Condorcet Domains: A Geometric Perspective

One of several topics in which Peter Fishburn [4, 5] has made basic contributions involves finding maximal Condorcet Domains. In this current paper, I develop a geometric approach that, at least for four and five alternatives, is equivalent to Fishburn’s clever alternating scheme (described below), a scheme that has advanced our understanding of the area. To explain “Condorcet Domains” and why they are of interest, start with the fact that when making decisions by comparing pairs of alternatives with majority votes, the hope is to have decisive outcomes in that one candidate always is victorious when she is compared with any other candidate. When this happens, the candidate is called the Condorcet winner. The attractiveness of this notion, where someone can beat anyone else in head-to-head comparisons, is why the Condorcet winner remains a central concept in voting theory. For a comprehensive, modern description of the Condorcet solution concept, see Gehrlein’s recent book [7]. But Condorcet also proved that pairwise rankings can lead to cycles, where a Condorcet winner cannot exist. His three voter example [3],1 now called the Condorcet triplet, has the preferences