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Kindly comments on The Doubling Box - Less Wrong Discussion
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I way to think about this problem to put you in near mode is to imagine what the utility might look like. Ex:
Day 1: Finding a quarter on the ground
Day 2: A child in Africa getting $5
.....
Day X: Curing cancer
Dax X+1: Curing cancer, Alzheimers, and AIDS.
On one hand, by waiting a day, more people would die of cancer. On the other, by not waiting, you'd doom all those future people to die of AIDS and Alzheimers.
You are adding a condition that was not present in the original problem. Namely, that every day you do not open the box, you lose some number of utilions.
Whoops, you're right.
Suppose instead of multiplying the utility by 2 each day, the box multiplied the utility by 1. Would it look like this?
Day 1: Curing cancer
Day 2: Curing cancer
Day 3: Curing cancer ...
Probably not - each of those "curing cancer" outcomes is not identical (cancer gets cured on a different day) so you'd assign them different utilities. In order to conform to the specification, the box would have to add an extra sweetener each day in order to make up for a day's worth of cancer deaths.