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Vaniver comments on Iterated Gambles and Expected Utility Theory - Less Wrong Discussion
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Comments (43)
Utility is approximately the logarithm of money. Pretend otherwise, and you will get results that go against the intuition, duh.
Utility is linear to money only if we take such a small part of the logarithmic curve that it is more or less linear at the given interval. But this is something you cannot extrapolate to situations where the part of the logarithmic curve is significantly curved. Two examples of linearity:
1) You are a millionaire, so you more or less don't give a fuck about getting or not getting $1000. In such case you can treat small money as linear and choose B. If you are not a millionaire, imagine that it is about certainty of 24¢ versus 25% chance of $1.
2) You are an effective altruist and you want to donate all the money to a charity that saves human lives. If $1000 is very small compared with the charity budget, we can treat the number of human lives saved as a linear function of extra money given. (See: Circular Altruism.)
To be clearer, utility is approximately the logarithm of your wealth, not of the change to your wealth. So there's a hidden number lurking in each of those questions--if you have $100k (5) of wealth, then option A brings it up to $100240 (5.00104) and option B brings it up to either $101000 (5.00432) with 25% probability and leaves it where it is with 75% probability, which works out to a weighted average log wealth of 5.00108, which is higher, so go with B.
But if your wealth is $1k (3), then option A brings you up to a weighted average of 3.09 and B brings you up to a weighted average of 3.07. So go with A!
(The breakeven point for this particular option is a starting wealth of $8800.)
I am confused. Pick B every time? Even although the weighted average of A is better in the second case? That's supposed to be "So go with A!" right?
Typo fixed; thanks!