It doesn’t mean anything to translate or scale a utility function. A utility function is just a mathematical way to encode certain relative relationships between outcomes, and it turns out if you add 42 to every value of the function, it still encodes those relationships faithfully. Or if you multiply everything by 9000.
This is why comparing two individuals’ utility functions in any naive way makes no sense: their functions are encoding relative preference relationships, but it doesn’t matter if one of them multiplies everything by 9000. So any comparison that breaks when one of them multiplies everything by 9000 or adds 42 to everything but the other doesn’t isn’t actually making correct use of the underlying relative preference relationships.
It doesn’t mean anything to translate or scale a utility function. A utility function is just a mathematical way to encode certain relative relationships between outcomes, and it turns out if you add 42 to every value of the function, it still encodes those relationships faithfully. Or if you multiply everything by 9000.
This is why comparing two individuals’ utility functions in any naive way makes no sense: their functions are encoding relative preference relationships, but it doesn’t matter if one of them multiplies everything by 9000. So any comparison that breaks when one of them multiplies everything by 9000 or adds 42 to everything but the other doesn’t isn’t actually making correct use of the underlying relative preference relationships.