 I'd like to bring up now Andy Niedermeyer. He's a friend that I met a number of years ago. He works at Jane Street Capital. And it also happens he's a friend of tonight's speaker. So I'm delighted that he's come to introduce the talk tonight, and so I will turn it over to Andy. Thank you. Hello. Hello. Thank you, Cindy, and thank you for having me. I'll try and keep this short so we can get to the cake. I think part of the reason why I'm here is because I've known Francis for almost 20 years. I was an undergraduate at Harvey Mudd. Francis was one of my professors. I did summer research with him way back in 2003, I believe. But back when I was looking at colleges back in the late 90s, back in the last century, there wasn't that many resources available for which college's math department was special in which ways and why I should choose one place over another. And those of you who remember what the internet was like in the 90s, there just wasn't a lot of information out there. But somebody had taken the trouble to build a website in his spare time detailing hundreds and hundreds of math fun facts. And that was Francis Suh, tonight's speaker. Starting as a graduate student, he had started presenting cool fun facts to his classes that he was TAing at Harvard as a way to kind of motivate mathematics simply to put it above and beyond simply, I have to do the thing that the professor put on the blackboard and I have to memorize this problem or I have to memorize this integral. And this is something that I basically copied when I was a grad student many, many years later, but when I was a teenager and I was looking at which college should I go to, this thing just popped right out at me and was like, wow, somebody's taken the time at this math department to kind of try and exhibit how awesome math is in these literally hundreds of different ways. And I think it's only appropriate that I get to introduce Francis at what I would call the living embodiment of the math fun facts web page where you're surrounded physically by many, many exhibits that detail just how awesome math is as opposed to just something you learn from a textbook. So I'm only gonna like take another few moments to detail some fun facts about Francis. He's a hymo winner from the MAA, that is the MAA's top teaching prize that only goes out to a few professors nationwide each year. He recently served as the president of the MAA for which he got to travel all across the world to different schools, undergraduate, grad schools. He got to go to conferences. He met literally hundreds if not thousands of students and professors all across the country. And finally, most importantly, he's a world-class pastry chef as we are about to find out. So I'll hand it over to Francis. Thank you very much. I'm not gonna take credit for the pastries, but we are gonna enjoy the pastries and I'm really excited to be talking tonight about how to cut cakes fairly. Now, how many people are like cake? You like cake? Awesome, good, good, cause that's the problem we're gonna solve tonight is how to divide cakes among people who actually like cake. One of the great things about this topic though is that it brings together so many different techniques from mathematics. And I know some of you don't even think of mathematics when you think about cutting cakes or questions that involve how people interact. But one of the ways that we encounter math, and this is a math encounter, is in the modeling of people's preferences. And one of the things that's kind of exciting in the last century is the fact that we are now using mathematics to model how people behave. If you think about the history of mathematics and the kinds of ways that people encountered math before, much of the mathematics of the 18th century and before, the 19th century and before was mathematics motivated by the physical sciences, planetary motion, things like that. But in the 20th century, we saw for the first time mathematics being modeled used to model the social sciences. Game theory, for instance, was born after the World War II. And so there are a couple takeaways that I want to highlight right now in this talk. And one is that math models are actually really useful in the social sciences. And the second takeaway is that the social sciences actually motivate interesting new mathematics. So those are the two takeaways that I hope you'll take from this talk. And what I want to talk about tonight are methods for dividing cakes or other things fairly among several people. And that'll happen in three parts. I'm gonna talk about three different kinds of classes of procedures for dividing things fairly. And I know you don't know what these words mean yet, but we will get there. Okay, so when we think about cake cutting, the classic cake cutting question, you often think about what happens between two people. So what is a two-person cake cutting procedure that you know? Anybody? Yes? Cut it in half. Cut it in half? Okay, one person cuts in potentially try to do it in half. And then what does another person do? Yes? Okay, okay. So the other person's checking if the other person's cutting it fairly, and what would you hope the second person does after they check that the pieces look good? Yes? I would have to thank you for seeing that the person who had been cutting correctly and both people would speed it. Yeah, you hope both people would eat it. Yes, thank you very much. Thanks for that answer. Anybody else know a procedure for cutting cake fairly among two people? Yes? Okay, good. So you're talking about a procedure that's really well-known. It's called the I Cut You Choose procedure. So it's a well-known procedure for cutting cake among two people. And it actually dates back in literature. We see it as far back as 800 BC in Hesiod's Theogony, for instance. There's an example where Prometheus cuts an ox into two pieces and offers Zeus a choice. Okay? Also in the Old Testament book of Genesis, Abraham offers lot a choice between the land on the left and the land on the right. Okay? So you see these kinds of procedures have been around a while. And so they're interesting from a historical perspective, but they're also interesting because we are really interested in dividing things today. For instance, what are some things that you would like to divide today between you and several people? Somebody named some examples? We see cake, but what else? My parents' estate. An estate? Good. What other things would you like to divide? Chocolate chip cookies. Chocolate chip cookies, very, very important. What other things do you think our people are interested in dividing? Yes? Halloween candy. Halloween candy? Good. So those are indivisible objects and that's a very different kind of division problem than cake. Yes? What else? I had a school or classes for the teachers to teach us. Classes for teachers to teach. Right, good. So this is a problem of task division, right? And sometimes that's related to what's called the problem of chore division where you're trying to divide a bunch of tasks and maybe you don't want the largest piece but you want the smallest piece, right? That's the problem of chore division. Yes? Attention. Attention. Dividing your attention. Oh wow, yes. Yes, and we want undivided attention. Is that right? Yes. Good, okay, lots of examples where you think division is important. And what I want to say right off the bat is one of the things that makes these kinds of division problems, they're called fair division problems, interesting is that people have different preferences. You see, if people didn't have different preferences this would be a very boring problem, right? Let's say I had four people and I suppose all four of us had the same preferences. Well, I could just take my cake, divide it into exact quarters in my opinion, but that's everybody's opinion. So then everybody would just take whatever piece because everybody agrees every piece is the same size. So it's not interesting if you have the same preferences. What's also interesting about this is when you have different preferences that means you can actually give people potentially more of what they want besides what you might think is a fair share. So an example is characterized here in this nursery rhyme by Mother Goose. It says Jack Spratt could eat no fat, his wife could eat no lean. So Twix them both, they cleared the cloth and licked the platter clean. So what do you see here? You see two people with different preferences for different parts of the meat and they're both able to get 100% of what they want because they have completely different preferences. Okay, so that's another interesting aspect of these fair division problems where people have different preferences is you actually might be able to give people more than what you expect. Okay, so there are lots of other examples of things that you might wanna divide. We've listed some already. I just wanna point out here this is an article from the New York Times that talks about how some of these division problems are actually pretty complicated. So what it says here, I'll read to you, it says, for birthday parties or legal parties, defining things fairly is not always a piece of cake. And you'll see that once we move from two people to more than two people, it's actually way more interesting and more complicated. And one of the ways that I got interested in this problem was the problem of rent division. So my friend Brad contacted me in graduate school and he said, you know me and my three friends just moved into a house. And the house has many different rooms with different features and one of the rooms is really large but it's really noisy because it's next to the street. Another one's really small but it's next to the kitchen so it's convenient. And he said, is there always a way to split the rent among the rooms so that we would each choose a different room? There's a question, what do you think the answer is? Is there always such a way? Raise your hand if you think the answer is yes. What if the answer is no? Okay and how many of you will think the answer depends on the hypotheses, right? Yes, okay good. So there's a question brewing in your mind and we're gonna return to this problem later. So the problem of fair division was probably first mentioned in an academic context in a paper in econometrica which is one of the leading economics journals by Hugo Steinhaus who was a mathematician in 1948. In fact, the paper's titled The Problem of Fair Division and the prototypical question is how do you divide a cake fairly? Now one of the great things about mathematics is when you have a question that seems as nebulous as this, mathematics actually helps us make a nebulous question much more precise, okay? Because we're forced to actually think about our assumptions, we're forced to make definitions. And so what I wanna explain now is exactly how math makes this question precise, how to cut a cake fairly. There's several words here that could be made more precise. For instance, cake, we've already discussed many of the different kinds of things you might wanna divide from Halloween candy which is indivisible to cake which is divisible. And so one of the things math does for us is it asks us to classify things, right? You have all these different problems and you say, okay, what are the salient features here that make these kinds of problems different? And so you're forced to look at properties of things. So for instance, you might look at the cake division problem differently if you have a divisible object versus an indivisible object. It might be a different problem if the object you're trying to divide is cake which is desirable, you want more cake versus chores which you don't want more, you want to get a smaller piece as possible, okay? And one of the ways we're forced to be precise is to wrestle with what does it mean for a person to submit their preferences? So if I am an arbiter and I'm trying to decide to make some decision on behalf of some group of people who are having some dispute, I need to think a little bit about what kind of information I'm gonna elicit from them. Am I going to ask them to submit a whole measure function which is basically a measure for every possible subset of cake? Probably not. But that's one way mathematicians think about this. If you're an economist, you might think about utility functions which is also a very similar concept, okay? But you might also just say, well, look, I'm just gonna present somebody with an option, here's a set, here's a cake divided, which piece do you want? And that's in the form of answering questions, okay? So these are all different ways you might elicit preferences and math helps us make choices about how we're gonna model such a situation. Okay, that's cake. What about cut? What do we mean by cutting a cake? Well, there's lots of, what do we mean by cutting a cake? Yes. Yeah, we're gonna take a knife, that's right, great. We're gonna take a knife and we're going to place the knife over the cake in certain ways. And it might vary depending on the kind of problem it is. So for instance, if it's cake, I might wanna cut the cake by parallel knives, right? If it is a Halloween candy, I might want to, I'm not gonna use a knife, I'm just gonna put the pieces in little piles, right? That's a division. If it is pie, it might actually be geometrically cutting radially from a point. And that actually turns out a different kind of problem than the classical cake cutting problem, okay? Then there's the word how. How to cut a cake fairly. Well, for a mathematician, the question is, does a solution exist? That's one kind of question you might ask. And then the other question is, if it exists, can we find it? Am I right? Everybody with me? Okay, those are mathematical questions, interesting questions. And then finally the word fair. What do we mean by fair? Let's see, throw out some answers here, some people. What do you mean, what do you think we could mean, possibly mean by fair? Yes, you. Everyone accepts it. Everybody accepts the solution? Yeah, they're, all the parties are happy with the solution. All parties are happy with the solution. They're happy with the piece they got. Okay, what do we mean by happy? There's a question. What do you mean by happy? What do we mean by happy? Happy. You like what you're eating? Yes, happy is you like what you're eating? Okay, let's be a little more refined. That was a great answer. What could we mean by happy? Yes, in the back. What fair it is, it's not exactly fair. One notion of fairness is you agree with the rules, even though you're not necessarily happy with the result. What about over here? Ah, you wouldn't refer anyone else's piece to the one you have. Okay, good. That's a notion. Now, let me just hear a couple more. Yes, you. A hungry person gets a larger piece and a full person gets a smaller piece. Oh, interesting. That's an interesting situation. Yeah, so maybe you're in a dividing something where the larger the piece gets, at some point you actually don't want more, okay? And when we're talking about, and that's actually the problem of mixtures where the cake, certain subsets of cake are actually negative and some are positive, right? You get too much, maybe it's negative. We're not gonna consider that when we talk about cake cutting, but it's definitely a problem that people study. Okay, lots of different notions of fairness. I'm just gonna point out a few that people have pointed out for special study. One is a notion of proportional division, okay? So to be proportionally fair means if I'm dividing a cake among four people, each person is happy if they get what they believe is one quarter of the cake. Are you with me? One over N if it's N people. That's not as strong as the notion that was mentioned over here of not envying anybody else. That's a notion of envy-free division where each person feels that her piece is the biggest piece. Are you with me? Notice that that's not, that's stronger than each person feels they got at least one quarter because I might feel I got a quarter, but still envy the person over there. Are you with me? So those are two different notions of fairness. There's lots of others. I'll just mention a couple more. There's a notion of equitable division. An equitable division is one which each person feels they got the exact same amount in their opinion as the other people feel they got in their opinion. Okay? So if you think about Jack Spratt and his wife, they each felt they got 100%. That's an equitable division, okay? It'd be inequitable if one felt they got 70% in their measure and 50% in their other measure, okay? Yes? Okay, so this, excuse me, in the cake that needs some picture. Yes, so the concern here is that because the cake is not homogeneous, it's got lots of different kinds of things, could you be in a situation where people are just not happy with anything that's possible that's presented to them? And that actually relates to the other notion, the last notion on this slide, which is the notion of efficiency. So an efficient division, sometimes called Pareto-Ofnival division, is one that isn't dominated by any other division. To be dominated means every, it's not dominated if there's no other division where everybody is better off, at least is good off and one person is better off, okay? And we're gonna be considering situations where if I present you a cake, I might say, hey, in this division, which piece do you prefer? You have to be able to answer one of those pieces. If you don't have that assumption, then that's not gonna be part of the model that we're considering. Now it could be the case, however, that by, for instance, switching some of the goodies on top of the cake, that I would produce a division that's actually dominating the one I have, and that's the notion of efficiency, okay? Yes, so great questions. Lots of different notions, solution concepts, as they're called, for a cake division, yes? Sorry, what about fair and random, like a lottery and a lottery? Excellent question. Are there notions of fairness related to randomness? And the answer is yes, I just did not put them on the slide, but it is something that people do consider, random, random fair, right? This is what you do when you roll a die and you say it's fair because everybody had an equal shot. Yeah, great question. Okay, so we've made some of these notions a little more precise in this question, how to cut a cake fairly. I just wanna point out there's, I was amused to see in an episode of numbers that cake division methods were mentioned in solving, I think they were trying to solve some kind of kidnapping case here. And they were trying to look at the ransom amount and how the ransom amount was divided in order to infer something. I don't know, they kind of made this stuff up, but it's clear they hired a math consultant because you'll see words like envy, free-ness and equitability here on these slides. Okay, so why, let's come back to our first example. Why is cut and choose fair? Why is it fair? I've just defined fair, so why is cut and choose fair? Before we can answer that question, I have to point out that when you give a fair division procedure, you have to specify what the rules are, but there's also the strategies that people have in their heads that they're playing. So for instance, a rule is something that an outside person can observe that you're following the rules. A strategy on the other hand is something that's going on in your head. And I'm gonna always try to notate those in green on these slides. So for instance, I cut is a rule, that's the rule, the first person cuts, yes? But the strategy is, I'm thinking in my head, I'm gonna cut it, I'm incentivized to cut it so that both pieces are equal to me. Why? Because I know the second rule is that you're gonna choose. And if I know that, then I'm incentivized to cut this so that both pieces are exactly the same size. Are you with me? Okay. Now, what is the second person going to do? They're gonna choose the piece they think is the best. Great. So now we have the rules and the strategies and why is this fair? Why is this fair? Yes? I left out an assumption that both people like cake. Yes, good question. I left out an assumption that both people like cake and usually when we talk about cake division problems, we're assuming that people actually like cake. Okay, yes, yes. I implicitly, but thank you for pointing that out, yes? If you have a situation where two people have the exact same preferences and there is literally no way of dividing something evenly and isn't it unfair to the person who's doing the cut? For example, there's three bags of M&Ms, you cannot provide a bag. If I have to, I can only do either three and zero or two and one. Excellent question. Also implicit in the notion of cake division is infinite divisibility, okay? So when I mentioned, there's all these different properties that lead us down different paths in this problem. When we talk about cake, we're focusing on objects that can be divided as finely as you like, okay? Great question. But it actually begs the question how you deal with individual objects and we're gonna get to that later in the talk. Yes, one more question here. So I was gonna answer your question, why is it fair? So at worst, player one can divide it to two pieces. That player one is indifferent which piece. So whichever piece they would consider fair, proportionally based on your definition of that. And then player two can do even better because they may have a preference of one over the other. Okay. And then they get to choose what they consider the best piece. But at worst, they're gonna get a piece that aren't different. Great, so what our guest here said is that since the first person's dividing it in what he feels is 50-50, no matter what the second person chooses, the first person's gonna be happy. And the second person's happy because they get their choice of piece. So what kind of happy are we talking about here? What kind of fairness? Which properties does this satisfy? Is it proportional? Yes, because each person feels they got more at least a half, correct? One over two? Is it envy free? Yes, because if you feel you got more than a half, you know the other person in your measure got less, okay? So envy free and proportionality are the same for two people, okay? But not beyond that. Is this division necessarily equitable? Would each person feel they got the same percentage in their measure as the other person in their measure? Not necessarily because the first person, you know it thinks it's a 50-50 split, but the second person has that choice of picking whichever one they want. They probably think one of those pieces is bigger, okay? So this does not have, doesn't satisfy equitable division. And if you ask me in the Q&A later, we can talk about equitable division for two people. But you might think about that question if you're bored at any point during this talk, think about a two-person equitable procedure. Okay, what about more than two people? Well, we're gonna talk about two different, these two notions of proportional and envy-free divisions. Let's start with proportional. Lots of methods for proportional division. I've just listed some of these here. You can see the dates on them. Only gonna talk about one of them. And that's the Dubin Spaniard moving knife procedure. I'll tell you what it is. Then we're actually gonna try it out. You're gonna enjoy some cake. Yes, cake, cake. I heard the gasp. Yay, awesome. All right, so what is the moving knife procedure? Well, the new thing knife procedure says, take a cake like this, and I'm gonna place a knife on the left edge of the cake. Yeah, and I'm gonna slowly move the knife to the right. Are you with me? And as I move the knife to the right, I'm gonna have everybody involved in the division. Let's say, tell me your name. Margaret. Laura. Laura. Adam. And? Yeah, what's your name? Jonathan. Okay, Margaret, Laura, Adam, and Jonathan are dividing this cake. So I'm the referee. So I'm playing the person who, I'm just there to hold the cake, the knife. I'm gonna move the knife over the cake like this. And as I move the knife over the cake, here's a knife. As I move the knife over the cake, I'm going to ask you guys to watch, pay attention, and as soon as the cake to the left is more than a quarter in your estimate, you're gonna call cut. Yes, you're thinking for yourself. That's correct. That's right. So you're going to call, you're going to call cut when, when the piece to the left is a quarter in your estimate. And then when you call cut, you get that piece. Okay? It has to be, as I move the knife over, there is some cake to the left, which right now is very small. But it's getting bigger and bigger in your estimation if you like cake, yes? And at some point it's gonna cross a quarter for one of you. And when that happens, you should call cut. Because if you wait any longer, someone else might call cut, and then you'll be in trouble, okay? So if you, so here's the strategy. The rule is any player can call cut at any time, but the strategy is when you think the piece to the left is exactly one over four, in this case with four people, you should call cut and then you get that piece, and then you leave the game. Are you with me? Okay, and then we'll do it again. So let's just, let's just do this with you four people. Okay? No, no, no, you guys are all going at the same time. Are you with me on that? Okay, so you call cut when you think this is at least a quarter, and now I'm gonna just slowly move it over. Okay, ready? Okay. Okay, who was first? It was you, right? It was you, it was Laura, yes? Okay, so we call cut and Laura gets this. Now, here's a question for you. Because Laura called cut, I know she thinks this is a quarter, yes? What do I know about everybody else? They think it's less than a quarter. So they think the piece to the right is bigger than what? Than three quarters. Okay, good, I cut the cake, Laura gets it, and now we have what remains. And now you can do the same thing, but with three people, okay? And one thing to keep in mind is when you do the three person procedure, that is the three remaining of you, one of you calls cut when you think what remains is, what's, the knife is sweeped out a third of what remains. What I wanna point out is that that's actually good because if you take a third of what you think is more than three quarters, that is more than a quarter, right? A third times three quarters works out to a quarter. So this is why this division method is proportionally fair because the player who calls cut thinks they got a one over N, and everybody else thinks what's left is bigger than N minus one over N. Now if you multiply that, if you do the remaining procedure on that, you'll get at least one over N minus one of what remains in their opinion. Those of you who know what induction is, we're basically inducting on a smaller number of players. Okay, so here's what we're gonna do. You guys are gonna do this for yourself with four people and I'm gonna pass out some cake. You're gonna enjoy some cake after you've done this division, okay? Proportional division. So what you need to do is form groups of four, meet somebody if you don't know them yet, okay? And then once you have a group of four, send one person up to get the cake and the plates. All right? No, it's for groups of four. You gotta start here, you gotta start here. Yeah, you gotta get plates and quarks. There's some pieces over there, just grab. Feel it, grab. All right, one person plays the referee and they're also the player as well, okay? And remember you're gonna move the knife over the cake until one person thinks it's a quarter of the cake. And then somebody should call cut when they think it's a quarter. Once you do the, once one person leaves, remember it's a three person problem, right? So then it's a third of what remains and then once you get down to two people, it's a half of what remains and you could just use cut and shoes at that point, right? Okay. How did it go for you guys? Did you get a fair piece, a proportional piece? Okay, yeah, so there's some issues here you might have noticed. For instance, one thing that can go wrong is you're not paying attention and the knife kind of moves too fast and suddenly you are, you didn't call cut fast enough and then what remains is really small to you, right? So that's one thing. And another thing problem is some of you might be trying to game the system. Like you're trying to like maybe see how much you can get away with before calling cut. And so one of the features of fair division algorithms that's really kind of neat is if you try to game the system you might actually end up not getting a good piece. So you have an incentive to play the game fairly, which is great. It's incentive compatible in that sense. The other thing that's great about this is if you tell the truth that is if you represent yourself fairly, accurately, then you're guaranteed to get a piece that you think is fair according to that notion. Okay, so this is one of just many different kinds of proportional procedures. There are a lot of other procedures and they each have different features, right? This one's nice because it's easy to describe for lots of people. You just sort of induct on one, take care of one person and they leave. You take care of another and they leave. Et cetera. One thing you'll notice is this is not an envy-free-free procedure, is it? Why? Because the person who leaves at the beginning might envy one of the later pieces that get cut. Good, so let's talk a little bit about envy-free methods. What can we do to get an envy-free division among just three people? Well, it turns out it's a much harder, much more interesting problem. I'll mention one method called the Selfridge-Conway method. There's a moving knife procedure as well but it's really, really complicated so I won't talk about that here. This one is actually less complicated but you'll see it's got some interesting aspects. So I'm gonna take a cake and I'm gonna try to divide it now in an envy-free way and here's how the Selfridge-Conway method goes. It says, let's start off. Let's suppose we have three players, Alejandro, Betty, and Carlos, okay? And the three of them are trying to divide a cake and what we're gonna do is I'm gonna say Alejandro, you're the first. Cut the cake into what you think are thirds. Are you with me? So Alejandro makes some division like this, yes? Okay, now the rest of us are looking at this and saying, well, these don't all look the same to me but they look the same to Alejandro, right? Because maybe Alejandro liked the cherries that were on this side, et cetera, right? Okay, so the rest of you look at this but in particular, Betty looks at this and says, these aren't the same size. So I'm gonna say, hey, Betty, tell me which two pieces you think are the biggest. And Betty says, oh, these look the biggest to me. Are you with me? Okay, so I'm gonna tell Betty, hey, Betty, can you cut the largest piece and trim it so that the two largest pieces are now the same size and we'll set aside the trimming. So that's what I'm gonna ask Betty to do. She's gonna trim and I'm gonna say, do it so that I'll suggest to her to make a two-way tie for largest, we'll set aside the trimming. So here's what she says, I'm gonna trim off this much and now these appear the same to Betty. Are you with me? Okay, and the trimmings are, I've just pushed aside, I'll deal with them later. Okay, so let's just make sure we're all on the same page. Let's ask, how does Alejandro view these three pieces? Well, blue and yellow are the same size, yes? But Alejandro thinks the red piece is smaller. Everybody with me? Okay, good, because it was trimmed off. What about Betty? What does Betty think? Well, Betty thinks these two are the what? Same size, and this piece is? Excellent, okay. So, Carlos, who we haven't even asked his opinion yet, now we're gonna say, hey Carlos, you pick any one of these three pieces you want and would you agree that he's gonna be happy with this part, one of these three pieces in this part of the division because he gets to pick first, yes? No matter what Carlos picks, I claim Betty will have something that she's happy with because there was a two-way tie for largest, so if Carlos picked this piece, Betty will pick this one, et cetera, yes? And if Betty picked this, Carlos picked this piece or this piece, Betty will pick this one. I'm gonna demand that Betty picks the trimmed piece if Carlos did it, because I know Betty's happy with the trimmed piece because she trimmed it, yes? You with me? Okay, so if I demand that Betty picks the trimmed piece if Carlos didn't, then would you agree that Alejandro is actually gonna be happy with whatever's left because he's never gonna get this piece, either Carlos or Betty got it, and these two he thought were the same size, yeah? Everybody happy with that? No. No, we didn't pick the trimmed piece. Oh, yes, good question. Okay, would you agree, are you happy that we divided this portion in an envy-free way if we disregard the trimmings? Okay, good. So what should we do with the trimmings? Do the same thing. Same thing, okay, if I did the same thing, then I'm gonna get a smaller piece of trimmings and then I'm gonna have to divide that, I'm gonna get a smaller piece of trimmings, I'm gonna have to divide that and we're gonna keep going, right? Yay, keep going, infinity and beyond. Oh no, wait, so actually what's great here is you know something about the trimmings that you didn't know about the whole cake and that allows you to make this a finite procedure. What do you know about the trimmings? Well, here's what you know. You know that no matter how much of the trimmings Betty gets, Alejandro will never envy her. Why? Because Alejandro thinks together these are the same size as his piece. So if Alejandro, if Betty gets any of this, in fact if she gets all of this, Alejandro will still be happy. Are you with me? So that means in the next division, Alejandro can pick after Betty. Are you with me? Okay, so how do we make Carlos happy? We will just ask Carlos to divide the trimmings into thirds and then have Alejandro pick and then Betty pick and then Carlos gets what's left. So that's what we do. If T here is a person who got the trimmed piece, call that, in our case it was Betty, and N is the other person who was Carlos, we're gonna ask Carlos to divide the cake into what he thinks are thirds. Then Betty chooses first, then Alejandro, and Alejandro is fine with whatever Betty gets, yes? And won't envy Carlos because he got to pick before Carlos. And now Carlos is happy because Carlos thought all three of those were the same size. Doesn't matter what the others get. Whew, all right. It's a little bit complicated, but there are some cool ingredients here. Now, here's one thing I want you to notice about this procedure. It didn't use just two cuts, which is what you would expect for three people to divide into three pieces. So actually you ended up taking pieces apart and pushing some of them together, yes? How many cuts did this procedure take? One, two, three, and possibly two more. Four, five pieces, five cuts, right? Okay, so this is actually a problem when you get to more numbers of people. And the end person procedure, the procedure that works for any number of people actually wasn't discovered until very recently. That was 1995, which I know is still before some of you were born. But this is relatively recently in historical terms, yes? This is an end person procedure. Bram, Steven Bram is actually here at NYU in the politics department. He's a political scientist. Alan Taylor's a mathematician, and they came up with this procedure, which I'm just gonna tell you it's features. It's too complicated to explain. It's an end person procedure, NV-free, but it takes a large number of steps to resolve, and possibly large number of pieces and cuts. It is a finite procedure, meaning it will always stop in a finite number of steps, but it's an unbounded procedure, meaning there's no a priori number you can name that all possible people's preferences would take less than that number of steps to resolve. So if you said a million, I could find you end people with very crazy preferences that might take more than a million steps to resolve, that's the problem with this procedure. It's finite, but it's unbounded. So there's an open question that was open for many years until just last year, and that's the question is, is there a bounded procedure for end person NV-free cake cutting? And believe it or not, last year somebody solved this question, a pair of people solved this question. What do you think the answer is for end people? Is there a bounded procedure? How many people say yes? How many people say no? How many people undecided? Yeah, okay. Well, here's what's known. It's actually, believe it or not, there is a bounded procedure. So they proved last year that there is a procedure that will terminate after a fixed number of steps. Only problem is the bound is really, really large for end the people, it is end of the end of the end of the end of the end of the end, a tower of ends, which if you just plug in the number two, this is still larger than the number of atoms in the universe or something like that. Okay, so it's not a very useful bound, but it does show theoretically that there is a bound. One question you might ask yourself, and this leads into the last portion of the talk, is can you cut cake in a, with a minimal number of cuts? If you have end people, it should, if every piece is connected, you expect end minus one, knife cuts. Is that possible? Stromquist showed that there is a three-person procedure that does just two cuts. That's the moving knife procedure I haven't told you about. And, but he also had a negative result just recently, a few years ago. He showed that in fact, if you, his original procedure, the moving knife procedure is really kind of crazy, and it's not finite in the sense that you always have to be on the edge of your seat, waiting for, waiting to look at, when the cake is a certain size. So is there a finite protocol, is what it's called? That's NV-free, minimal number of pieces, and he showed it's actually provably impossible. Yay! Okay. But, this then points to, well, what can we do? Well, we could look for approximate divisions. Because after all, maybe you don't really care up to crumbs, right? Whether you're NV-free. And maybe you can't even decide up to crumbs. Many of you were trying this in the last procedure and you're like, I can't even decide, you know, a small epsilon. So why not get in approximate NV-free divisions? So this is the last portion of my talk. I was just talking about approximate NV-free divisions. And what's cool here is it brings in a whole set of other techniques from mathematics that are interesting. So here what we're gonna do is we're gonna look at the space of possible divisions of cake as itself some kind of geometric space in a way I'll describe. And then we'll break that space into pieces. And then we'll kind of search our way through the pieces where the pieces have certain labels given by preferences. Okay, I'm just giving you a general feeling of how this goes. And then you find a good solution that way. And what this relates to is something called topological combinatorics. Okay, and they're both big words here. Combinatorics you can think of as clever ways of counting things. Topology you can think of as a study of the connectedness of various shapes. That's one way to say it very loosely. And there's an interesting connection between both topology and combinatorics that's exemplified in something known as the Broward-Fick's point theorem, which is an amazing theorem useful for all sorts of cool math. And Sperner's lemma, which is, it looks like a very simple theorem and I'm about to describe, but it turns out to be equivalent to the Broward-Fick's point theorem. Some of you may know the Broward-Fick's point theorem because it's the theorem that says if I have a cup of coffee and I slosh it around and I take a picture before the sloshing and after the sloshing, some point in the coffee will be in the same position, both before and after the sloshing. That's known as the Broward-Fick's point theorem. And it's useful for proving solutions to differential equations exist. It's useful for proving the Nash equilibrium theorem. John Nash was famous for. Lots of other applications. Okay, so what is Sperner's lemma? It's a simple combinatorial lemma that says something about a triangle that is broken into lots of little triangles, baby triangles, and every vertex is labeled by a number, either one, two, or three. And it's labeled according to these rules. So I'm going to take a corner and label it one, take this corner two, this corner three. They all have to have different labels. And then along the one, two edge, I only allow the labels one or two. Are you with me? So this could either be one or two, one or two, one or two. Along this side, what am I going to allow? Two or three, and along this side. And then on the inside, I allow anything. Okay, so this is what's called a Sperner labeling. And here's an example. Do you see that this is one, two, and three on the big corners, yes? And all along the bottom, it's either one or two, two or three, and one or two. It could be all ones, okay? And then on the inside, you allow one, two's or threes. Are you with me? This is what's called a Sperner labeling. And Sperner's lemma says something amazing happens when you have a Sperner labeling. In particular, there has to be a one, two, three triangle. Do you see one? A baby triangle that's labeled one, two, three. Or on the top. Actually, do you see any others? Yeah, there's another one here. Is there any other one? There's another one right here, yes? Okay, so Sperner's lemma actually says there has to be an odd number of baby one, two, three triangles. Okay? And what's great about this lemma is that an odd number means it can't be zero. So there has to be at least one, okay? Okay, let's see why this is true, and then I'll show you what it has to do with cake cutting. So here's a proof of Sperner's lemma. It goes like this. Think of this as a house with many rooms. This triangular house with lots of little rooms, yes? And every time you see a one, two edge, think of that as a door in the house. So here's a door, here's a door, here's a door, and here's a door, and here's a door. This is not a door, because it's not a one, two edge. Are you with me? So when I see a one, two edge, I think of it as a door. Okay, so here's what I'm gonna do. I'm gonna run through the house and I'm going to lay down pebbles next to the doors, a pebble on either side of a door, yes? Except if it's on the boundary of the house, and I just lay one pebble here. But all the interior doors, I'm gonna lay down two pebbles, one on either side. Are you with me? Okay, here's a question for you. How many pebbles did I lay down? And I'm actually not interested in the total number. Just tell me if it's even or odd. Odd, how did you see that so quickly? You counted them all? But some of us can't count that fast, yes? So there were lots of even groups and then there were three singles. Oh, lots of even groups, lots of pairs. And if you add up a bunch of pairs, you get an even number, yes? And then three singles, yes? Ah, so it has to, in this case, it's odd. I claim any Sparner-Label triangle will have an odd number of pebbles because even if I divided this more finely, if I start at one and I end at two, would you agree there have to be an odd number of switches from one to two to back to one? If I ever switch back to one, I have to switch back to two again? So there have to be an odd number of singles boundary pebbles, okay? Okay, so what have we done? We've just counted the number of pebbles. We've decided that they're odd. So now let's switch our perspective. Instead of counting door by door, I'm gonna do what's called a combinatorial proof. I'm gonna count the same thing in two different ways. I'm now gonna count the pebbles room by room. So here's a question for you. Is it possible for a room to have no pebbles? Yes, there's one right here, no one, two edges. Is it possible to have two pebbles? Yes, there are two one, two edges here, yes? Is it possible to have three pebbles? No. No. It's possible, we're pretty unlikely. Mm, let's see, let's see. Let's see. Okay, let's think through this together. Would you agree that if you'd had two doors, then there's a one, two, there have to be two one, two edges, yes? So then there actually has to be a repeated label, either one, one, or two, two. So the third edge cannot be a one, two door as well. Are you with me? Good, every room has at most two doors, yes? And when does it have one door? One pebble when it's a one, two, three triangle, yes? Good, so the total number of pebbles is odd and every room has either zero, one, or two. So if it's odd and every room has zero, one, or two, it has to have some rooms with one pebble in it. In fact, it has to have an odd number and those are the one, two, three doors. Oh, nifty. That's a combinatorial proof of this fact and if you want another proof, this one actually turns out to be useful in applications. So this one goes similarly. It says, think of this as a house with many rooms and the one, two edges are doors. It starts off the same way but now we're gonna actually walk through those doors. So I'm gonna start at the boundary. If I see a door, I'm gonna walk in and every time I see a door, I just keep walking until I can't go any further. Either I end on the boundary or if I follow doors, I might end in a one, two, three triangle, yes? So why must there be a one, two, three triangle? Well, how many doors on the boundary? An odd number. How many are matched up by paths? An even number. So some of one, at least one, has to match to a one, two, three triangle and if there are any others, they come in pairs as well. Now what I want you to remember from this proof is in fact, this is a very, a nice, efficient way of finding a one, two, three triangle and they're more sophisticated methods for doing this. I've just showed you one but this becomes really useful in applications which we're gonna go to now. I'm gonna mention the same idea works in higher dimensions. I'll let you think about after the talk how you would show this for a one, two, three, four tetrahedron labeled in a spurner way and I claim that the notion of door here you want is a one, two, three face on a tetrahedron and that's a hint I'm gonna give you if you wanna think about it later. Okay, so what does this have to do with cake? Well, I can think of cake as a division like this by parallel knives. I'm gonna imagine the length of the pieces as being summing up to one and maybe the length here is x, y and z. So would you agree that every division corresponds to a triple of numbers that add up to one and are non-negative? If you think about what that is that actually represents a piece of a plane that's in the first orthent in the x, y, z where x, y, and z are all positive and it cuts out a triangle here, right? So this triangle, for instance, this is in three dimensions this is x, y, and z going up out of vertically. Do you see that its corner here is where the piece one which is where the blue piece is as big as possible the entire cake and the other two pieces are empty. Do you see that? Do you see here in this corner represents the division where piece two is the whole cake and piece one and three are empty, yes? Some point in the middle is a, if I point here this corresponds to a division of cake something looks like this, for instance. Are you with me? Okay, so just to see if you're paying attention what kind of division does a point down here represent? Where the point at the bottom here is the z coordinate is zero. There's no red, the red's as small as possible zero and these two form the whole cake. Are you with me? Okay, this is a way to describe every division as a point in this triangle. That's kind of the beautiful abstract way of thinking about this. And now I claim that this is, we can understand we can understand how to divide a cake fairly using this model. Why? I'm going to break up this big triangle into little pieces. Okay? And then I'm going to assign labels to each of these in a certain way. First thing is I'm gonna ask Alejandro. I'm gonna assign Alejandro. I'm gonna say give him this vertex. I'm gonna give Betty this vertex and I'll give Charlie Carlos this vertex, et cetera. Just go through and give everybody an owner. I'll tell you how I do that in a minute but just assume every vertex here has an owner. And now ask for this division, ask the owner which piece do you prefer if the cake were cut like this? Remember this is some division and I'm gonna ask the players which piece do you want? So here's a question for you. No matter who owns this corner where blue is as big as possible and the other pieces are empty, if they're hungry what are they gonna answer? Piece one, piece two or piece three? They're gonna say piece one because it has all the mass. At this corner, what is the owner of this corner going to say? They're gonna say piece one, two or three? Yellow, two, piece two, yes? And at this corner they're gonna answer which piece are they gonna want? Three. And what about all along this edge? Any of these vertices, the red piece is empty. So if they're hungry, what are they gonna answer if I ask them which piece do you want? Piece one or two? And so if you ask the owners what they want you're gonna get spurner labels naturally, right? One, two, three, one or two, one or three, two or three and anything in the middle. Okay, that means this answers to these questions are spurner labels so there's a one, two, three triangle somewhere in here and I don't know where it is. But I can find it if I want by asking players questions. Are you with me? Okay, so now we're gonna finish this up by saying look if I just assign owners very carefully I'm gonna assign owners in such a way that every triangle has Alejandro, Betty and Carlos that involved at every triangle. Yay. So if there's a one, two, three triangle that corresponds to one in which Alice preferred piece one, Carlos preferred piece two and Betty preferred piece three for three very nearby cake divisions. So if I just pick any point in that triangle that's a division where people are approximately envy-free, okay? Isn't that awesome? That's kind of awesome. Yes, that's worth a clap. And if you actually want exact envy-free you can get there if you use a little bit of analysis some higher level calculus. You can take subsequences of finer and finer triangles and converge to a solution. We don't talk about that but this is what's under the hood. Okay, great. It is beautiful. So this is a version of the cake cutting theorem. The approach I just outlined is due to somebody named Forest Simmons who came up with this idea in the 1980s to use Sperner's Lama to solve cake cutting. And the only assumptions we've made is that all the players are hungry, that they actually want cake, and that the pieces are, preferences are closed. That is if you prefer a piece for a limiting sequence of cuts you prefer it in the limit, okay? Closed under limits. Okay, great. So I'm just gonna finish my talk now by returning to the problem that got me interested in this problem, the rent division problem. To do that, we're actually thinking about how you divide indivisible goods because after all, a house with rooms, the rooms you can't, they're just given to you. You can't divide those. They're already divided in some ways. So how do you divide indivisibles? There's a question. In this episode of The Simpsons, Bart and his friends are trying to divide a comic book which is also indivisible and watch how they accomplish this division. You know better. Turn the page, Bart. Careful. Listen, you guys are welcome to come over and read it any time you like it. Oh, why can't we keep it in my house? Your house. That's crazy talk. Well, the comic's ours as much as it is yours. How about this, guys? Bart can have it one day from Thursdays. Milhouse will get it Tuesdays and Fridays. And yours truly will take it Wednesdays and Saturdays. Perfect. Wait a minute. What about Sunday? Yeah, what about Sunday? Well, Sunday provision will be determined by a random number generator. I will take the digits one through three. Milhouse will have four through six. And Bart will have seven through five. Perfect. Wait a minute. What about zero? Yeah, what about zero? Yeah. Well, this is our paper competition. Best three out of five house that. This is very good. Okay, all right. Yeah. Well, today, I guess I'll be taking my comic and the... Nice try, Martin. It almost worked, but tonight this comic book stays right here. If the comic book stays right here, then so do I. Me too. Fine. We're all going to stay here with the comic book. It'll be like a sleepover. Yeah, a sleepover. That's what pals do, right? Real friendly like. OK, so they divided not the comic book, but what? The time spent on the comic book. So that's another lesson from Fair Division. If you can't divide the object itself, maybe there's some other thing you can divide that would mediate the fairness. And similarly with the problem of housemates, you can't change the rooms, but what can you divide? The rents. OK, so how are we going to divide the rent? Well, it turns out it's a question people ask all the time, right? So this is actually a question for Marilyn Vossavant, who is, I think, the world's highest IQ, person of the world's highest IQ. And if you read this column, you'll see someone asked this question. It's college roommates trying to divide a house. And if you read the answers carefully, you'll realize she didn't take into account that people have different preferences. So ask Marilyn or ask the pet psychic? No. No, let's ask mathematics. Math has something to say here. The space of rent divisions is also a space. x plus y plus z add to the total rent. And each division, then, is a point in a triangle. And you can do the same thing, assign owners. You can ask the owners, what room would you prefer at this set of prices? And now, what are they going to answer at this set of prices? Which room are they going to pick if two of them are free? Room two or three here? What about here? Room one or three? Room one or two? And all along the bottom, where piece three is zero rent, they're going to choose piece three. So you get a labeling that's not a Spurner labeling, but it is something called dual Spurner. And SCARF proved that, in fact, this also has a one, two, three triangle. And there's a constructive way of finding it. And if you want something to do after the talk, you can think about seeing if you can prove this version of Spurner's lemma. And so what that means is you have a theorem. OK, so this is a theorem that I proved after my friend Brad came to me and said, how do you do this? I said, math can help you. So the only thing we've assumed here is that the house is good. In every division of rent, you are able to answer the question, I want one of the rooms. That wouldn't be true if the house were priced too high, for instance, that all the rooms were $1 million a month. The second condition is we've assumed that tenants are miserly, meaning they'd never pass up a free room. That wouldn't be the case if you had a free closet, right? Nobody would pick. The third is, again, room preferences are unchanged by limits. And if you have these things, that's enough to guarantee there's an envy-free rent division. And just to show you that people are still thinking about variations of this kind of thing, just this year, there's a result by a group of mathematicians, which shows that, in fact, you can accomplish a division for both cake and rent, even if one person's preferences are secret. And by secret, that means they don't reveal the answer until after the division is made, they get to pick a piece. OK, lots of ways you can take this. And in lots of different directions, there's a problem of entitlements. If people are supposed to get more than somebody's supposed to get more than the others, there's a problem of complexity procedures, a whole area of computational social sciences that has just popped up in the last 10 to 15 years, that are looking at how long these algorithms take. And you can vary the meanings of the words. I'm spending a semester at the Mathematical Sciences Research Institute, and there's a group of mathematicians there who are thinking about dividing multiple cakes where the preferences are linked over the cakes. And there's some interesting, some cool results. And the way to think about these is to actually think about the space of divisions. The space of divisions isn't a triangle anymore. Guess what? It's a high-dimensional polyhedron. And there are versions of Sparner's Lemma that hold for higher-dimensional polyhedra. Who would have thought, right? Beautiful stuff. So takeaways. Math is great for modeling problems in the social sciences. And the social sciences actually motivate new mathematics. And if you're interested in trying out any of these algorithms, I have this thing called a fair division calculator. This is an old applet. But a couple years ago, a New York Times reporter contacted me. He said, guess what? I have a division. I've been trying to divide rent. And I came across your papers. And I found your fair division calculator. Can we redo this for the New York Times Interactive? And I said, yes, it'd be awesome, because the applet doesn't work anymore. And so if you look it up, you'll see there's an article to divide the rent, start with a triangle. And they coded up the fair division calculator that asks you the series of questions necessary to divide your rent fairly. Thank you very much for your time. I enjoyed being here. So we do have time for some questions. So if you could raise your hand, I'll bring the mic to you if you could stand up. What did you mean by linked cakes? Yeah, good question. So what did I mean by linked cakes? If you have two cakes and there's no relationship between the cakes, then the problem is simple. You just do each cake separately. Like I can divide this cake among three people and then the next cake among three people. But if the cakes are linked, then that means what I decide I want in this cake depends on what I get in the other cake. That's what I mean by linked. So that's related to the idea that if I, in a discrete division, I don't want the sofa unless I can get the love seat. You know what I mean? So yeah, that's what I mean by that. So it's a harder problem. OK. Can you give a hint on how the Browler-Fick's theorem and the Spirner-Lemmerer are equivalent? Yeah, great question. So the Browler-Fick's point theorem says that if you have a function from something that's a ball that's like a blob to another to the same blob, then there's a point that stays put. Yes? OK. And Spirner's Lemma has this funny 1, 2, 3 thing. So the way you get there is if you think about what the function is doing, that will determine the labels on the triangle. So let's imagine that my blob is a triangle. And now suppose that this function, whatever it is, it moves points around. Yes? So if I look at the arrow that points from where a point is to where it goes, that points in one of the directions of the corners of the triangle, for instance. And so if I use those directions to form a label in a way that I won't go into now, you will get naturally a Spirner labeling. And the 1, 2, 3 triangle corresponds to points very nearby that are pointing in different directions. And so that's close to a fixed point. That's sort of the idea. Sorry, what about Congress? The converse. Oh, converse. It's not a political question. I was going to stay away from politics. Yeah. So you want to know how you can prove the Baratheon from Spirner's Lemma. Yes? Great question. So if you have a labeled triangulation, then you can construct a continuous function that uses those labels to decide where points go. And so in particular, here's what you do. Here's one way to think about this. If I have a label triangulation, I'm basically going to use the labels to point in, use the labels to decide where the function takes the point to another point in the triangle. In particular, one thing you can try, and some variant of this will work, is moving a point to one of the main corners. And I'll let you figure out which corner to move it to based on the labeling in such a way that the existence of a fixed point corresponds to the existence of a 1, 2, 3 triangle. So let's give Francis Sue another round of applause. Thank you for coming.