it was incomplete? Are you saying that it can be considered complete without specifying the alternatives?
I think that sorting this muddled conversation out would not be worth the effort required.