RUI: Combinatorial fixed point theorems, polytopes, and preference sets

RUI：组合不动点定理、多胞形和偏好集

• 批准号: 0701308
• 负责人: Francis Su
• 金额: $ 11.45万
• 依托单位: Harvey Mudd College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701308&HistoricalAwards=false
• 关键词: RUI; Combinatorial; fixed; point; theorems

RUI: Combinatorial Fixed Point Theorems, Polytopes, and Preference SetsThe investigator's prior work has introduced methods from combinatorial topology and discrete geometry to the study of certain "fair division" questions in mathematical economics. The current proposal will support the development of the mathematics behind these methods and the solution of several combinatorial questions that have been motivated by his prior work, including: (1) the study of minimal triangulations and minimal simplicial covers of polytopes, (2) the further development of combinatorial fixed point theorems and associated simplicial algorithms, and (3) the development of convexity theorems that have bearing on social choice problems. Broader impacts of the proposed work include its interdisciplinary applications to group decision making, fair division, and voting problems, and the active participation and training of undergraduates in the proposed research.Informally speaking, a "fair division" problem is concerned with finding methods for dividing a set of goods among players in such a way that everyone can be satisfied according to some notion of fairness. This topic is of interest to economists, political scientists, and game theorists, and it often motivates interesting mathematical questions. The set of possible solutions in such problems is often a polyhedron of high dimension (a polytope), and one can usually find a solution by breaking the polytope into many pieces (tetrahedra), and walking around the polytope in a way that is guided by player preferences. Thus the mathematics of how polytopes can be built up from its tetrahedral building blocks using ideas from three areas (combinatorics, topology, and discretegeometry) as well as related classical convexity theorems will have direct application to important problems in the social sciences involving human strategy and decision making.

RUI：组合不动点定理、多面体和偏好集研究者之前的工作已将组合拓扑和离散几何的方法引入到数理经济学中某些“公平划分”问题的研究中。当前的提案将支持这些方法背后的数学发展以及由他之前的工作激发的几个组合问题的解决，包括：（1）多面体的最小三角剖分和最小单纯覆盖的研究，（2）组合不动点定理和相关单纯算法的进一步发展，以及（3）与社会选择有关的凸性定理的发展 问题。 拟议工作的更广泛影响包括其在群体决策、公平分配和投票问题中的跨学科应用，以及本科生在拟议研究中的积极参与和培训。非正式地说，“公平分配”问题涉及寻找在参与者之间分配一组商品的方法，以便每个人都可以根据某种公平概念感到满意。这个话题引起了经济学家、政治学家和博弈论学家的兴趣，并且经常引发有趣的数学问题。此类问题的一组可能的解决方案通常是高维多面体（多面体），通常可以通过将多面体分解为许多块（四面体）并以玩家偏好引导的方式在多面体周围行走来找到解决方案。因此，如何使用来自三个领域（组合学、拓扑学和离散几何）的思想以及相关的经典凸性定理从四面体构建块构建多面体的数学将直接应用于涉及人类策略和决策的社会科学中的重要问题。