It is true that adding different utility functions is in general an error. However, for agents bound to follow Rationality (and Rationality alone), the different utility functions are best thought of as the same utility function conditioned on different hypotheses, where the different hypotheses look like “The utility to P2 turns out to be what really matters”.
After all, if the agents are making their decisions on the basis of Rationality alone, then Rationality alone must have a utility function. Since Rationality is universal, the utility function must be universal. What alternative does Rationality have, given the constraints of the problem, other than a weighted sum of the utility functions of the different individuals who might turn out to matter?
Error: Adding values from different utility functions.
See this comment.
[Resuming my tongue-in-cheek argument...]
It is true that adding different utility functions is in general an error. However, for agents bound to follow Rationality (and Rationality alone), the different utility functions are best thought of as the same utility function conditioned on different hypotheses, where the different hypotheses look like “The utility to P2 turns out to be what really matters”.
After all, if the agents are making their decisions on the basis of Rationality alone, then Rationality alone must have a utility function. Since Rationality is universal, the utility function must be universal. What alternative does Rationality have, given the constraints of the problem, other than a weighted sum of the utility functions of the different individuals who might turn out to matter?
“Rationality” seems to give different answer to the same problem posed with different affine transformations of the players’ utility functions.
[Still arguing with tongue in cheek...]
That’s where the measures p and q come in.