Is Four Enough? Automated Reasoning Approaches and Dual Bounds for Condorcet Dimensions of Elections

TL;DR

This paper uses automated reasoning and linear programming to investigate bounds on the size of Condorcet winning sets in elections, providing empirical evidence that smaller committees than previously known may always suffice.

Contribution

It introduces a MILP-based framework and dual LP analysis to tighten bounds on Condorcet dimensions, suggesting that smaller winning sets are always possible.

Findings

No elections requiring more than size 3 committees were found in moderate-sized searches.

Dual LP analysis led to a conjecture that size 4 committees always exist.

The framework enhances the search for counterexamples and bounds in ranked voting scenarios.

Abstract

In an election where $n$ voters rank $m$ candidates, a Condorcet winning set is a committee of $k$ candidates such that for any outside candidate, a majority of voters prefer some committee member. Condorcet's paradox shows that some elections admit no Condorcet winning sets with a single candidate (i.e., $k=1$), and the same can be shown for $k=2$. On the other hand, recent work proves that a set of size $k=5$ exists for every election. This leaves an important theoretical gap between the best known lower bound $(k\geq 3)$ and upper bound $(k \leq 5)$ for the number of candidates needed to guarantee existence. We aim to close the gap between the existence guarantees and impossibility results for Condorcet winning sets. We explore an automated reasoning approach to tighten these bounds. We design a mixed-integer linear program (MILP) to search for elections that would serve as counter-examples to conjectured bounds. We employ a number of optimizations, such as symmetry breaking, subsampling, and constraint generation, to enhance the search and model effectively infinite electorates. Furthermore, we analyze the dual of the linear programming relaxation as a path towards obtaining a new upper bound. Despite extensive search on moderate-sized elections, we fail to find any election requiring a committee larger than size 3. Motivated by our experimental results in this direction, we simplify the dual linear program and formulate a conjecture which, if true, implies that a winning set of size 4 always exists. Our automated reasoning results provide strong empirical evidence that the Condorcet dimension of any election may be smaller than currently known upper bounds, at least for small instances. We offer a general-purpose framework for searching elections in ranked voting and a new, concrete analytical path via duality toward proving that smaller committees suffice.