Properties of Position Matrices and Their Elections - Boehmer, Niclas; Cai, Jin-Yi; Faliszewski, Piotr; Fan, Austen Z.; Janeczko, Łukasz; Kaczmarczyk, Andrzej; Wąs, Tomasz (Proceedings of the AAAI Conference on Artificial Intelligence, 2023)
We study the properties of elections that have a given position matrix (in such elections each candidate is ranked on each position by a number of voters specified in the matrix). We show that counting elections that generate a given position matrix is #P-complete. Consequently, sampling such elections uniformly at random seems challenging and we propose a simpler algorithm, without hard guarantees. Next, we consider the problem of testing if a given matrix can be implemented by an election with a certain structure (such as single-peakedness or group-separability). Finally, we consider the problem of checking if a given position matrix can be implemented by an election with a Condorcet winner. We complement our theoretical findings with experiments.