Mathematical Sciences: Mathematics and the Theory of Voting

这个项目将致力于研究“什么样的投票系统是最公平的？”这个基本问题的数学方面。本研究将运用同源性理论的方法来研究多候选人选举中选民偏好的流动。各种各样的数学领域，如“凸几何”NP完备性、维数理论和同调理论，将在这种分析中发挥作用。本项目旨在澄清“选民悖论”的本质，这是阿罗不可能定理意义上的完全公平选举制度存在的主要障碍。为了决定一个系统的公平性，将建立一些新的绝对和相对的“权力”衡量标准，即参与者的影响力。

This project will be devoted to the mathematical aspects of the fundamental question, "what voting systems are most fair?". The investigation will use methods of homology theory to study the flow of voter preference in a multicandidate election. Mathematical fields as diverse as "convex geometry" NP- completeness, dimension theory, and homology theory will play a role in this analysis. This project expects to clarify the nature of the "voter's paradox" which is the main impediment to the existence of completely fair election systems in the sense of Arrow's Impossibility Theorem. Several new absolute and relative measures of "power", or influence of the players, will be established in order to decide the fairness of a system.