Won’t you get behavior practically indistinguishable from the ‘slightly risk averse agent with sqrt(x) utility function’ by simply using x^0.499 as utility function?
Also, by the way. The resulting final utility function for any sort of variable needs not be smooth, monotonously growing, or inexpensive to calculate.
Consider my utility function for food obtained by me right now. Slightly more than is optimal for me to eat before it spoils, in the summer without fridge, would give no extra utility what so ever over the right amount; or results in the dis-utility (more trash). A lot more may make it worth it to invite a friend for dinner and utility starts growing again.
Essentially the utility peaks then starts going down, then at some not very well defined point utility suddenly starts growing again.
There can be all sorts of really odd looking ‘irrational’ heuristics that work as a better substitute for true utility function which is expensive to calculate (but is known to follow certain broken line pattern), than some practical to compute utility function.
WRT utility of extra money… money themselves are worth nothing, it’s the changes to your life you can make with them, that matter.
As it is, I would take 10% shot at 10 millions $ over 100 000 for certain; 15 years ago I would take 10 000 for certain over 10% shot at 10 million (of course in the latter case it ought to be possible to partner up with someone who has big capital to get say 800 000 for certain).
Ultimately, attaching utility functions to stuff is like considering a fairly bad chess AI that just sums values of pieces and positional features perhaps. That sort of AI, running on same hardware, is going to lose big time to AIs with more clever board evaluation than that.
Won’t you get behavior practically indistinguishable from the ‘slightly risk averse agent with sqrt(x) utility function’ by simply using x^0.499 as utility function?
Also, by the way. The resulting final utility function for any sort of variable needs not be smooth, monotonously growing, or inexpensive to calculate.
Consider my utility function for food obtained by me right now. Slightly more than is optimal for me to eat before it spoils, in the summer without fridge, would give no extra utility what so ever over the right amount; or results in the dis-utility (more trash). A lot more may make it worth it to invite a friend for dinner and utility starts growing again.
Essentially the utility peaks then starts going down, then at some not very well defined point utility suddenly starts growing again.
There can be all sorts of really odd looking ‘irrational’ heuristics that work as a better substitute for true utility function which is expensive to calculate (but is known to follow certain broken line pattern), than some practical to compute utility function.
WRT utility of extra money… money themselves are worth nothing, it’s the changes to your life you can make with them, that matter. As it is, I would take 10% shot at 10 millions $ over 100 000 for certain; 15 years ago I would take 10 000 for certain over 10% shot at 10 million (of course in the latter case it ought to be possible to partner up with someone who has big capital to get say 800 000 for certain).
Ultimately, attaching utility functions to stuff is like considering a fairly bad chess AI that just sums values of pieces and positional features perhaps. That sort of AI, running on same hardware, is going to lose big time to AIs with more clever board evaluation than that.