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nshepperd comments on Pinpointing Utility - Less Wrong
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I mean set as in set-theory. As in the utility function is a set of equivalent functions. If I'm disagreeing with math use, please correct me. (on second thought that wording is pretty bad, so I might change it anyway. Still, are my set-intuitions wrong?)
I mean space as in a 1-dimensional space (with a non-crazy metric, if crazy metrics even exist for 1d). By "measure distance" I mean go into said space with a tape measure and see how far apart things are.
I call it a space because then when I visualize it as such, it has all the right properties (scale/shift agnosticism).
If I call it a real-valued function, I imagine the real number line, which has a labeled axis, so to speak, so it tempts me to do numerology.
You can think of a utility function as defining a measure of "signed distance" on its domain.
Utilities have some similarity to distance in physical space, in that to give coordinates to all objects you need to select some origin and system of units for your coordinate system, but the physical reality is the same regardless of your coordinate system. A member of a particular utility function's equivalence class, can then be thought of as a function that gives the coordinates of each thing in the domain (world-states, presumably), in some particular coordinate system.
For an example, if I prefer to have three ice creams over zero, three times as much as I prefer one ice cream over zero, then we can write that as a "utility function"
u(no ice cream) = 0; u(one ice cream) = 1; u(three ice creams) = 3. In this case we have chosen arbitrarilyno ice creamas the origin of our coordinate system, and "distance betweenone ice creamandnone" as the basic unit of distance.Is this what you mean by a 1-dimensional space?
That's exactly what I mean.
Four servings of ice cream would have me ill.