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Simmons–Su Protocols
The Simmons–Su protocols are several protocols for envy-free division. They are based on Sperner's lemma. The merits of these protocols is that they put few restrictions on the preferences of the partners, and ask the partners only simple queries such as "which piece do you prefer?". Protocols were developed for solving several related problems: Cake cutting In the envy-free cake-cutting problem, a "cake" (a heterogeneous divisible resource) has to be divided among ''n'' partners with different preferences over parts of the cake. The cake has to be divided to ''n'' pieces such that: (a) each partner receives a single connected piece, and (b) each partner believes that his piece is (weakly) better than all other pieces. A protocol for solving this problem was developed by Forest Simmons in 1980, in a correspondence with Michael Starbird. It was first publicized by Francis Su in 1999. Given a cut-set (i.e. a certain partition of the cake to ''n'' pieces), we say that a partne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free
Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent. In other words, no person should feel envy. General definitions Suppose a certain resource is divided among several agents, such that every agent i receives a share X_i. Every agent i has a personal preference relation \succeq_i over different possible shares. The division is called envy-free (EF) if for all i and j: :::X_i \succeq_i X_j Another term for envy-freeness is no-envy (NE). If the preference of the agents are represented by a value functions V_i, then this definition is equivalent to: :::V_i(X_i) \geq V_i(X_j) Put another way: we say that agent i ''envies'' agent j if i prefers the piece of j over his own piece, i.e.: :::X_i \prec_i X_j :::V_i(X_i) 2 the problem is much harder. See envy-free cake-cutting. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sperner's Lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an simplex contains a cell whose vertices all have different colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms. Finding a Sperner coloring or equivalently a Brouwer fixed point is now believed to be an intractable computational problem, even in the plane, in the general case. The problem is PPAD-complete, a complexity class invented by Christos Papadimitriou. According to the Soviet ''Mathematical Encyclopaedia'' (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free Cake-cutting
An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation. When there are only two partners, the problem is easy and was solved in antiquity by the divide and choose protocol. When there are three or more partners, the problem becomes much more challenging. Two major variants of the problem have been studied: * Connected pieces, e.g. if the cake is a 1-dimensional interval then each partner must receive a single sub-interval. If there are n partners, only n-1 cuts are needed. * General pieces, e.g. if the cake is a 1-dimensional interval then each partner can receive a union of disjoint sub-intervals. Short history Modern research into the fair cake-cutting problem started in the 1940s. The first fairness criterion studied was proportional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest Simmons
A forest is an area of land dominated by trees. Hundreds of definitions of forest are used throughout the world, incorporating factors such as tree density, tree height, land use, legal standing, and ecological function. The United Nations' Food and Agriculture Organization (FAO) defines a forest as, "Land spanning more than 0.5 hectares with trees higher than 5 meters and a Canopy (biology), canopy cover of more than 10 percent, or trees able to reach these thresholds ''in situ''. It does not include land that is predominantly under agricultural or urban use." Using this definition, ''Global Forest Resources Assessment (FRA), Global Forest Resources Assessment 2020'' (FRA 2020) found that forests covered , or approximately 31 percent of the world's land area in 2020. Forests are the predominant terrestrial ecosystem of Earth, and are found around the globe. More than half of the world's forests are found in only five countries (Brazil, Canada, China, Russia, and the United S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Starbird
Michael P. Starbird (born 1948) is a Professor of Mathematics and a University of Texas Distinguished Teaching Professor in the Department of Mathematics at the University of Texas at Austin. He received his B.A from Pomona College and his Ph.D. in mathematics from the University of Wisconsin–Madison. Starbird's mathematical specialty is topology. He joined the University of Texas at Austin as a faculty member in 1974, and served as an associate dean in Natural Sciences from 1989 to 1997. He serves on the national education committees of the Mathematical Association of America and the American Mathematical Society. He directs UT's Inquiry Based Learning Project and works to promote the use of Inquiry Based Learning methods of instruction nationally. Awards He has received over fifteen teaching awards including the Mathematical Association of America's 2007 national teaching award; the Minnie Stevens Piper Professor award, which is a Texas statewide award given to professors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis Su
Francis Edward Su is an American mathematician. He joined the Harvey Mudd College faculty in 1996, and is currently Benediktsson-Karwa Professor of Mathematics. Su served as president of the Mathematical Association of America from 2015–2017 and is serving as a Vice President of the American Mathematical Society from 2020-2023. Su has received multiple awards from the MAA, including the Henry L. Alder Award and the Haimo Award, both for distinguished teaching. He was also a Phi Beta Kappa Visiting Scholar during the 2019-2020 term. Su received his B.S. in Mathematics from the University of Texas, graduating Phi Beta Kappa in 1989. He went on to receive his Ph.D. from Harvard University, where his advisor was Persi Diaconis. His research area is combinatorics, and he is particularly known for his work on fair division. Su and Michael Starbird are co-authors of the book "Topology Through Inquiry". His book, "Mathematics for Human Flourishing", was released on 7 January 2020. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologically Closed
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For S as a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r ( is allowed). Another way to express thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair - each person should receive a piece that he or she believes to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brouwer Fixed-point
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place among the fundamental theorems of topology. The theorem is also used for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PPAD
In computer science, PPAD ("Polynomial Parity Arguments on Directed graphs") is a complexity class introduced by Christos Papadimitriou in 1994. PPAD is a subclass of TFNP based on functions that can be shown to be total by a parity argument. The class attracted significant attention in the field of algorithmic game theory because it contains the problem of computing a Nash equilibrium: this problem was shown to be complete for PPAD by Daskalakis, Goldberg and Papadimitriou with at least 3 players and later extended by Chen and Deng to 2 players.*. Definition PPAD is a subset of the class TFNP, the class of function problems in FNP that are guaranteed to be total. The TFNP formal definition is given as follows: :A binary relation P(''x'',''y'') is in TFNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(''x'',''y'') holds given both ''x'' and ''y'', and for every ''x'', there exists a ''y'' such that P(''x'',''y'') holds. Subclas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rental Harmony
Rental harmony is a kind of a fair division problem in which indivisible items and a fixed monetary cost have to be divided simultaneously. The housemates problem and room-assignment-rent-division are alternative names to the same problem. In the typical setting, there are n partners who rent together an n-room house for cost fixed by the homeowner. Each housemate may have different preferences — one may prefer a large room, another may prefer a room with a view to the main road, etc. The following two problems should be solved simultaneously: * (a) Assign a room to each partner, * (b) Determine the amount each partner should pay, such that the sum of payments equals the fixed cost. There are several properties that we would like the assignment to satisfy. * Non-negativity (NN): all prices must be 0 or more: no partner should be paid to get a room. * Envy-freeness (EF): Given a pricing scheme (an assignment of rent to rooms), we say that a partner ''prefers'' a given room if h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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