army1987 comments on Checklist of Rationality Habits - Less Wrong

117 Post author: AnnaSalamon 07 November 2012 09:19PM

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Comment author: Kindly 08 November 2012 04:34:05AM *  16 points [-]

To keep the limits of the log argument in mind, log 50k is 10.8 and log (50k+70k) is 11.69 and log 1 billion is 20.7

Comparing these numbers tells you pretty much nothing. First of all, taking log($50k) is not a valid operation; you should only ever take logs of a dimensionless quantity. The standard solution is to pick an arbitrary dollar value $X, and compare log($50k/$X), log($120k/$X), and log($10^9/$X). This is equivalent to comparing 10.8 + C, 11.69 + C, and 20.7 + C, where C is an arbitrary constant.

This shouldn't be a surprise, because under the standard definition, utility functions are translation-invariant. They are only compared in cases such as "is U1 better than U2?" or "is U1 better than a 50/50 chance of U2 and U3?" The answer to this question doesn't change if we add a constant to U1, U2, and U3.

In particular, it's invalid to say "U1 is twice as good as U2". For that matter, even if you don't like utility functions, this is suspicious in general: what does it mean to say "I would be twice as happy if I had a million dollars"?

It would make sense to say, if your utility for money is logarithmic and you currently have $50k, that you're indifferent between a 100% chance of an extra $70k and a 8.8% chance of an extra $10^9 -- that being the probability for which the expected utilities are the same. If you think logarithmic utilities are bad, this is the claim you should be refuting.

Comment author: [deleted] 08 November 2012 05:12:50PM 1 point [-]

In particular, it's invalid to say "U1 is twice as good as U2". For that matter, even if you don't like utility functions, this is suspicious in general: what does it mean to say "I would be twice as happy if I had a million dollars"?

This is what I immediately thought when I first read about the Repugnant Conclusion on Wikipedia, years ago before having ever heard of the VNM axioms or anything like that.