We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show that computing the winner for proportional approval voting is NP-hard, closing a long standing open problem. As none of the rules are strategyproof, even for dichotomous preferences, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents. In many settings, we show that it is NP-hard for an agent or agents to compute how best to vote given a fixed set of approval ballots from the other agents.
The goal of this paper is to propose and study properties of multiwinner voting rules which can be consider as generalisations of single-winner scoring voting rules. We consider SNTV, Bloc, k-Borda, STV, and several variants of Chamberlin--Courant's and Monroe's rules and their approximations. We identify two broad natural classes of multiwinner score-based rules, and show that many of the existing rules can be captured by one or both of these approaches. We then formulate a number of desirable properties of multiwinner rules, and evaluate the rules we consider with respect to these properties. We develop a general framework for social choice problems in which a limited number of alternatives can be recommended to an agent population. In our budgeted social choice model, this limit is determined by a budget, capturing problems that arise naturally in a variety of contexts, and spanning the continuum from pure consensus decision making (i.e., standard social choice) to fully personalized recommendation. Our approach applies a form of segmentation to social choice problems-- requiring the selection of diverse options tailored to different agent types--and generalizes certain multiwinner election schemes. We show that standard rank aggregation methods perform poorly, and that optimization in our model is NP-complete; but we develop fast greedy algorithms with some theoretical guarantees. Experiments on real-world datasets demonstrate the effectiveness of our algorithms.