The key will be a combinatorial lemma called sperners lemma, in this talk, well prove sperners lemma and elaborate on some of its applications. Although many exact and approximative methods for finding fair solutions already exist. Equivalence between the existence of an approximate. Undergraduate seminar in discrete mathematics, sperners. Sperners lemma and its variants continue to be useful in applications. Sperners lemma and its applications combinatorics and graph. Abstract this paper is a seminar paper that summarizes francis edward sus paper rental harmony. What does it take to ensure a cake is cut fairly, so that no player prefers their neighbors piece to their own. He and his housemates were moving to a house with rooms of various sizes and features, and were having trouble deciding who should. Sperners lemma in fair division september 28, 2016 simplices are a recurring construction in math, because theyre the simplest example of an ndimensional solid triangles, tetrahedra, etc. Let tbe a triangulation of some polygon pand let fbe the number of full triangles.
For this reason, sperners lemma can also be used in rootfinding algorithms and fair division algorithms. Since then, sperners lemma has seen numerous applications, notably in the proof of existence of mixed nash equilibria 11, in fair division, and recently it played an important role in the study of computational complexity of nding a nash equilibrium 3, 2. Fair division is an active research area in mathematics, economics, computer science, etc. Let t be a triangulation of a ddimensional polytope p with n vertices v 1, v 2,v n. One can check that every subsimplicial complex of kis simplex joinclosed.
A tutorial on the proof of the existence of nash equilibria. Proceedings of the twentysixth annual acmsiam symposium on discrete algorithms 10. This method is based on sperners lemma, a famous combinatorial lemma proved by emanuel sperner in 1928. Sperners lemma defeats the rental harmony problem youtube. Sperners colorings, hypergraph labeling problems and fair. Simmons approach to cakecutting, sus approach to room assignment rent division and scarfs method for computation of equilibrium prices. We prove three results about colorings of the simplex reminiscent of sperners lemma, with applications in hardness of approximation and fair division. Sperners colorings and optimal partitioning of the simplex. First, have students play around with this nyt interactive feature to get a sense of how the fair division algorithm mentioned in the article works. Let pbe a polygon in the plane, and consider a triangulation of this polygon.
Sperners lemma in fair division on kakutanis fixed point theorem, kkms thoerem, and the core of a balanced game achieving rental harmony with a secretive roommate thieves can make sandwiches equitable rent division rental harmony with roommates which is the fairest rent division of them all. It bears a strong resemblance to one of the most popular proofs of brouwers. The moving wall represents the time period between the last issue available in jstor and the most recently published issue of a journal. Let us consider v and let k be a simplexjoinclosed simplicial complex structure of v. Sperners lemma and fair division problems pdf by suthee ruangwises. Sperners lemma implies kakutanis fixed point theorem. Check out 3blue1browns video on another fair division problem here. Proof of sperners lemma we create a graph out of the triangles.
This paper is a seminar paper that summarizes fran cis edward sus paper rental harmony. This lemma holds for every dimension, but the most common and simplest one is the twodimensional case. Sperners lemma in fair division by francis edward su addresses how to divide the rent for a house among several housemates, each of whom will have a different room with different features. In the rental division problem, a group of housemates must decide a fair allocation of rents for rooms which may have different features. Hence, by sperners lemma there must be at least one subsimplex in which the labels are all different. Pdf this article was awarded the maas 2001 merten m. More recently, sperners lemma and related combinatorial theorems have been used to show the existence of envyfree divisions for a variety of fair division problems, including the classical cake. Notesonspernerslemma rich schwartz february 18, 2016 1 spernerslemma sperners lemma applies to any cleanly triangulated triangle. Undergraduate seminar in discrete mathematics, sperners lemma. In sperners lemma, we colour the vertices of a triangulation t with the colours 1, 2 and. A polytopal generalization of sperners lemma sciencedirect. In this video the mathologer sets out to explain a very elegant new solution to this and related hard fair division problems that even made it into the new york times.
One of its most wellknown applications is in john nashs paper 8, where the theorem is used to prove the existence of an equilibrium strategy in nperson games. Brouwers fixed point theorem cannot be constructively proved, so the existence of an equilibrium in a competitive economy also cannot be constructively proved. The main purpose of this paper is to present a solution of the following conjecture. Fair division and the many facets of sperners lemma meetup. The techniques also give generalizations of sperners lemma. Sperners lemma in fair division francis edward su my friends dilemma was a practical question that mathematics could answer, both elegantly and constructively. An edge is called a 12edge if its endpoints is colored 1 and 2, and a triangle is said to be complete if each of its vertices. However, do not be fooledthis little lemma is as powerful as it is simple. Sperners lemma 193 let us consider v and let kbe an a. Therefore, by sperners lemma, there is a complete triangle in t.
It can, for instance, be used to give a short, elementary proof of the brouwer xed point theorem 7. Two divisions of the triangle spanned by v1,v2 and v3. For our triangulation, for each abedge put a dot on each side of the edge and let nbe the number of. Sperners lemma, the brouwer fixed point theorem, the. The result about splitting rent fairly has actually become. Some authors have presented a constructive or an approximate version of brouwers fixed point theorem using. Sperners lemma in fair division 3 we encountered earlier for n 2. Our proof, in particular, yields an algorithm to find such a fair division of rent. My friends dilemma was a practical question that mathematics could answer, both elegantly. Alice may prefer windows and bob may like hardwood floors and the goal is to find a way to decide how best to split the rent based on their preferences. Most research works on fair division assume that the manna the resource to divide. For other low dimensions, figures 2 and 3 show examples of a spernerlabelled 1simplex and 3simplex.
Let abe the number of abedges on the boundary of p. Fair division and generalizations of sperner and kkmtype. Although many exact and approximative methods for finding fair solutions. Sperners lemma for triangles our fair division approach is based on a simple combinatorial lemma, due to sperner 14 in 1928. Color each vertex of this triangulation by one of three colors 1, 2, or 3. I \outside vertex has odd degree by line segment observation. On the other hand, sperners lemma which is used to prove brouwers theorem is constructively proved. In rare instances, a publisher has elected to have a zero moving wall, so. Label the vertices of t by 1,2,n in such a way that a vertex of t belonging to the interior of a face f of p can only be labelled by j if v j is on f. There are many different kinds of fair division problems. A beautiful combinatorical proof of the brouwer fixed. Sperners lemma readily follows from a simple parity argument in graph theory. Fixed points and fair divisions berkeley math circle.
The vast majority of material presented here, including all of the gures except those in the last section, is borrowed from 1. A powerful approach to fairdivision questions that provides new methods for achieving approximate envyfree divisions, in which each person feels he or she has received the best share. For example, they have recently been used to solve fair division problems in game theory 18, 21. Generalizations and closing thoughts why do we care about equiareal triangulation.
In sperners lemma, we colour the vertices of a triangulation twith the colours 1, 2 and 3. Projects undergraduate seminar in discrete mathematics. One can verify that every subsimplicial complex of kis a. First, we will start by stating and proving sperners lemma.
Sperners lemma, on the other hand, is a combinatorial result concerning the labelling of the vertices of simplices and their triangulations. By degreesum formula, there are an odd number therefore, at least 1 of degree 3 inner vertices \completely colored triangles. We treat problems of fair division, their various interconnections, and their relations to sperners lemma and the knasterkuratowskimazurkiewicz kkm theorem as well as their variants. The overt purpose of this talk is to illustrate the close relationship between these two results, and in particular their common genesis from a combinatorial result known as sperners lemma. He is an assistant professor at harvey mudd college, a blue.
It is a proper improvement apart from the case of stacked polytopes. We prove extensions of alons necklace splitting result in certain regimes and relate it to hyperplane mass partitions. Fair division problems are concerned with finding ways of dividing a resource. The goal is to nd a multicoloured triangle in the subdivision, which is a triangular. An example is shown on the left hand side of figure 1. Sperners lemma is one of the key ingredients of the proof of monskys theorem, that a square cannot be cut into an odd number of equalarea triangles. We prove the following conjecture of atanassov studia sci. A polytopal generalization of sperners lemma request pdf. By clicking on a vertex, one can change the room preference for the given player at the given set of prices, and then see how the other vertexes in the triangle change as a result. Maliwal, ayesha, sperners lemma, the brouwer fixed point theorem, the kakutani fixed point theorem, and their applications in social sciences 2016. Every sperner covering of a triangulation that is, unique colors on exterior vertices, edges inherent the colors between. This issue is resolved in the division on the right by further dividing v 2afinto two further triangles, v 2acand v 2cf.
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