All of us have wants. It’s like we each have a list, with the things we think best at the top, and the worst things down low. (A list can have ties, by the way.)
But can we, as a group, have wants? Does it make sense to say that a group has a list of wants that comes from the lists of those in the group?
Ken proved this does not make sense.1 Let’s zoom out and see what his point was.
First, we need to say a word or two on “rules.” A rule takes in the lists of each in a group and spits out the list of the group as a whole. Ken’s proof says no rule can have all the good traits we want it to have.
Here are the traits Ken had in mind:
All Lists are Fine - we can’t say “you may not have those lists” to those in the group. Each may have a list of their true wants, as long as it is a list of the things at stake, and as long as the list ranks those things from high to low (ties are fine, as I said).
When Wants are the Same - if A beats B on the lists of each, then A beats B on the list of the group.
Just the Ranks - does A beat B on the group’s list? You can tell just from how each in the group ranks A and B. (The group’s rank for that pair can’t change due to a change in how folks rank a third thing C.)
No Boss - there is no Boss of the group, in the sense that no one can get what they want for sure. (In clear words: for each in the group, it’s not true that if A beats B on their list, A must beat B on the group’s list.)
And you could say there’s a fifth trait, which comes out of what we mean by “rules”:
The Group Has a List - the group’s wants can be put on a list. (To see what I mean, think of a group for whom A beats B, B beats C, and C beats A. We can’t put these wants on a list. What would go at the top?)
Ken proved no rule has all five traits. Wow. That’s nuts!
But are all of these traits as good as Ken thought?
1. All Lists are Fine
It would be odd to say, “I can tell you what the group wants, but not if those in the group have these wants.” To say that is to give up the game!
But here’s a nice thought. If we have luck, those in our group may have lists that work by sheer chance, and our rule will act as if it had the next four good traits.
Let me show you what I mean.
Say we can line up the things on the lists from left to right, so that each in the group has a spot on the line they like best, and each wants the things close to that spot more than things far from that spot.
If you graph this out, so that each has a line that goes up for things one wants more and down for things that one wants less, each line looks like a range of big rock hills that has just one peak.
When you can line up the things left to right in this way, we say that folks in the group have “one peak wants.”
When folks have one peak wants, then there is a rule that has the next four traits, known as Most Folks Rule.2 It says A beats B for the group iff3 on most lists, A beats B.
Note that Most Folks Rule does not have all the good traits if All Lists are Fine, since then a group might not have one peak wants. We might end up with a group that has wants that do not fit on a list.
Say we have three folks with these wants:
Ann: A > B > C.
Bert: B > C > A.
Cate: C > A > B.
Then most folks rank A > B, B > C, and C > A. So there’s no list for the group, since their wants go round and round. This is known as French Guy’s Loop.4
2. When Wants are the Same
It’s hard not to love this one. How can the group not want what each wants?
But let me share a thing you might not know.
If our goal is just to give folks what they want, then sure, it’s bad when all the folks in a group want A more than B, and yet the group wants B more than A. That makes no sense. But what if we don’t just want to give folks what they want? What if we want to give each the right to say what goes on in their own life?
Then we get Sen’s Loop.
Think of two guys: Lewd likes smut, and Prude does not. They have to choose who will read a book of smut. So they have three things to rank:
Lewd reads the book (call this L).
Prude reads the book (call this P).
No one reads the book (call this N).
And since each has a right to say what goes on in his life, we’ll say that each gets to say what the group wants when it comes to a pair of things:
Lewd’s rights: L and N.
Prude’s rights: P and N.
Now, if we say that All Lists are Fine, Lewd and Prude might have these lists:
Prude: N > P > L.
Lewd: P > L > N.
Of course, for Lewd, the worst thing is that no one reads the book, while for Prude, that’s the best thing. But note that the two men both want Prude to read it more than they want Lewd to read it. (Lewd wants Prude to cut loose, let’s say, while Prude wants to “fall on the sword.”)
But now we get a loop!
The group ranks P > L, since that’s how Prude and Lewd both rank them.
The group ranks L > N, thanks to Lewd’s rights.
The group ranks N > P, thanks to Prude’s rights.
Some say this loop is less cool than the last one. But I think it’s so cool. Katz lays out how Sen’s Loop shows up in the law, and my friend Luc lays out how the loop shows up when you choose things and there’s a group of ways in which things are good.
Such a group of goods is a bit like a group of folks, and a rule for choice when there are n goods is a bit like a rule to find the wants of a group with n folks in it.
The math for Sen’s Loop—like that of Ken’s proof—can mean a lot of things.
3. Just the Ranks
This is the big one. Let me just say one main point.
You should know this trait means more than “If the group wants A more than B, then it would still want A more than B if we got rid of C.” This is what the trait’s full name makes you think. But watch out! The rule says more. It says that the sole way to change a group’s rank for A and B is to change how folks in the group rank A and B.
So it’s just the ranks of each that count. That’s a big deal!
Think of Prude and Lewd once more. Here’s how they rank P (Prude reads the smut book) and N (no one reads it):
Prude: N > P.
Lewd: P > N.
Now, let’s say you’re like my friend Rich, who writes Good Thoughts. You think the right rule will give the group what’s best for it as a whole.5 This means that, if it’s best for Prude if he reads the book, and best for Lewd if Prude won’t read the book, then we have to know more than how these guys rank things. We have to know: “How much does each man lose, if he does not get the best thing for him?” If Lewd will lose a lot (he loves smut!), and Prude will lose just a small bit, then it’s best for the group if Lewd gets to read the book. But if Lewd will lose just a small bit, and Prude will lose a lot (he hates smut!), then it’s best for the group if Lewd can’t read the book.
This kind of rule is old and loved. It has us ask, “Who has most to lose?” This seems like a good thing to ask, when we want to know what’s best for a group as a whole.
So is “Just the Ranks” not such a good trait, in the end?
Well, here’s one way in which it’s good. If you hold a vote to find out what a group wants, it’s not hard to get folks to tell you how they rank things. But it’s hard to find out who has more to lose. What will you do—just ask them? Each might ham it up so that they get the thing they most want. Worse, it’s hard to know how to weigh a loss to me and a loss to you. Sure, if I lose an arm, that’s worse for me than if you lose a thumb. But what if I say a loss of my arm is a loss of 10 points, while you say a loss of your arm is a loss of 9 points. How can we be sure that my points and your points count the same? What could make that true?
What a mess!
You can see why, when we vote for who will be in charge, we tend to just ask folks to rank the ones who run. We don’t ask how much folks have to lose. But this makes “Just the Ranks” a nice trait for votes in the real world. And so we should not be so quick to say it’s bad—which means Ken’s proof still has bite.
4. No Boss
Why not just pick a Boss for the group? Then we can say that the groups’s list is just the list of the Boss (at least, we say this when it comes to pairs of things where the Boss has a view).
But that would be bad. We want the group’s list to come from the lists of all of us, not just one Boss.
Let me give one case, though, where a rule has a Boss. It’s called the Word Rule, since it fits how we list words in the book where we look up what words mean.
To know if “Axe” or “Zap” comes first in the book, we start with the first slot. “A” beats “Z.” So “Axe” comes first.
If we have a tie in the first slot, then we go to the next. Take “Ape” and “Axe.” These tie in the first slot. But “P” beats “X.” So “Ape” comes first.
If we have a tie in two slots, then we go to the third. And if we have a tie there, we go to the fourth. And so on. If there’s a tie in each slot, the two words are tied. But most times there’s a slot where one word wins.
You can see how there’s a kind of “Boss” here—the first slot! The sole case where the rest of the slots have a chance to chime in is when there’s a tie in the first slot.
What if we line up the folks in the group like the slots in a word? Then the one in front is the Boss, and the next one is the boss of the ones that come next, and so on. This is how things might be in a place of work, or a land with a king. But it’s not how we want things to go in a place where each counts for one.
5. The Group Has a List
This one seems cut and dry.
But not so fast! There’s a proof by a guy named Al. This proof shows that, if we get loose, and we just say the group’s wants are sort of like a list, we don’t end up with a Boss—but we do end up with a part of the group that’s sort of like a Boss group.
I can’t say the name for such a Boss group, but here’s a hint from the news.
To say that the group’s wants are sort of like a list is to be fine with this: for the group, A beats B, and B beats C, but A does not beat C.
To say that a part of the group is sort of like a Boss is to say this: if at least one who’s in this part of the group ranks A > B, then the whole group can’t rank B > A. And if all in the part of the group rank A > B, so must the group as a whole.
Deep stuff. As Al shows, when we find a way round Ken’s old proof, a new “can’t do that” proof may soon pop up. We can’t just tweak one of Ken’s traits and call it a day.
We Stan Ken
I’ll leave you with a neat fact.
When Al wrote his proof, he was in a class taught by Ken, Sen, and a guy named Rawls—rock stars, in my line of work!
It’s so great that a young kid could prove a big thing. And that’s why I wrote this. Lots of smart folks have spent lots of time on Ken’s “can’t do that” proof. But there’s so much left to learn! You may still be in school, like Al was. But you could find a way to tweak Ken’s traits and prove new things. Or you could use Ken’s proof in a new field, where no one knows who Ken is. The lists don’t have to be lists of wants. Lists and rules are just math, and math can mean lots of things. The proofs may be old, but we can use them to think new thoughts.
There you have it. I hope you come to love Ken’s proof as much as I do!
Ken was asked this by a guy at RAND, where smart folks like games, while the two men had a drinks break. The guy, whose name I can’t say here, thought, “If we know your list of wants, we can guess what you will do next. It would be nice to have a list of the Reds’ wants, since we’re in a Cold War.” Ken was not, as far as I know, all that big on war. His proof has made its splash not with war guys, but with folks who ask how we should count votes.
This holds as long as there are 3 folks, or 5 folks, or 7 folks, and so on. Not if there are 2 folks, or 4 folks, or 6 folks, and so on.
If you read this blog, you may know this, but “p iff q” means “if p then q, and if q then p.”
My friend the Swede, whose work made me want to write this, once wrote a piece on how you lose cash if your “likes form a loop.” A cool phrase!
So great! But I think there is a small fix I can find. You said: “what if I say a loss of my arm is a loss of 10 points, while you say a loss of your arm is a loss of 11 points.” It seems to me you want to say “9 points” and “10 points”, not “10 points” and a thing I don’t know how to say.
As a matching and social choice theorist, I am deeply disappointed that I did not write this post. Here, take my re-stack you filthy scooper!
(Excellent work.)