Finding Preference Profiles of Condorcet Dimension $k$ via SAT

TL;DR

This paper encodes the problem of finding preference profiles with a specific Condorcet dimension as a SAT problem, enabling the use of SAT solvers to discover minimal examples and explore their properties.

Contribution

It introduces a novel SAT-based encoding for the Condorcet dimension problem, providing new minimal examples and advancing understanding of preference profiles with higher Condorcet dimensions.

Findings

Found a minimal preference profile with Condorcet dimension 3

Improved existing examples in terms of agents and alternatives

Open question remains on the existence of profiles with dimension 4

Abstract

Condorcet winning sets are a set-valued generalization of the well-known concept of a Condorcet winner. As supersets of Condorcet winning sets are always Condorcet winning sets themselves, an interesting property of preference profiles is the size of the smallest Condorcet winning set they admit. This smallest size is called the Condorcet dimension of a preference profile. Since little is known about profiles that have a certain Condorcet dimension, we show in this paper how the problem of finding a preference profile that has a given Condorcet dimension can be encoded as a satisfiability problem and solved by a SAT solver. Initial results include a minimal example of a preference profile of Condorcet dimension 3, improving previously known examples both in terms of the number of agents as well as alternatives. Due to the high complexity of such problems it remains open whether a preference profile of Condorcet dimension 4 exists.