voting system designers

Abstract: Motivated by applications in reinforcement learning from human feedback, we consider a setting where a given collection of rankings needs to be compressed into a smaller collection of rankings. This can also be seen as a combination of two fundamental social choice frameworks: ranking aggregation and multiwinner voting. We propose three proportionality notions that capture different types of representation in this setting: positional proportionality, pairwise proportionality, and proportionality for solid coalitions (PSC). On the one hand, we show that positional proportionality and PSC are always satisfiable for any input rankings and target compression size, and a desired output can be found in polynomial time. On the other hand, we prove that pairwise proportionality cannot be satisfied in general, but can nevertheless be attained when the input rankings are single-peaked or single-crossing.

Abstract: A Condorcet domain is a set of rankings over a given candidate set, such that every election that consists only of (an odd number of) votes from the domain has a transitive majority relation. We study outer diversity of Condorcet domains, i.e., a measure that quantifies expected swap distance from a random vote to a closest one in the domain. We numerically analyze outer diversity for maximal Condorcet domains with few candidates, and then we establish its asymptotic behavior for several special domains, mostly obtaining theoretical results.

Abstract: We study the problem of computing how close a given election is to being group-separable, measuring proximity by swaps of adjacent candidates in the votes. We also consider several other domains, including caterpillar group-separable, balanced group-separable, single-peaked, and single-crossing ones. Our problem is generally intractable, but we find practical FPT algorithms parameterized by the number of candidates or swaps. For the latter case, our algorithm applies to all domains characterized by finite forbidden subelections, resolving a well-established open problem. We supplement our theoretical findings with experimental analysis.

Abstract: A tournament T is k-inducible if there are k linear orders on its vertex set such that, for every arc $i \to j$ of T, a majority of the orders rank i above j. For odd k, let N(k) be the least order at which some tournament is not k-inducible. Only N(3) = 8 is known exactly; for N(5) the best bounds were $12 \le N(5) \le 38$, from our previous paper [2], which also gave the first explicit example of moderate order, the Paley tournament $P_{43}$. Results. A bespoke search algorithm improves both ends: $13 \le N(5) \le 23$. The upper bound comes from proving that $P_{23}$ is not 5-inducible, the case Bachmeier et al. [1] reported they could not decide, their SAT solver not having terminated within a cumulative six weeks; ours takes 22 hours on one laptop. The lower bound comes from an analysis at order 12. We also show that $P_{31}$ is not 5-inducible, while $P_{19}$ is 5-inducible but not with unit margin, that is, not by a profile in which every arc is carried by exactly three voters against two. Both $P_{19}$ and $P_{23}$ are arc-critical for their respective properties, whereas $P_{31}$ and $P_{43}$ are not vertex-critical: deleting a vertex leaves a tournament that is still not 5-inducible. Method. The search places one vertex at a time, always choosing the vertex with the fewest options left, and propagates the consequences. Together with the automorphisms of the tournament, this decides on a single laptop instances that neither integer programming nor a general-purpose SAT solver can settle. The refutations for $P_{19}$ and $P_{23}$ are certified as well: the search is split into independent subproblems, a SAT solver emits a machine-checkable proof for each, and a separate program rechecks every proof. All results, subject to two human-checked lemmas, are reproducible from https://github.com/Leonardini/TournamentsBeyond5Voters.

Abstract: Proportional Approval Voting (PAV) chooses committees that maximize a sum of harmonic utilities. We study the set of maximizing committees: whether a candidate belongs to some or all of them, whether the optimum is unique, and how many optima exist. When the committee size is part of the input, the three decision problems are $Δ_2^\mathrm{P}$-complete. Uniqueness remains hard for instances with at most two optimal committees. Counting optimal committees is $\#\!\cdot\!\mathrm{OptP}$-complete under metric reductions: every function $f$ in this class reduces to an election with exactly $f(x)+1$ optimal committees. The reductions encode satisfying assignments directly as committees and use harmonic marginal rewards to realize binary objectives with polynomially many voters. Each satisfying assignment has a unique committee representation, and fixed clause ballots give these representations the same clause score. We also prove Turing equivalence with $\#\mathrm{SAT}$ and show that membership of the counting problem in $\#\mathrm{P}$ would imply $\\mathrm{NP}=\mathrm{coNP}$.

Abstract: An open problem in approval-based multiwinner voting concerns whether we can efficiently compute committees that satisfy both justified representation and Pareto optimality. We answer this question negatively by proving that, on the domain of all profiles, outputting a committee satisfying both axioms is NP-hard. An initial proof was found by ChatGPT Astra. This was then verified and rewritten by the author.