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Viliam 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!
Yep, for me 100 dollars provides a nice chunk of utility. 10000 does not provide 100 times as much utility and anything more than a couple hundred K provides little utility at all.
In theory that 1.6 billion powerball lottery had a (barely) positive expected return (depending on how taxes work out) and thus rationalists should throw money at it but in reality a certainty of 1 dollar is better than a 1 in a billion chance of getting 1.6 billion. (I know these numbers aren't exact)
How do you know?
Because at that point I'm tapdancing on the top of Maslow's Hierarchy of Needs, extremely financially secure with lots of reserves.
It doesn't go to zero but it's like the difference between the utility of an extra portion of truffle desert when I'm already stuffed vs the utility of a few bags of plumpynut when I have a starving child.
With a couple of hundred thousand dollars? They don't make you financially independent (defined as "don't have to work"), you can't even buy an apartment in SF or NYC, etc...
Ya but I don't want to buy an apartment in new York.
Again, I didn't say utility goes to zero. it just drops off dramatically. The difference between 0 and 250K is far bigger in terms of utility than the difference between 250k and 500k. You still can't buy a new york apartment and having 500K is better than having only 250K but in terms of how it changes your life the first increment is far more significant.
Yes, of course, no one is arguing against that. However you make a couple of rather stronger statements e.g.
which don't strike me as true for most people.
There are large individual differences -- for a monk who has taken a vow of poverty even your "first increment" is pretty meaningless, while for someone with a life dream of flying her own plane around the world a few hundred $K aren't that much. But I would expect that a large majority of people would be able to extract significant utility out of a few hundred thousands of dollars beyond the first couple.
You might want to read again the second and third word of HungryHobo's original comment.
Yes and my question was how does he know? If he never had that amount of money available to him, his guesstimate of how much utility he will be able to gain from it is subject to doubt. People do change, especially when their circumstances change.
If you did realize you two were talking specifically about HungryHobo rather than/as well as about people in general, you might want to edit your comments to make it clearer.
Of course, but in relative terms he's still right, it's just easier to see when you are thinking from the point of the hungry hobo (or peasant in the developing world).
Standing from the point of view of a middle class person in a rich country looking at hypothetical bets where the potential loss is usually tiny relative to our large net worth+human capital value of >4-500k, then of course we don't feel like we can mostly dismiss utility over a few hundred thousand k, because we're already there.
Consider a bet with the following characteristics: You are a programmer making 60k ish a year a couple years out of school. You have a 90% probability of winning. If you win, you will win 10 million dollars in our existing world. If you lose (10%) you will swapped into parallel universe where your skills are completely worthless, you know no-one, and you would essentially be in the position of the hungry hobo. You don't actually lose your brain, so you could potentially figure out how to make ends meet and even become wealthy in this new society, but you start with zero human capital -- you don't know how to get along in it, any better than someone who was raised in a mumbai slum to typical poor parents does in this world.
So do you take that bet? I certainly wouldn't.
Is there any amount of money we could put in the win column that would mean you take the bet?
When you start considering bets where a loss actually puts you in the Hungry hobo position, it becomes clearer that utility of money over a few hundred thousand dollars is pretty small beer, compared to what's going on at the lower tiers of Maslow's hierarchy.
Which is another way of saying that pretty much everyone who can hold down a good job in the rich world has it really freaking good. The difference between $500k and $50 million (enough to live like an entertainer or big-time CEO without working) from the point of view of someone with very low human capital looks a lot like the famed academics having bitter arguments over who gets the slightly nicer office.
This also means that even log utility or log(log) utility isn't risk averse enough for most people when it comes to bets with a large probability mass of way over normal middle class net worth + human capital values, and any significant probability of dropping below rich-country above-poverty net worth+ human capital levels.
Fortunately, for most of the bets we are actually offered in real life, linear is a good enough approximation for small ones, and log or log-log utility is a plenty good enough approximation for even the largest swings (like starting a startup vs. a salaried position), as long as we attach some value to directing wealth we would not consume, and there is a negligible added probability of the kind of losses that would take us completely out of our privileged status.
In most real life cases any problems with the model are overwhelmed by our uncertainties in mapping the probability distribution.
I don't think this works out, if you think you are agreeing with Villiam. Suppose your net worth is $20,000. Then the utility increase represented by $100 is going to be [proportional to] 0.00498. On the other hand, the utility increase represented by $10,000 is going to be [proportional to] 0.40546. That is, $10,000 will be 81 times as valuable as $100.
In other words, it is less than 100 times as valuable. But not by that much, and certainly not by enough to explain the degree to which people prefer the certain $100.
Using your net worth as part of the calculation doesn't feel right.
Even if my net worth is quite high much of that may be inaccessible to me short term.
If I have 100,000 in liquid cash then 100 has lower utility to me than if I have 100,000 in something non liquid like a house and no cash.