Condorcet domains on at most seven alternatives

Abstract

A Condorcet domain is a collection of linear orders which avoid Condorcet’s paradox for majority voting. We have developed a new algorithm for complete enumeration of all maximal Condorcet domains and, using a supercomputer, obtained the first enumeration of all maximal Condorcet domains on $n\le 7$ alternatives.

We investigate properties of these domains and use this study to resolve several open questions regarding Condorcet domains, and propose several new conjectures. Following this we connect our results to other domain types used in voting theory, such a non-dictatorial and strategy-proof domains. All our data are made freely available on the web.