It’s not always possible to say A ≻ B or A ≺ B. Sometimes
- neither A nor B is smaller. A≹B
- neither A nor B is more successful. A≹B
- neither A nor B is prettier. A≹B
- neither A nor B is smarter. A≹B
- you don’t love A any more or any less than you love B. ℒ(A)≹ℒ(B)
- neither A nor B is tastier. A≹B
- neither A nor B is closer. |A−x| ≹ |B−x|
- neither A nor B is more fair. A≹B
- neither A nor B is better. A≹B
I’ve argued this before using posets. And I intend to argue it further later, when I claim that the concept of Pareto superiority was a major step forward in ethics.
*[The concept of Pareto dominance allows you to make, at least in theory, a valid, fully general comparison between two states of the world. A≻B in full generality iff aᵢ ≻ bᵢ ∀ aᵢ ∈ A and ∀ bᵢ ∈ B, by the individual standards of ∀ ≻ᵢ.]
For now, though, I’ll draw some examples of functionals that don’t beat one another. That is, ƒ≹g nor g≹ƒ. (You might assume ≹ has to be 2-symmetric but I’m just stating it for clarity.)
In this drawing, green wins sometimes and purple wins other times. Is it more important to win the “righthand” cases or the “lefthand” cases? How much better for each scenario? (see integrating kernel) Is it better for the L₂ norm to be higher? Or just for the ∫ mass to be greater?
In this drawing, orange wins sometimes and blue wins other times. Is it more important to win the “interior” cases or the “extremal” cases? How much better for each scenario? (see kernel of integration)
How about a function that measures the desirability of a particular boyfriend / girlfriend in various scenarios. How about the function g measures boyfriend B in the various scenarios (domain) and the function ƒ measures boyfriend A in the various scenarios. By measures, I mean the function’s codomain is some kind of totally ordered set where it does make sense to talk about better ≻ and worse ≺.
at dinner) ≻ g(
career) ≺ g(
in bed) ≫ g(
with your family) ≺ g(
with your family)
at the beach) ≺ g(
at the beach)
- …and so on…
So how do you decide whether A≺B or B≺A? Perhaps you have your own priorities sorted so well that you can apply a kernel. Or perhaps A≹B in the final analysis.
I could make a comparable list for
- comparing two houses or apartments (well, this one’s closer to the park, but that one has that cozy breakfast nook),
- comparing two societies (one where the top marginal tax rate is 41% and one where the top marginal tax rate is 40%),
- and on and on.
Sometimes it’s hard to compare. Sometimes — like which of your kids do you love the best — it’s impossible to compare.