arXiv subjects
Publications and source records attributed to Luis Sánchez-Fernández.
We propose an alternative algorithm to compute the MMS voting rule. Instead of using linear programming, in this new algorithm the maximin support value of a committee is computed using a sequence of maximum flow problems.
We present a novel approach to the core set/instance selection problem in machine learning. Our approach is based on recent results on (proportional) representation in approval-based multi-winner elections. In our model, instances play a double role as voters and candidates. The approval set of each instance in the training set (acting as a voter) is defined from the concept of local set, which already exists in the literature. We then select the election winners by using a representative voting rule, and such winners are the data instances kept in the reduced training set. We evaluate our approach in two experiments involving neural network classifiers and classic machine learning classifiers (KNN and SVM). Our experiments show that, in several cases, our approach improves the performance of state-of-the-art methods, and the differences are statistically significant.
The goal of multi-winner elections is to choose a fixed-size committee based on voters' preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. In an influential paper, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). We observe that EJR is incompatible with the highly desirable Perfect Representation (PR) criterion, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with PR, as well as with a stronger variant of this axiom, which we term Fractional Perfect Representation (FPR). Moreover, just like EJR, PJR can be used to characterise the classic Proportional Approval Voting (PAV) rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.
We provide the first large-scale data collection of real-world approval-based committee elections. These elections have been conducted on the Polkadot blockchain as part of their Nominated Proof-of-Stake mechanism and contain around one thousand candidates and tens of thousands of (weighted) voters each. We conduct an in-depth study of application-relevant questions, including a quantitative and qualitative analysis of the outcomes returned by different voting rules. Besides considering proportionality measures that are standard in the multiwinner voting literature, we pay particular attention to less-studied measures of overrepresentation, as these are closely related to the security of the Polkadot network. We also analyze how different design decisions such as the committee size affect the examined measures.
In this paper we study several monotonicity axioms in approval-based multi-winner voting rules. We consider monotonicity with respect to the support received by the winners and also monotonicity in the size of the committee. Monotonicity with respect to the support is studied when the set of voters does not change and when new voters enter the election. For each of these two cases we consider a strong and a weak version of the axiom. We observe certain incompatibilities between the monotonicity axioms and well-known representation axioms (extended/proportional justified representation) for the voting rules that we analyze and provide formal proofs of incompatibility between some of these axioms and perfect representation.
We propose the maximin support method, a novel extension of the D'Hondt apportionment method to approval-based multiwinner elections. The maximin support method is based on maximizing the support of the least supported elected candidate. It can be computed efficiently and satisfies (adjusted versions of) the main properties of the original D'Hondt method: house monotonicity, population monotonicity, and proportional representation. We also establish a close relationship between the maximin support method and Phragmén's voting rules.
We identify a whole family of approval-based multi-winner voting rules that satisfy PJR. Moreover, we identify a subfamily of voting rules within this family that satisfy EJR. All these voting rules can be computed in polynomial time as long as the subalgorithms that characterize each rule within the family are polynomial. One of the voting rules that satisfy EJR can be computed in $O(n m k)$.
In this short note, we describe an approval-based committee selection rule that admits a polynomial-time algorithm and satisfies the Extended Justified Representation (EJR) axiom. This rule is based on approximately maximizing the PAV score, by means of local search. Our proof strategy is to show that this rule provides almost optimal average satisfaction to all cohesive groups of voters, and that high average satisfaction for cohesive groups implies extended justified representation.