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.
