Note: The LaTeX is not rendering properly on this reply. Does anyone know what the reason could be?
I chose x2+y2=2 because the optimal point in that case is the set of integers x=y=1, but the argument holds for any positive real constant, and by using either equality, less than or not greater than.
There is one thing we assumed which is that, given the utility function x+y, our proxy utility function is x .This is not necessarily obvious, and even more so if we think of more convoluted utility functions: if our utility was given by u(x,y)=xy, what would be our proxy when we only know x?
To answer this question generally my first thought would be to build a function T that maps a vector space V, a utility function u:V→R+, the manifold S of possible points and a map from those points s∈S to a filtration Fs that tells us the information we have available when at point s to a new utility function u'.
However this full generality seems a lot harder to describe.
The problem with x2+y2=1 is that it’s not clear why x would seem like a good proxy in the first place. With an inequality constraint, x has positive correlation with the objective everywhere except the boundary. You get at this idea with u(x,y)=xy knowing only x, but I think it’s more a property of dimensionality than of objective complexity—even with a complicated objective, it’s usually easy to tell how to change a single variable to improve the objective if everything else is held constant.
It’s the “held constant” part that really matters—changing one variable while holding all else constant only makes sense in the interior of the set, so it runs into Goodhart-type tradeoffs once you hit the boundary. But you still need the interior in order for the proxy to look good in the first place.
Note: The LaTeX is not rendering properly on this reply. Does anyone know what the reason could be?
I chose x2+y2=2 because the optimal point in that case is the set of integers x=y=1, but the argument holds for any positive real constant, and by using either equality, less than or not greater than.
There is one thing we assumed which is that, given the utility function x+y, our proxy utility function is x .This is not necessarily obvious, and even more so if we think of more convoluted utility functions: if our utility was given by u(x,y)=xy, what would be our proxy when we only know x?
To answer this question generally my first thought would be to build a function T that maps a vector space V, a utility function u:V→R+, the manifold S of possible points and a map from those points s∈S to a filtration Fs that tells us the information we have available when at point s to a new utility function u'.
However this full generality seems a lot harder to describe.
Best, Miguel
The problem with x2+y2=1 is that it’s not clear why x would seem like a good proxy in the first place. With an inequality constraint, x has positive correlation with the objective everywhere except the boundary. You get at this idea with u(x,y)=xy knowing only x, but I think it’s more a property of dimensionality than of objective complexity—even with a complicated objective, it’s usually easy to tell how to change a single variable to improve the objective if everything else is held constant.
It’s the “held constant” part that really matters—changing one variable while holding all else constant only makes sense in the interior of the set, so it runs into Goodhart-type tradeoffs once you hit the boundary. But you still need the interior in order for the proxy to look good in the first place.
Fixed the LaTeX for you. You were in WYSIWYG editor mode, where you type LaTeX by pressing CTRL/CMD and 4 at the same time.
Thank you habryka!