I don’t see why 2 is necessary given that any point on the Pareto frontier is a mixture of pointy points (intuition for this: any point on the face of a polyhedron is a mixture of that face’s corners). In any case, I agree with the basic mathematical point that you can get any Pareto optimal mixture of outcomes by mixing between non-negative linear combinations of utility functions.
Well, I was imagining a Pareto frontier that changes smoothly from flat to curved. Then we can’t quite get a pointy point exactly on the edge of the flat part. That’s what 2 is for, it gives us some of these points (though not all). But I guess that doesn’t matter if things are finite enough.
I don’t see why 2 is necessary given that any point on the Pareto frontier is a mixture of pointy points (intuition for this: any point on the face of a polyhedron is a mixture of that face’s corners). In any case, I agree with the basic mathematical point that you can get any Pareto optimal mixture of outcomes by mixing between non-negative linear combinations of utility functions.
Well, I was imagining a Pareto frontier that changes smoothly from flat to curved. Then we can’t quite get a pointy point exactly on the edge of the flat part. That’s what 2 is for, it gives us some of these points (though not all). But I guess that doesn’t matter if things are finite enough.
Ok, that seems right.