If you take a utility function and multiply all the utilities by 0.01, is it the same utility function? In one sense it is, but by your measure it will always win a “most pessimistic” contest.
Update: thinking about this further, if the only allowable operations on utilities are comparison and weighted sum, then you can multiply by any positive constant or add and subtract any constant and preserve isomorphism. Is there a name for this mathematical object?
In particular, this means that nothing has “positive utility” or “negative utility”, only greater or lesser utility compared to something else.
ETA: If you want to compare two different people’s utilities, it can’t be done without introducing further structure to enable that comparison. This is required for any sort of felicific calculus.
There’s a name I can’t remember for the “number line with no zero” where you’re only able to refer to relative positions, not absolute ones. I’m looking for a name for the “number line with no zero and no scale”, which is invariant not just under translation but under any affine transformation with positive determinant.
I’m in an elementary statistics class right now and we just heard about “levels of measurement” which seem to make these distinctions: your first is the interval scale, and second the ordinal scale.
The “number line with no zero, but a uniquely preferred scale” isn’t in that list of measurement types; and it says the “number line with no zero and no scale” is the interval scale.
A utility function is just a representation of preference ordering. Presumably those properties would hold for anything that is merely an ordering making use of numbers.
You also need the conditions of the utility theorem to hold. A preference ordering only gives you conditions 1 and 2 of the theorem as stated in the link.
Good point. I was effectively entirely leaving out the “mathematical” in “mathematical representation of preference ordering”. As I stated it, you couldn’t expect to aggregate utiles.
How about “judge by both utility functions and use the most pessimistic result”?
If you take a utility function and multiply all the utilities by 0.01, is it the same utility function? In one sense it is, but by your measure it will always win a “most pessimistic” contest.
Update: thinking about this further, if the only allowable operations on utilities are comparison and weighted sum, then you can multiply by any positive constant or add and subtract any constant and preserve isomorphism. Is there a name for this mathematical object?
Affine transformations. Utility functions are defined up to affine transformation.
In particular, this means that nothing has “positive utility” or “negative utility”, only greater or lesser utility compared to something else.
ETA: If you want to compare two different people’s utilities, it can’t be done without introducing further structure to enable that comparison. This is required for any sort of felicific calculus.
There’s a name I can’t remember for the “number line with no zero” where you’re only able to refer to relative positions, not absolute ones. I’m looking for a name for the “number line with no zero and no scale”, which is invariant not just under translation but under any affine transformation with positive determinant.
I’m in an elementary statistics class right now and we just heard about “levels of measurement” which seem to make these distinctions: your first is the interval scale, and second the ordinal scale.
The “number line with no zero, but a uniquely preferred scale” isn’t in that list of measurement types; and it says the “number line with no zero and no scale” is the interval scale.
A utility function is just a representation of preference ordering. Presumably those properties would hold for anything that is merely an ordering making use of numbers.
You also need the conditions of the utility theorem to hold. A preference ordering only gives you conditions 1 and 2 of the theorem as stated in the link.
Good point. I was effectively entirely leaving out the “mathematical” in “mathematical representation of preference ordering”. As I stated it, you couldn’t expect to aggregate utiles.
You can’t aggregate utils; you can only take their weighted sums. You can aggregate changes in utils though.