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We investigate two systems of fully proportional representation suggested by Cham-berlin & Courant and Monroe. Both systems assign a representative to each voter so that the " sum of misrepresentations " is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of… (More)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting.… (More)

The concept of <i>distance rationalizability</i> has several applications within social choice. In the context of voting, it allows one to define ("rationalize") voting rules via a consensus class (roughly, a set of elections in which it is obvious who should win) and a distance function: namely, a candidate is said to be an election winner if it is ranked… (More)

We study the complexity of (approximate) winner determination under Monroe's and Chamberlin-Courant's multiwinner voting rules, where we focus on the total (dis)satisfaction of the voters (the utilitarian case) or the (dis)satisfaction of the worst-off voter (the egalitarian case). We show good approximation algorithms for the satisfaction-based utilitarian… (More)

In elections, a set of candidates ranked consecutively (though possibly in different order) by all voters is called a clone set, and its members are called clones. A clone structure is the family of all clone sets of a given election. In this paper we study properties of clone structures. In particular, we give an axiomatic characterization of clone… (More)

The goal of this paper is to propose and study properties of multiwinner voting rules (with a particular focus on rules based in some way on single-winner scoring rules). We con-of Chamberlin–Courant's and Monroe's rules, identify two natural approaches to defining multiwinner rules, and show that many of our rules can be captured by one or both of these… (More)

A voting rule is an algorithm for determining the winner in an election, and there are several approaches that have been used to justify the proposed rules. One justification is to show that a rule satisfies a set of desirable axioms that uniquely identify it. Another is to show that the calculation that it performs is actually maximum likelihood estimation… (More)

We provide experimental evaluation of a number of known and new algorithms for approximate computation of Monroe's and Chamberlin-Courant's rules. Our experiments, conducted both on real-life preference-aggregation data and on synthetic data, show that even very simple and fast algorithms can in many cases find near-perfect solutions. Our results confirm… (More)

We explore the relationship between two approaches to rationalizing voting rules: the maximum likelihood estimation (MLE) framework originally suggested by Condorcet and re-The former views voting as an attempt to reconstruct the correct ordering of the candidates given noisy estimates (i.e., votes), while the latter explains voting as search for the… (More)

This paper contributes to the program of numerical characterization and classification of simple games outlined in the classic monograph of von Neumann and Morgenstern. We suggest three possible ways to classify simple games beyond the classes of weighted and roughly weighted games. To this end we introduce three hierarchies of games and prove some… (More)