While this is a basic point, it’s one people seem to screw up around here a lot, so I’m glad someone wrote an article going over this in detail. Upvoted.
I have one nitpick: You say, “We have to take the ratio between two utility differences”, but really, because only positive affine transformations are OK, what we really have to take is the ratio between a utility difference and the absolute value of a utility difference.
Tangentially, I’d also like to point out the article Torsors Made Easy by John Baez. OK, to be honest, I’m not sure how understandable this really is to someone who doesn’t already know a bit. But “torsor” is a useful concept to have when thinking about things like this, and there probably isn’t a better quick explanation out there.
Tangentially, I’d also like to point out the article Torsors Made Easy by John Baez. OK, to be honest, I’m not sure how understandable this really is to someone who doesn’t already know a bit.
Having read that article years ago, without any previous exposure to the concept of torsors (other than the implicit exposures Baez notes, that everyone’s had), torsors also came to mind for me when reading nyan_sandwich’s article.
I have one nitpick: You say, “We have to take the ratio between two utility differences”, but really, because only positive affine transformations are OK, what we really have to take is the ratio between a utility difference and the absolute value of a utility difference.
Why? Positive affine transformations are OK, and they don’t affect the sign of utility differences.
While this is a basic point, it’s one people seem to screw up around here a lot, so I’m glad someone wrote an article going over this in detail. Upvoted.
I have one nitpick: You say, “We have to take the ratio between two utility differences”, but really, because only positive affine transformations are OK, what we really have to take is the ratio between a utility difference and the absolute value of a utility difference.
Tangentially, I’d also like to point out the article Torsors Made Easy by John Baez. OK, to be honest, I’m not sure how understandable this really is to someone who doesn’t already know a bit. But “torsor” is a useful concept to have when thinking about things like this, and there probably isn’t a better quick explanation out there.
Having read that article years ago, without any previous exposure to the concept of torsors (other than the implicit exposures Baez notes, that everyone’s had), torsors also came to mind for me when reading nyan_sandwich’s article.
Why? Positive affine transformations are OK, and they don’t affect the sign of utility differences.
Yes; the point of making this change is to exclude negative affine transformations.
Ooops, you are totally right. Your units have to be absolute value. Thank you, I’ll maybe fix that.