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gwern comments on Checklist of Rationality Habits - Less Wrong
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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.
This is what I did, without the pedantry of the C.
I don't follow at all. How can utilities not be comparable in terms of multiplication? This falls out pretty much exactly from your classic cardinal utility function! You seem to be assuming ordinal utilities but I don't see why you would talk about something I did not draw on nor would accept.
The point is that because the constant is there, saying that utility grows logarithmically in money underspecifies the actual function. By ignoring C, you are implicitly using $1 as a point of comparison.
A generous interpretation of your claim would be to say that to someone who currently only has $1, having a billion dollars is twice as good as having $50000 -- in the sense, for example, that a 50% chance of the former is just as good as a 100% chance of the latter. This doesn't seem outright implausible (having $50000 means you jump from "starving in the street" to "being more financially secure than I currently am", which solves a lot of the problems that the $1 person has). However, it's also irrelevant to someone who is guaranteed $50000 in all outcomes under consideration.
Then how do you suggest the person under discussion evaluate their working patterns if log utilities are only useful for expected values?
By comparing changes in utility as opposed to absolute values.
To the person with $50000, a change to $70000 would have a log utility of 0.336, and a change to $1 billion would have a log utility of 9.903. A change to $1 would have a log utility of -10.819.
I see, thanks.
"The utility of A is twice the utility of B" is not a statement that remains true if we add the same constant to both utilities, so it's not an obviously meaningful statement. We can make the ratio come out however we want by performing an overall shift of the utility function. The fact that we think of utilities as cardinal numbers doesn't mean we assign any meaning to ratios of utilities. But it seemed that you were trying to say that a person with a logarithmic utility function assesses $10^9 as having twice the utility of $50k.
Kindly says the ratios do have relevance to considering bets or risks.
Yes, I think I see my error now, but I think the force of the numbers is clear: log utility in money may be more extreme than most people would intuitively expect.