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 arbitrarily no ice cream as the origin of our coordinate system, and “distance between one ice cream and none” as the basic unit of distance.
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.