Coherent domains and improved lower bounds for the maximum size of Condorcet domains

Abstract

In this paper, we study Condorcet domains, sets of linear orders from which majority ranking produces a linear order. We introduce a new class of Condorcet domains, called coherent domains, which is natural from both a voting theoretic and combinatorial perspective. After studying the properties of these domains we introduce set-alternating schemes. This is a method for constructing well-behaved coherent domains. Using this we show that, for sufficiently large numbers of alternatives $n$, there are coherent domains of size more than $2.1973^n$. This improves the best existing asymptotic lower bounds for the size of the largest general Condorcet domains.