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pragmatist comments on The Doubling Box - Less Wrong Discussion

13 Post author: Mestroyer 06 August 2012 05:50AM

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Tags: decision infinity paradox unbounded utility busy_beaver

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Comment author: pragmatist 06 August 2012 06:23:52AM *  1 point [-]

You have given reasons why requiring bounded utility functions and discounting the future are not adequate responses to the problem if considered individually. But your objection to the bounded utility function response assumes that future utility isn't discounted, and your objection to the discounting response assumes that the utility function is unbounded. So what if we require both that the utility function must be bounded and that future utility must be discounted exponentially? Doesn't that get around the paradox?

I remember reading a while ago about a paradox where you start with $1, and can trade that for a 50% chance of $2.01, which you can trade for a 25% chance of $4.03, which you can trade for a 12.5% chance of $8.07, etc (can't remember where I read it).

The problem statement isn't precisely the same as what you specify here, but were you thinking of the venerable St. Petersburg paradox?

Comment author: Mestroyer 06 August 2012 06:52:57AM 0 points [-]

If your utility function is bounded and you discount the future, then pick an amount of time after now, epsilon, such that the discounting by then is negligible. Then imagine that the box disappears if you don't open it by then. at t = now + epsilon * 2^-1, the utilons double. At 2^-2, they double again. etc.

But if your discounting is so great that you do not care about the future at all, I guess you've got me.

This isn't the St. Petersburg paradox (though I almost mentioned it) because in that, you make your decision once at the beginning.

Comment author: pragmatist 06 August 2012 08:42:39AM 0 points [-]

If your utility function is bounded and you discount the future, then pick an amount of time after now, epsilon, such that the discounting by then is negligible. Then imagine that the box disappears if you don't open it by then. at t = now + epsilon * 2^-1, the utilons double. At 2^-2, they double again. etc.

Perhaps I am misinterpreting you, but I don't see how this scheme is compatible with a bounded utility function. For any bound n, there will be a time prior to epsilon where the utilons in the box will be greater than n.

When you say "At 2^-2...", I read that as "At now + epsilon * 2^-1 + epsilon * 2^-2...". Is that what you meant?

Comment author: Mestroyer 06 August 2012 02:08:50PM 0 points [-]

yeah, that's what I meant. Also, instead of doubling, make it so they exponentially decay toward the bound.

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