arXiv subjects

Magdaléna Tydrichová

Publications and source records attributed to Magdaléna Tydrichová.

4 recordsLinked to original sources

arXiv2024

Comparing Ways of Obtaining Candidate Orderings from Approval Ballots

To understand and summarize approval preferences and other binary evaluation data, it is useful to order the items on an axis which explains the data. In a political election using approval voting, this could be an ideological left-right axis such that each voter approves adjacent candidates, an analogue of single-peakedness. In a perfect axis, every approval set would be an interval, which is usually not possible, and so we need to choose an axis that gets closest to this ideal. The literature has developed algorithms for optimizing several objective functions (e.g., minimize the number of added approvals needed to get a perfect axis), but provides little help with choosing among different objectives. In this paper, we take a social choice approach and compare 5 different axis selection rules axiomatically, by studying the properties they satisfy. We establish some impossibility theorems, and characterize (within the class of scoring rules) the rule that chooses the axes that maximize the number of votes that form intervals, using the axioms of ballot monotonicity and resistance to cloning. Finally, we study the behavior of the rules on data from French election surveys, on the votes of justices of the US Supreme Court, and on synthetic data.

cs.GT↗

arXiv2022

Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_\infty$ norms

We present various results about Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_{\infty}$ norms. When there are four candidates, we show that the maximal size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in the plane under norm $\ell_1$ or $\ell_{\infty}$ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in last position of a two-dimensional Euclidean preference profile under norm $\ell_1$ or $\ell_\infty$, which generalizes the case of one-dimensional Euclidean preferences (for which it is well known that at most two candidates can be ranked last). We generalize this result to $2^d$ (resp. $2d$) for $\ell_1$ (resp. $\ell_\infty$) for $d$-dimensional Euclidean preferences. We also establish that the maximal size of a two-dimensional Euclidean preference profile on $m$ candidates under norm $\ell_1$ is in $Θ(m^4)$, i.e., the same order of magnitude as under norm $\ell_2$. Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm $\ell_2$ for four candidates can be characterized by three voter-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. in Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47(2):379-400.

math.MG↗

arXiv2022

Weighted majority tournaments and Kemeny ranking with 2-dimensional Euclidean preferences

The assumption that voters' preferences share some common structure is a standard way to circumvent NP-hardness results in social choice problems. While the Kemeny ranking problem is NP-hard in the general case, it is known to become easy if the preferences are 1-dimensional Euclidean. In this note, we prove that the Kemeny ranking problem remains NP-hard for $k$-dimensional Euclidean preferences with $k\!\ge\!2$ under norms $\ell_1$, $\ell_2$ and $\ell_\infty$, by showing that any weighted tournament (resp. weighted bipartite tournament) with weights of same parity (resp. even weights) is inducible as the weighted majority tournament of a profile of 2-Euclidean preferences under norm $\ell_2$ (resp. $\ell_1,\ell_{\infty}$), computable in polynomial time. More generally, this result regarding weighted tournaments implies, essentially, that hardness results relying on the (weighted) majority tournament that hold in the general case (e.g., NP-hardness of Slater ranking) are still true for 2-dimensional Euclidean preferences.

cs.DM↗

arXiv2020

Recognizing Single-Peaked Preferences on an Arbitrary Graph: Complexity and Algorithms

This paper is devoted to a study of single-peakedness on arbitrary graphs. Given a collection of preferences (rankings of a set of alternatives), we aim at determining a connected graph G on which the preferences are single-peaked, in the sense that all the preferences are traversals of G. Note that a collection of preferences is always single-peaked on the complete graph. We propose an Integer Linear Programming formulation (ILP) of the problem of minimizing the number of edges in G or the maximum degree of a vertex in G. We prove that both problems are NP-hard in the general case. However, we show that if the optimal number of edges is m-1 (where m is the number of candidates) then any optimal solution of the ILP is integer and thus the integrality constraints can be relaxed. This provides an alternative proof of the polynomial-time complexity of recognizing single-peaked preferences on a tree. We prove the same result for the case of a path (an axis), providing here also an alternative proof of polynomiality of the recognition problem. Furthermore, we provide a polynomial-time procedure to recognize single-peaked preferences on a pseudotree (a connected graph that contains at most one cycle). We also give some experimental results, both on real and synthetic datasets.

cs.GT↗