Zero-sum conversion: a cute trick for decision problems

by Manfred1 min read26th Jul 201215 comments

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A while ago, we were presented with an interesting puzzle, usually just called "Psy-kosh's non-anthropic problem."  This problem is not, as is made clear, an anthropic problem, but it generates a similar sort of confusion by having you cooperate with people who think like you, and you're unsure which of these people you are.

In the linked post, cousin_it declares "no points for UDT," which is why this post is not called a total solution, but a cute trick :)  What I call zero-sum conversion is just a way to make the UDT calculations (that is, the things you do when calculating what the actual best choice is) seem obvious - which is good, since they're the ones that give you the right answer.  This trick also makes the UDT math obvious on the absent-minded driver problem and the Sleeping Beauty problem (though that's trickier).

The basic idea is to pretend that your decision is part of a zero-sum game against a non-anthropic, non-cooperating, generally non-confusing opponent.  In order to do this, you must construct an imaginary opponent such that for every choice you could make, their expected utility for that choice is the negative, the opposite of your expected utility.  Then you simply do the thing your opponent likes least, and it is equivalent to doing the thing you'll like best.

 

Example in the case of the non-anthropic problem (yes, you should probably have that open in another tab):

Your opponent here is the experimenter, who really dislikes giving money to charity (characterization isn't necessary, but it's fun).  For every utilon that you, personally, would get from money going to charity when you say "yea" or "nay," the experimenter gets a negative utilon.

Proof that the experimenter's expected utilities are negative yours is trivial in this case, since the utilities are opposites for every possible outcome, including cases where you're not a decider.  But things can be trickier in other problems, since expected utilities can be opposites without the utilities being exactly opposite for all outcomes.  For example, what happens in the case where the participants in the non-anthropic problem get individual candybars instead of collective money to charity?

Anyhow, now that we have our opponent whose expected utilities are the opposite of yours for every decision you make, you just have to make the decision that's worst for your opponent.  This is pretty easy, since our opponent doesn't have to deal with any confusing stuff - they just flip a coin, which to them is an ordinary 50/50 situation, and then pay out based on your decision.  So their expected value of "yea" is -550, while their expected value of "nay" is -700.

This valuation already takes into account cooperation and all that stuff - it's simply correct.  It's merely a coincidence that this seems like you didn't update the evidence of whether you're a decider or not.  Though, now that you mention it, it's a general fact that in cooperate problems like this, you can construct a suitable opponent by just reversing your utility in all situations, giving you this "updatelessness."

 

Disclaimer: I haven't looked very hard for people writing up this trick before me.  Katja or someone quite possibly already has this on their blog somewhere.

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