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Economy & Business

A Lesson In Classic Fair Division Problems And The Solutions


The recent government shutdown was in part a battle over how to fairly divide federal resources. This problem is one of fair division. Fair division is its own area of study, from how to fairly split rent between roommates to how to pick a movie and a group. Problems of fair division crop up all the time. Sally Herships from our Planet Money podcast walks us through some classic fair division problems and their solutions.

SALLY HERSHIPS, BYLINE: Recently, I ran an informal economic study. I presented a sealed cardboard box to two 10-year-old girls, then I backed way up.

UNIDENTIFIED CHILD #1: Oh, my God. It's got candy on it.

UNIDENTIFIED CHILD #2: Marshmallows.

HERSHIPS: Inside the box with a cake with a lot of candy on it. I asked the girls to figure out how to fairly divide it.

How do you both feel about the marshmallows?

UNIDENTIFIED CHILD #1: I love marshmallows. I don't really like Skittles, though. But they do...

UNIDENTIFIED CHILD #2: I love Skittles. But they do make different flavors if you combine them.

HERSHIPS: This problem is called fair division. Luckily, there is a solution - divide and choose.

UNIDENTIFIED CHILD #1: If one person cuts it and the other person decide - what - gets what, then it could be equal.

HERSHIPS: We see fair division problems all the time. How do you split a property after a divorce, profits in a business, runway space at an airport? Constantinos Daskalakis, or Costis, is an expert in game theory at MIT. He says there are some basics like the rental harmony problem. He asks Planet Money reporter Kenny Malone and I to imagine we have an apartment with two bedrooms. One is big, but it has no closet; the other does, but it's small.

CONSTANTINOS DASKALAKIS: Maybe, you know, I value a smaller room that has a closet, but you value more a big room that - because you like space.

HERSHIPS: I'm taking the closet.

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

DASKALAKIS: OK. Thanks for sharing. So, like, yeah, so the question is, you know, who gets what? But also, how is rent split?

HERSHIPS: Say the rent is $2,000. The roommates need to figure out what percentage of that they think each room is worth.

DASKALAKIS: Maybe for you, two rooms are equal. But 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 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 they consider to be the right values of the two rooms.

HERSHIPS: And what they found, Costis' roommate was willing to pay more for the big room - $1,400. Costis was willing to pay 1,000 for either.

DASKALAKIS: So each of us get the room where they're the highest bidder. However, how much do we pay? We pay the average of the two prices.

HERSHIPS: So Costis only has to pay $800 even though he was willing to pay a thousand.

Are you free tonight? Because I feel like next time my boyfriend and I decide who gets to pick what movie to watch, I would like you to be part of the decision.

There's a solution for this, too - fair random assignment, or coin flipping - for when you're dealing with something that can't be divided like what movie to watch. If you have more than two people, you can use a similar protocol, but it has a way cooler name.

DASKALAKIS: This has the eerie name of random dictatorship. Say you want to - say you have a group of five friends and you want to decide what to do tonight. OK. One way to be fair about the decision is to say, I'm going to flip a five-faced dice. OK.


DASKALAKIS: And, you know, whoever is elected will decide the whole plan.

HERSHIPS: Costis says there are other solutions - auctions, lotteries, time sharing. But, of course, if you want something badly enough and you can afford it, you can always buy yourself a cake and eat it too. Sally Herships, NPR News. Transcript provided by NPR, Copyright NPR.