The mathematical object in question is a pair of utility functions. We have to chose how we lift the concept of an affine transformation of a utility function to an affine transformation of a pair of utility functions.
One choice is to define an affine transformation of a pair of utility functions (u,v) as pair of affine transformations (f,g) which take (u,v) to (fu, gv). With this choice we cannot compare the component utility functions within a pair.
Another choice is to define an affine transformation of a pair of utility functions (u,v) by applying a single transformation f to both components getting (fu, fv). This preserves comparisons within the pair.
The key point is that our inability to do interpersonal comparisons of utility is a modeling assumption. It is something we put in to the analysis, not something that we get out of the analysis.
sketerpot is asking “why can’t we just compare the utilities?” and in the same comment noticing that there are problems with discovering utilities. What is to stop people exaggerating their utilities in order to game the bargaining system?
sketerpot’s comment pretty much nails the situation. Since permitting interpersonal comparisons of utility opens a huge can of worms an important leg of the broader project is to say: Let us assume that interpersonal comparison of utility is impossible, and press on with the analysis to find what solutions to the bargaining problem are available under this assumption.
The mathematical object in question is a pair of utility functions. We have to chose how we lift the concept of an affine transformation of a utility function to an affine transformation of a pair of utility functions.
One choice is to define an affine transformation of a pair of utility functions (u,v) as pair of affine transformations (f,g) which take (u,v) to (fu, gv). With this choice we cannot compare the component utility functions within a pair.
Another choice is to define an affine transformation of a pair of utility functions (u,v) by applying a single transformation f to both components getting (fu, fv). This preserves comparisons within the pair.
The key point is that our inability to do interpersonal comparisons of utility is a modeling assumption. It is something we put in to the analysis, not something that we get out of the analysis.
sketerpot is asking “why can’t we just compare the utilities?” and in the same comment noticing that there are problems with discovering utilities. What is to stop people exaggerating their utilities in order to game the bargaining system?
sketerpot’s comment pretty much nails the situation. Since permitting interpersonal comparisons of utility opens a huge can of worms an important leg of the broader project is to say: Let us assume that interpersonal comparison of utility is impossible, and press on with the analysis to find what solutions to the bargaining problem are available under this assumption.