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Fair division problems

NPR Planet Money Episode on fair division problems.

These problems have always interested me. Part of the reason I’m taking some economics classes to supplement my Philosophy is that I’m interested in how the systems we use to divide resources “fairly” work in the real world. When there are limited resources, who are the people that get them and do they deserve it?

I would have liked the episode to get a little into how the mathematicians that study fair division classify a particular solution as “fair,” especially since they hint that they intent to leave a rigorous explanation of the term to the Philosophers.

The main issue in the episode is how to divide the limited dock space in Santa Barbara between the wealthy and those who need it to earn a living, but I’m curious why they don’t just expand the dock. This is a bit of an econ 1 answer, but if the price of a space is $100,000 like the episode reports, shouldn’t this be a signal to the city that they can build more and people would be willing to pay? I understand municipalities aren’t in the dock construction business, and there may be environmental concerns (preserving the natural beauty of the coast, dock is already as large as it can get, etc…), but in theory we can use market forces to get the price lower rather than relying on what might seem like contrived schemes.

The episode is also worth checking out for the “rental harmony” problem. A guest speaker, Constantinos Daskalakis, gives a pretty concise summary of an ingenious potential solution.

MALONE: Costis says imagine that you've got an apartment with two bedrooms. One of those bedrooms is big, but it has no closet.

HERSHIPS: The other is small, but it does have that magical closet space.

DASKALAKIS: That's right. And how do you split the rent? Maybe you know, I value a small room that has a closet, but you value more a big room that - because you like space.

HERSHIPS: I'm taking the closet.

MALONE: Yeah, I don't need the closet. It's fine. I wear the same jeans every day for two weeks in a row. I don't need the closet.

DASKALAKIS: OK. Thanks for sharing (laughter).

MALONE: Yeah, you know.

DASKALAKIS: So like, yeah. So the question is, you know - who gets what? What's the allocation? But also, how is the rent split?

MALONE: So what is the protocol here? Costis says, well, let's say the rent is 2,000 bucks. First thing, everybody needs to figure out what percentage of the rent they think each room is worth.

DASKALAKIS: Maybe for you, two rooms are equal. But then for Kenny, big space is so much more valuable that he says, look; you know, I don't care about the closet at all. I don't even have clothes. OK? So what I care about is the space. So...

MALONE: I have some clothes. I just was saying that I wear the same pants. I have clothes. OK. Go ahead, though.

(LAUGHTER)

DASKALAKIS: So the protocol that we use with my roommate was - and it's a classical one. Each of the two roommates, in a sealed envelope, writes what they consider to be the right split - what do they consider to be the right values of the two rooms.

HERSHIPS: You guys actually did this?

DASKALAKIS: We did that, yeah.

HERSHIPS: Is he also a fair division guy, your roommate?

DASKALAKIS: So he's also a mathematician.

HERSHIPS: Just checking.

DASKALAKIS: Now, you know, we get together, and we open those envelopes.

MALONE: And when they opened those envelopes, they found out that Costis' roommate thought the person with the big room should pay $1,400 and the person with the smaller room with the closet should pay 600 bucks. But Costis, he valued both those rooms equally. He was willing to pay $1,000 for either room.

HERSHIPS: And in case you haven't been following along with a calculator, each roommate's total bid for both rooms has to add up to $2,000.

DASKALAKIS: So he valued the big room more than I valued the big room.

MALONE: Yeah.

DASKALAKIS: On the other hand, I value the small room more than he values the small room. So each of us gets the room where they're the highest bidder. However, how much do we pay? We pay the average of the two prices.

MALONE: Yeah.

DASKALAKIS: So for the big room, he said 1,400. I said a thousand. So he gets it, but he pays 1,200. So he's happy. Right?

HERSHIPS: And Costis pays 800 for the small room, and he's happy because he was willing to pay a thousand.

DASKALAKIS: So everybody's happy, no?