Decision Theory 101, With Special Guest Star: Webcomics!
“Economists think everything can be boiled down to money.” Well, yes. We do. But we have better reasons for it than you might think. I say “we,” but I of course turned in my supply and demand curves six months ago to be an engineer, and nobody yells at engineers for thinking mainly in numbers. But the perfectly accurate slander I was subject to during my time at Northwestern still grates a little, so I plan to set it straight in two parts. Today I will explain why economists use money’s dorky cousin, utility, or in other words why we think it’s legit to express people’s desires in numbers. Next time (hopefully before June, but no promises) I will chart the difficulties you run into if you try to compute utility by measuring money and not-money (people’s feelings, personal freedom, human lives, what have you) on different scales.
Suppose you’re aware of fifty webcomics and have opinions about which of them are better than others. You make a utility function over webcomics by giving each comic a number so that for any two comics, the one with the higher number is the one you like better. Can you even do that? Let’s assume you can and see where it takes us.
Numbers are easy to arrange from highest to lowest. If I sort my list of webcomics* by utility number, I wind up with a ranking of webcomics from best (is there really a “best” webcomic?) to worst**. Conversely, if I start with a ranking of fifty webcomics, I can give the top one a utility of 50, the second a utility of 49, and so on in that fashion until everyone is eaten…er…numbered. Higher numbers indicate better comics, like a utility function should. So if you can make a utility function, you can make a ranking and vice versa.
If you can make a ranking of your favorite webcomics, what then? First, if Sinfest is above Sluggy on the ranking, Sluggy is not above Sinfest. Second, if Order of the Stick is above PVP, there are no comics below PVP but above OOTS. The list doesn’t have double entries and doesn’t wrap around. If we substitute “better” and “worse” for “above” and “below” and boring ol’ letters for strip names, we get:
Asymmetry: If A is better than B, B is not better than A.
Negative Transitivity: If A is better than B, everything being considered is either worse than A, better than B, or both.
Why “negative” transitivity? Because regular transitivity was taken by a similar but not quite identical property. But why mention these obvious facts and give them silly names at all? Because these two properties are the only things that need to be true for us to build a ranking. Here, I’ll show you. This is the practical process for building a ranking:
“What’s better than Sluggy? Well, Penny Arcade is, I think. What’s better than Penny Arcade? Maybe Perry Bible Fellowship? What’s better than PBF? Um….XKCD. What’s better than XKCD? I don’t know. I can’t think of anything better than XKCD. Ok, that goes at the top of the ranking. Now, other than XKCD, what’s better than Sluggy?”
This will take a long time. But so long as every chain of thought ends with “I can’t think of anything better than that (except for what’s already on my list),” we’ll eventually get a ranking. Now there’s only fifty webcomics on the list we’re trying to rank, so to keep our chain of thought running forever we’ll have to repeat ourselves. Fortunately, Asymmetry and Negative Transitivity keep us from doing that. Observe:
“What’s better than Sluggy? Well, Penny Arcade is, I think. So by Negative Transitivity, everything else on this list is either worse than Penny Arcade or better than Sluggy. Let’s cross off everything worse than Penny Arcade, so that everything on the list is better than Sluggy. What’s better than Penny Arcade? Maybe Perry Bible Fellowship? It’s not crossed off, of course, because Asymmetry says nothing can be both better than PA and worse than PA. So now everything is either worse than PBF or better than Penny Arcade. We’ll cross off everything worse than PBF (including PA), so now everything left is better than PA and Sluggy…”
“I can’t think of anything better than XKCD. Did I cross out anything that’s better? No, because XKCD is better than (Sluggy/PA/PBF) and I only crossed out things for being worse than (Sluggy/PA/PBF), so by Negative Transitivity nothing I’ve crossed out is better than XKCD. Ok, that goes at the top of the ranking.”
And so it goes. Since at least one thing is crossed out every step of the way, and since the only crossed out comics are those you would never choose in the next step, you can't keep naming comics forever whether you actually go through the trouble of crossing them out or not. Moral of the story: if your value judgments display Asymmetry (they’d better) and Negative Transitivity (at least plausible), then it’s perfectly okay to assign utility numbers to whatever you’re trying to make judgments about. Economists pretend that people make decisions based on such numbers not because we think they actually do, but because very little seems to be lost in the translation and numbers are a convenient way to do math.
* And by “my list of webcomics,” I mean “my list of webcomics-not-written-by-my-LJ-Friends,” for obvious reasons.
** If there are two equally good comics, they will have equal utility numbers, and I’ll just jam them into the same list entry. If I forget about ties later, trust me that the argument works just as well with ties as without.
Suppose you’re aware of fifty webcomics and have opinions about which of them are better than others. You make a utility function over webcomics by giving each comic a number so that for any two comics, the one with the higher number is the one you like better. Can you even do that? Let’s assume you can and see where it takes us.
Numbers are easy to arrange from highest to lowest. If I sort my list of webcomics* by utility number, I wind up with a ranking of webcomics from best (is there really a “best” webcomic?) to worst**. Conversely, if I start with a ranking of fifty webcomics, I can give the top one a utility of 50, the second a utility of 49, and so on in that fashion until everyone is eaten…er…numbered. Higher numbers indicate better comics, like a utility function should. So if you can make a utility function, you can make a ranking and vice versa.
If you can make a ranking of your favorite webcomics, what then? First, if Sinfest is above Sluggy on the ranking, Sluggy is not above Sinfest. Second, if Order of the Stick is above PVP, there are no comics below PVP but above OOTS. The list doesn’t have double entries and doesn’t wrap around. If we substitute “better” and “worse” for “above” and “below” and boring ol’ letters for strip names, we get:
Asymmetry: If A is better than B, B is not better than A.
Negative Transitivity: If A is better than B, everything being considered is either worse than A, better than B, or both.
Why “negative” transitivity? Because regular transitivity was taken by a similar but not quite identical property. But why mention these obvious facts and give them silly names at all? Because these two properties are the only things that need to be true for us to build a ranking. Here, I’ll show you. This is the practical process for building a ranking:
“What’s better than Sluggy? Well, Penny Arcade is, I think. What’s better than Penny Arcade? Maybe Perry Bible Fellowship? What’s better than PBF? Um….XKCD. What’s better than XKCD? I don’t know. I can’t think of anything better than XKCD. Ok, that goes at the top of the ranking. Now, other than XKCD, what’s better than Sluggy?”
This will take a long time. But so long as every chain of thought ends with “I can’t think of anything better than that (except for what’s already on my list),” we’ll eventually get a ranking. Now there’s only fifty webcomics on the list we’re trying to rank, so to keep our chain of thought running forever we’ll have to repeat ourselves. Fortunately, Asymmetry and Negative Transitivity keep us from doing that. Observe:
“What’s better than Sluggy? Well, Penny Arcade is, I think. So by Negative Transitivity, everything else on this list is either worse than Penny Arcade or better than Sluggy. Let’s cross off everything worse than Penny Arcade, so that everything on the list is better than Sluggy. What’s better than Penny Arcade? Maybe Perry Bible Fellowship? It’s not crossed off, of course, because Asymmetry says nothing can be both better than PA and worse than PA. So now everything is either worse than PBF or better than Penny Arcade. We’ll cross off everything worse than PBF (including PA), so now everything left is better than PA and Sluggy…”
“I can’t think of anything better than XKCD. Did I cross out anything that’s better? No, because XKCD is better than (Sluggy/PA/PBF) and I only crossed out things for being worse than (Sluggy/PA/PBF), so by Negative Transitivity nothing I’ve crossed out is better than XKCD. Ok, that goes at the top of the ranking.”
And so it goes. Since at least one thing is crossed out every step of the way, and since the only crossed out comics are those you would never choose in the next step, you can't keep naming comics forever whether you actually go through the trouble of crossing them out or not. Moral of the story: if your value judgments display Asymmetry (they’d better) and Negative Transitivity (at least plausible), then it’s perfectly okay to assign utility numbers to whatever you’re trying to make judgments about. Economists pretend that people make decisions based on such numbers not because we think they actually do, but because very little seems to be lost in the translation and numbers are a convenient way to do math.
* And by “my list of webcomics,” I mean “my list of webcomics-not-written-by-my-LJ-Friends,” for obvious reasons.
** If there are two equally good comics, they will have equal utility numbers, and I’ll just jam them into the same list entry. If I forget about ties later, trust me that the argument works just as well with ties as without.