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#890: The Division Problem

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#890: The Division Problem

#890: The Division Problem

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A few weeks ago, reporter Sally Herships decided to run an informal economic study.


So who am I talking to? Will you introduce yourself?

ESME: My name is Esme (ph). I'm 10 years old.

MALONE: Esme, 10 years old, she was with her friend.

KATHERINE: I'm Katherine (ph). I'm 10 years old.

HERSHIPS: OK. Here we go.

MALONE: Sally put a box on the table and opened it up for the girls.

ESME: (Yelling unintelligibly).

KATHERINE: Oh, my God. It's got candy on it.

ESME: And marshmallows.

HERSHIPS: It was a cake.

MALONE: It was an intentionally elaborate cake. There was, like, a section with a pile of marshmallows, another section with a pile of gummy bears, another section with a pile of Skittles and then, like, a pastry in the middle.

KATHERINE: Oh, I know what that - it's a cannoli. It's a cannoli.

ESME: Yeah.

HERSHIPS: Wait - no touching yet. We have to figure out how to fairly divide the cake.

MALONE: This is an actual academic problem. There is a whole genre of literature dedicated to the cake-cutting problem.

HERSHIPS: This is a surprisingly complicated problem. The cake has all these different parts, and the girls have really different tastes.

How do you both feel about the marshmallows?

KATHERINE: I love marshmallows.

ESME: I love marshmallows.

KATHERINE: I don't really like Skittles, though.

HERSHIPS: You don't...

ESME: I like...

KATHERINE: But they do make different flavors if you combine them.

KATHERINE: Luckily, many mathematicians have thought about this for years, and there is a solution. It's called divide and choose.

MALONE: Or if you're a kid, it's called - I cut, you pick.

ESME: If one person cuts it and the other person decides who - what gets what, then it could be equal.

KATHERINE: I think that works, too, because I'm going to want to cut them equally so that then at least we - at least I don't get a tiny piece and she gets a big piece.

MALONE: Hello, and welcome to PLANET MONEY. I'm Kenny Malone.

HERSHIPS: And I'm Sally Herships. And today on the show, we walk you through some of our favorite fair division problems and their solutions.

MALONE: We will tell you how to fairly split the rent between roommates...

HERSHIPS: ...How to pick a movie when you're in a group of people with strong opinions.

MALONE: And we go to California, where one of the most complicated fair division problems we've ever seen exists. It is a mess.


MALONE: OK. Problems of fair division - you see examples of these all the time. Take your morning commute. Maybe your highway has an express lane.

HERSHIPS: Yeah. How do you decide which drivers get to use it? Should you only allow car poolers, or should people be allowed to pay to bypass traffic?

MALONE: Or even think about a swimming pool. Should there be a special adult swim? Should some lanes be dedicated just for swimming laps?

HERSHIPS: Yeah, but then it's just this one guy. It's always this guy in the little...

MALONE: (Laughter).

HERSHIPS: ...Rubber bathing cap and those, like, blue plastic goggles.

MALONE: He's ruining it for everybody.


MALONE: Yeah - well, yeah. Anyway, these are all resources that someone needs to cut and divide. And the whole reason we got interested in fair division is because of this mess we heard about in Santa Barbara, Calif.

HERSHIPS: So I headed there to check it out.

Is this where I find Mick Kronman?


HERSHIPS: Mick Kronman.


HERSHIPS: ...Harbor operations.

I'm at Santa Barbara's public harbor. Mick Kronman helps run things here. He's the harbor master. Mick...

MICK KRONMAN: Welcome to Santa Barbara Harbor.

HERSHIPS: Hi. Nice to meet you.

KRONMAN: Come on in.

HERSHIPS: Thank you. Oh, my God. It's so beautiful.

KRONMAN: Welcome.

HERSHIPS: No one told me California was so pretty. (Laughter) I had no idea.

MALONE: Spoiler, Sally - it's beautiful.

HERSHIPS: I know. I'm such a lame ass.


HERSHIPS: Look out of the water here - on one side, mountains and palm trees; on the other, boats - sailboats, commercial fishing boats, yachts.

KRONMAN: So this big white boat over here, Sally, it's a catamaran.

HERSHIPS: That looks super fancy.

KRONMAN: It is super big.

HERSHIPS: Mick says that boat belongs to Warren Buffet's business partner Charlie Munger. He says David Crosby used to keep a boat here. Kevin Costner and Kathy Ireland's husband still do.

MALONE: And here is the problem Santa Barbara has run into. Over the last 30 years, it has become a destination for the rich and famous. And rich and famous people love really big boats.


MALONE: And so the harbor has run into this classic dilemma. It needs to parcel out these limited spaces at the public harbor so that everyone, from a fisherman to a billionaire, could have a shot at getting one of these spots. Very quickly, demand for places to keep a boat got out of hand.

HERSHIPS: The waiting list for a boat slip, which is a parking space for a boat, was so long - get ready for it - you could be waiting 200 years for a spot.

MALONE: However, there was this one loophole; you could buy somebody else's boat. And as part of that transaction, they could transfer the rights to their, like, boat parking spot to you. And so this market sprang up. You overpay for somebody's boat, you get their spot in the dock, and you don't have to worry about that giant waiting list. It's like buying somebody's crappy car to get their parking spot.

HERSHIPS: And as soon as you have something increasing in value and a way to trade it, you get speculators.

KRONMAN: Because back in the day - back in 1999 and 2000, a lot of people, they were buying boats and getting slip permits as investments.

HERSHIPS: Except, of course, the city owns all those dock spots - so investors weren't buying real estate; they were just paying for the right to park their boats there and then hoping they could sell those rights to make a profit. Prices got so high, people were paying as much as a hundred grand.

MALONE: But there was this one rule speculators had to abide by. You couldn't just get a permit, leave it empty and let its value appreciate. You had to have a boat there - any boat.

HERSHIPS: Were people just buying, like, some sort of crappy old boat and parking it?

KRONMAN: Yep. Yep, to sit on the investment. Oh, absolutely.

HERSHIPS: Could you describe it for me?

KRONMAN: A stick, basically - a wooden stick that was 60 feet long.

HERSHIPS: What, a stick? No.

KRONMAN: Basically a log with some, you know - I mean, I'm exaggerating but not by much. Just a - yeah, OK - a log marries a canoe, and their offspring is what I'm talking about (laughter).

MALONE: It was clear to Mick that his harbor's way of divvying up these spots was not working.

KRONMAN: It gets completely out of whack.

HERSHIPS: This whole mess at the harbor is an example a fair division at its very worst.

MALONE: And this was how we got fascinated with the world of fair division problems. We figured - certainly, there must be someone who studies all the different ways harbors or cities or cake-loving people can split stuff up fairly. And of course, there was.


HERSHIPS: Yeah, yeah.


DASKALAKIS: Yeah. So hi. I'm Constantinos Daskalakis, and I'm a professor of computer science at MIT.

MALONE: What do your colleagues call you?

DASKALAKIS: Currently, Costis.

MALONE: Costis is an expert in game theory. And he says, of course, not every fair division problem is as simple as dividing a cake.

HERSHIPS: Yeah. Like, what about things you can't cut? What happens when money gets involved? Or what about when there are more than two people trying to share something?

MALONE: So Costis is going to walk us through a range of fair division problems. And we're going to start with one of the most practical examples, something called the rental harmony problem.

HERSHIPS: And man, I wish I knew about this one in my 20s. This is a solution for a perplexing issue many of us face sometimes in our lives - how to fairly split the rent between roommates.

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.


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.


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.


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?

HERSHIPS: Everyone should just live with mathematicians. 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 discussion.

MALONE: And this gets us to our next tier of classic fair division problems. Let's say you're with a friend. It's Friday night. You cannot agree what you should do.

HERSHIPS: Kenny, I know what you're going to say. You want to bake a cake.

MALONE: I wish I could bake a cake. What I actually would like to do is watch "The Great British Bake Off." Have you seen that show?

HERSHIPS: Oh, my God, yes - every single season multiple times.


HERSHIPS: I'm embarrassed to say how many.

MALONE: So I'm, like, four years behind on this?

HERSHIPS: Kind of. And sorry, Kenny, I really seriously do not want to watch that again. What I would like to watch, since you asked, is this awesome new nature series from David Attenborough and the BBC. It is called "Dynasties" - or denasties (ph) if you re British - and it's awesome.

MALONE: I don't care how it's pronounced. I want to watch baking, and here we are at an impasse.

HERSHIPS: Luckily, fair division has a solution for this - random assignment, aka coin flipping.

MALONE: Now of course, we all know about coin flipping. But Costis says, think about what this is actually doing. You pull out a coin when you're dealing with something that cannot be divided, like what movie to watch or what TV show to watch.

HERSHIPS: Yeah. And flipping a coin gives two people the same chance of getting to choose - thus fair, random assignment.

MALONE: And if you have more than two people, you can use a similar protocol, but it has a way cooler name.

HERSHIPS: Way cooler.

DASKALAKIS: You know, sometimes this has the eerie name of random dictatorship.

HERSHIPS: Random dictatorship - oh, I like that.

DASKALAKIS: Like, 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 five-face, you know, dice. OK? Yeah. And you know, whoever is elected will decide the whole plan - will decide what restaurant to go to, what movie to go to, what bar to go to. So that's called a - it has a name. It sounds bad - random dictatorship. But you know, ultimately...

HERSHIPS: Sounds good if you win - sounds good if you win the coin toss (laughter).

MALONE: Costis says the random dictatorship is more common than you might think. In fact, it is a key part of our judicial system. Take the jury.

DASKALAKIS: The jury is random. So you have...


DASKALAKIS: ...You know, random guys who will make a decision. So that's a...


DASKALAKIS: ...Random dictatorship kind of situation where you...

MALONE: It is.


DASKALAKIS: ...Randomly pick a subset of, you know, U.S. citizens, and you ask them to make a decision.

HERSHIPS: How would we apply this to the boats?

DASKALAKIS: Right, yeah. Let's go back to the boats. I guess this was a small interlude to say that different people have different perspectives of what is fair in different situations.

HERSHIPS: After the break, we go back to the boats.


MALONE: Our fair division expert, Costis, says it is not his job to define fair. That's a question for the philosophers or, in the case of the public harbor, the City of Santa Barbara.

HERSHIPS: But he says let's go through some of the systems the harbor could put in place. There could be a lottery.

MALONE: As in a full reset of the harbor - cancel everybody's existing permits, then reallocate those permits randomly.

DASKALAKIS: There's nothing bad with the lottery. It's going to be inconvenience for people. But ultimately, that is super fair. It treats everybody equally. OK?

MALONE: That makes sense.

HERSHIPS: Yeah, that seems fair.

MALONE: And that's great if that is how the city wants to define fair. Of course, there would be a lot of very angry boat owners who got their permits canceled.

HERSHIPS: Another solution Costis mentioned - timesharing. When you have limited resources, one way you can allow more people to use them is by letting people take turns. But with boats, that is not so practical - because where exactly are boaters supposed to put their boats when it's not their turn?

MALONE: And then of course, there is the money version of this. You use money as a proxy for who values these slips the most. If you want a permit badly enough - and you can afford it - you can buy one. As for the voters who can't afford a spot at the dock, well, the system is supposed to benefit them, too.

DASKALAKIS: You know, it depends on your definition, I guess. So like, if I - if my city can extract, you know, a million dollars from the billionaire - OK? - and build a school next to the dock - OK? - or, you know, build a kindergarten or whatever next to the dock so that people who then go to cruise can, you know, leave their kids play in the kindergarten. Like, you see what I'm saying? You know, like, fairness is very complex to define mathematically.


HERSHIPS: Back at the Santa Barbara Harbor, Mick Kronman, the harbor master, said when he started his job, he was already stuck. The investors had already begun sinking money into their spots. And they were counting on getting it back again with a profit. Mick says now, years later, if the harbor changed that system, they would sue.

MALONE: And if the harbor made selling slip permits illegal, it wouldn't fix the problem either. It would just create a whole other kind of problem.

KRONMAN: Because this whole scene would go underground - in other words, the underground economy has so many tens of millions, if not hundreds of millions, of dollars invested in it, it's not going away. There's no way it's going away. Whether you do away with slip permit transfer fees or not - or slip permit transfers - people will take out life insurance policies on each other. They'll set up phony corporations. And functionally, the whole thing will continue the way it does now without any oversight from municipal control.

HERSHIPS: So if you opened the waiting list today, how many names do you think would go on it?

KRONMAN: Thousands.

MALONE: Thousands, Mick said.

HERSHIPS: Mick decided, if this multimillion dollar economy for public dock space was going to exist, then the public should find a way to get its fair share.

MALONE: So Mick started to make it more and more expensive to sell your dock permit. It used to cost 25 bucks to transfer your spot. Now they can go for as high as $500 per foot.

HERSHIPS: So if you own a 60-foot boat, we are talking 30 grand.

MALONE: Now, this is not a solution that all boat owners are jazzed about. For sure, it creates a situation where you need money to get a boat slip. That is less fair for the Santa Barbarans who have boats but not as much money. Arguably, though, it is fairer for all Santa Barbarans.

KRONMAN: The transfer fee goes into Waterfront Department coffers, if you will - into our operating budget. And we use that to maintain the harbor.

HERSHIPS: For the public.

KRONMAN: For the public - to accommodate boating and make sure the facilities are kept up. Right? We're walking on a wooden dock, all the planks of which have been replaced over the years. You can see it's pressure treated. You can even - you know, you can see where - you know, it's a high-quality dock. They aren't full of splinters and what have you - though people shouldn't be walking on wooden docks in their bare feet.

HERSHIPS: Mick says this money also helps pay for fire boats, for law enforcement, for first responders. The Harbor Patrol gets a lot of medical calls if someone has a heart attack or gets sick on the water.

MALONE: Mick says the other thing that these massive transfer fees do is make it less appealing to invest in a dock slip because when you sell your permit, this giant chunk of money is going to cut into your profits.

HERSHIPS: And sure enough, when Mick and I strolled through the harbor, the rusty canoes, they were almost all gone. There were just a couple of exceptions that looked like they were floating placeholders protecting an investment, a spot at the Santa Barbara dock.

Sorry. You said here's an example. Let's describe what we're looking at here. That is...

KRONMAN: It's a mold-ridden, decrepit, floating thing that looks not used very much. It may be operable, you know?

HERSHIPS: There is a kayak tied to it.

KRONMAN: Yeah. Yeah, it may be used for, you know, the kayaking, nothing else. But the boat itself probably isn't worth, I'm guessing, more than $500, $750.

MALONE: But again, those investment boats are rare. And Mick says this solution seems to be working as best as possible. Most people are actually using the harbor for commercial fishing or recreational boating. And the public harbor is now getting a pretty nice cut of the millions of dollars sloshing around its docks.

HERSHIPS: Let me ask you kind of a weird question, which is - do you think that this is - like, in a perfect world - in a perfect harbor...


HERSHIPS: ...Is this the most fair solution? Or is this the solution that had to be adapted given all of the history and the - like, if you were going to start again...

KRONMAN: Right. Right, right.

HERSHIPS: ...And build a harbor from scratch, would you do things the same way?


HERSHIPS: No, Mick said. If he could start from scratch, he'd take the money out of it. He'd find a different way to define fairness. But this is the system he's stuck with. And that's something to keep in mind if you ever find yourself fairly dividing something. Think hard about the system you pick at the very beginning - the way you define fairness - because you may be stuck with that definition for a very, very long time.


MALONE: Is there a fair division problem ruining your life? We definitely want to know about that. You can email us, planetmoney@npr.org.

HERSHIPS: Bryant Urstadt edits PLANET MONEY. Alex Goldmark is supervising producer.

MALONE: Today's episode was produced by Alexi Horowitz-Ghazi. And a note about elaborate cake...

ESME: (Yelling unintelligibly).

KATHERINE: Oh, my God. It's got candy on it.

ESME: And marshmallows.

MALONE: If you would like to see a picture of our Skittles-gummy bear-marshmallow-cannoli fair division cake, we're going to post that on Instagram. We are @planetmoney. I'm Kenny Malone.

HERSHIPS: And I'm Sally Herships.

MALONE: Thanks for listening.

HERSHIPS: Oh - and do you guys have any advice for grown-ups when they're trying to share things and they're having a hard time figuring out how to do it fairly?

KATHERINE: Just leave it to us. We'll eat the cake for them.

HERSHIPS: OK. That's pretty nice of you.

KATHERINE: (Laughter).

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