RUI: Combinatorial Fixed Point Theorems, Polytopes, and Fair Division

摘要：SuDMS-0301129“RUI：组合不动点定理，多面体，公平除法”研究者之前的工作已经将组合拓扑和离散几何的方法引入到数学经济学中某些“公平除法”问题的研究中。目前的提案将支持这些方法背后的数学的发展和几个组合问题的解决方案，这些问题是由经济学问题引起的，包括：(1)新的组合不动点定理的发展，(2)相关的简单算法，(3)最小三角剖分和多边形的最小简单覆盖的研究。本研究的广泛影响包括其在群体决策谈判支持方面的跨学科应用，以及本科生在本研究中的积极参与和培训。非正式地说，“公平分配”问题是指找到在玩家之间分配一组商品的方法，使每个人都能根据某种公平的概念得到满足。经济学家、政治学家和博弈论家都对这个话题感兴趣，它经常引发有趣的数学问题。这类问题的可能解决方案通常是一个高维的多面体（多面体），人们通常可以通过将多面体分解成许多块（四面体），并根据玩家的偏好在多面体上行走来找到解决方案。因此，利用三个领域（组合学、拓扑学和离散几何）的思想，如何从四面体构建多面体的数学将直接应用于涉及人类战略和决策制定的社会科学中的重要问题。该奖项由数学科学司和社会、行为和经济科学理事会共同资助，作为数学科学优先领域的一部分。

Abstract for Proposal of SuDMS-0301129"RUI: Combinatorial fixed point theorems, polytopes, and fair division"The investigator's prior work has introduced methods from combinatorialtopology and discrete geometry to the study of certain "fair division" questions in mathematical economics. The current proposal will supportthe development of the mathematics behind these methods and the solutionof several combinatorial questions that have been motivated from economicproblems, including: (1) the development of new combinatorial fixed pointtheorems, (2) associated simplicial algorithms, and (3) the study ofminimal triangulations and minimal simplicial covers of polytopes. Broader impacts of the proposed work include its interdisciplinaryapplication to negotiation support for group decision making, and theactive participation and training of undergraduates in the proposedresearch.Informally speaking, a "fair division" problem is concerned with findingmethods for dividing a set of goods among players in such a way thateveryone can be satisfied according to some notion of fairness. Thistopic is of interest to economists, political scientists, and gametheorists, and it often motivates interesting mathematical questions. The set of possible solutions in such problems is often a polyhedron ofhigh dimension (a polytope), and one can usually find a solution bybreaking the polytope into many pieces (tetrahedra), and walking aroundthe polytope in a way that is guided by player preferences. Thus themathematics of how polytopes can be built up from its tetrahedral buildingblocks using ideas from three areas (combinatorics, topology, and discretegeometry) will have direct application to important problems in the socialsciences involving human strategy and decision making.This award is being jointly funded by the Division of Mathematical Sciences and the Directorate of Social, Behavioral, and Economic Science as part of the Mathematics Sciences Priority Area.