Utility functions aren’t composable! Utility functions aren’t composable! Sorry to shout, I’ve just realized a very specific way I’ve been wrong for quite some time.
VNM utility is completely ignores that structure and of outcomes and “similarities” between outcomes. U(1 apple) doesn’t need to have any relation to U(2 apples). With decision scenarios I’m used to interacting in, there are often ways in which it is natural to things of outcomes as compositions or transformation of other outcomes or objects. When I think of outcomes, they can be more or less similar to each other, even if I’m not talking about value. From facing a lot of scenarios like this, it’s easy to think it terms of “Find some way to value the smaller set of outcomes that can compose to make all outcomes”, which makes it easy to expect such composability to be a property of of VNM utility works. But it’s not! It really really isn’t.
I’ve recently been reading about ordinal numbers, and getting familiar with the idea that you can have things that have order, but no notion of distance. I had that in the back of my mind when going through the wikipedia page for VNM utility, and I think that’s what made it click.
Yes, indeed quite important. This is a common confusion that has often lead me down weird conversational paths. I think some microeconomics has most made this clear to me, because in there you seem to be constantly throwing tons of affine transformations at your utility functions to make them convenient and get you analytic solutions, and it becomes clear very quickly that you are not preserving the relative magnitude of your original utility function.
I think one of the reasons it took me so long to notice was that I was introduced to VNM utility I’m the context of game theory, and winning at card games. Most of those problems do have the property of the utility of some base scoring system composing well to generate the utility of various end games. Since that was always the case, I guess I thought that it was a property of utility, and not the games.
Utility functions aren’t composable! Utility functions aren’t composable! Sorry to shout, I’ve just realized a very specific way I’ve been wrong for quite some time.
VNM utility is completely ignores that structure and of outcomes and “similarities” between outcomes. U(1 apple) doesn’t need to have any relation to U(2 apples). With decision scenarios I’m used to interacting in, there are often ways in which it is natural to things of outcomes as compositions or transformation of other outcomes or objects. When I think of outcomes, they can be more or less similar to each other, even if I’m not talking about value. From facing a lot of scenarios like this, it’s easy to think it terms of “Find some way to value the smaller set of outcomes that can compose to make all outcomes”, which makes it easy to expect such composability to be a property of of VNM utility works. But it’s not! It really really isn’t.
I’ve recently been reading about ordinal numbers, and getting familiar with the idea that you can have things that have order, but no notion of distance. I had that in the back of my mind when going through the wikipedia page for VNM utility, and I think that’s what made it click.
Yes, indeed quite important. This is a common confusion that has often lead me down weird conversational paths. I think some microeconomics has most made this clear to me, because in there you seem to be constantly throwing tons of affine transformations at your utility functions to make them convenient and get you analytic solutions, and it becomes clear very quickly that you are not preserving the relative magnitude of your original utility function.
I think one of the reasons it took me so long to notice was that I was introduced to VNM utility I’m the context of game theory, and winning at card games. Most of those problems do have the property of the utility of some base scoring system composing well to generate the utility of various end games. Since that was always the case, I guess I thought that it was a property of utility, and not the games.