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cousin_it comments on What Cost for Irrationality? - Less Wrong
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I suspect that risk-loving humans are just humans who don't understand probability. If you threaten to kill someone unless they double their wealth in twenty-four hours, they should still prefer a 50:50 double or nothing gamble to a 25:75 quadruple or nothing one. Beyond a certain point, the utility of money always has diminishing returns; what varies between people is the location of that point.
That's correct... kind of... but let's make things a bit clearer.
"Risk-averse" and "risk-loving" are shorthand words that describe someone's curve of utility against money. Risk-loving means the curve bulges downward ("convex"), risk-averse means the curve bulges upward ("concave"). You're just pointing out that the curve may be locally convex in the vicinity of the person's current wealth, but concave elsewhere. It is probably true that most people won't desire the extra million as much after they get their first billion. But if you have no opportunity to make huge bets, you may well be risk-loving on small bets without being ignorant or irrational.
Do you have any examples of real economic circumstances under which a sane person (someone who isn't solely concerned with maximizing the number of Porsches they own, e.g.) would have a convex utility/money curve?
(If there is a way to phrase this question so that it seems more curious and less confrontational, please assume that I said that instead.)
I read somewhere that the reason we don't see these people is that they all immediately go to Vegas, where they can easily acquire as many positive value deals as they want.
Human beings don't eat money. Your utility/money curve depends on the prices of things you can buy with the money, and the relative utilities of those things. Both factors can vary widely. I know no law of nature saying a $1000 gadget can't give you more than twice the utility of a $500 gadget. For the most direct example, the $1000 gadget could be some kind of money-printing device (e.g. a degree of higher education).
That is (or should be) the reason why people to borrow money. You borrow if the utility gain of having more money now outweighs the loss of utility by having to pay back more money later.
But note that utility becomes more complicated when time gets involved. The utility of a dollar now is not the same as the utility of a dollar next week.
This can explain locally convex curves. But is it imaginable to have a convex curve globally?
It's imaginable for an AI to have such a curve, but implausible for a human having a globally convex curve.
That's what I think. Anything is imaginable for AI.
Yes. y = log(x) is convex globally. A logarithmic utility function makes sense if you think of each additional dollar being worth an amount inversely proportional to what you have already.
No, your example is concave. The above posters were referring to functions with positive second derivative.
The mnemonic I was taught is "conve^x like e^x"
I learned "concave up" like e^x and "concave down" like log x.
How in jubbly jibblies did this get voted down? The obvious way to resolve the ambiguity in "convex" and "concave" for functions is to also specify a direction.