Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence (AAAI-10) Approximation Algorithms and Mechanism Design for Minimax Approval Voting Ioannis Caragiannis Dimitris Kalaitzis Evangelos Markakis RACTI & Department of Computer RACTI & Department of Computer Department of Informatics Engineering and Informatics Engineering and Informatics Athens University of Economics University of Patras, Greece University of Patras, Greece & Business, Greece [email protected] [email protected] [email protected] Abstract fined size k. Then the minisum solution consists of the k candidates with the highest number of approvals. We consider approval voting elections in which each voter votes for a (possibly empty) set of candidates and the out- This solution however may ignore some voters’ prefer- come consists of a set of k candidates for some parameter ences in certain instances and does not take fairness issues k, e.g., committee elections. We are interested in the min- into account. We demonstrate this with the following exam- imax approval voting rule in which the outcome represents ple with four voters, five candidates, and k = 2. Each row a compromise among the voters, in the sense that the max- represents the preference of the corresponding voter. The imum distance between the preference of any voter and the minisum solution contains the candidates {a, b}. The dis- outcome is as small as possible. This voting rule has two tances of the voters from this outcome are 1, 0, 2, and 5 for main drawbacks. First, computing an outcome that minimizes voters 1, 2, 3, and 4, respectively (counting the number of the maximum distance is computationally hard. Furthermore, alternatives in which the voter disagrees with the outcome). any algorithm that always returns such an outcome provides Instead, the solution {a, c} has distances 3, 2, 2, and 3, re- incentives to voters to misreport their true preferences. spectively, and suggests a better compromise among the vot- In order to circumvent these drawbacks, we consider approx- ers since everybody is relatively close to the outcome. imation algorithms, i.e., algorithms that produce an outcome that approximates the minimax distance for any given in- stance. Such algorithms can be considered as alternative vot- a b c d e ing rules. We present a polynomial-time 2-approximation al- 1 1 1 0 0 1 gorithm that uses a natural linear programming relaxation for 2 1 1 0 0 0 the underlying optimization problem and deterministically 3 1 1 1 1 0 rounds the fractional solution in order to compute the out- 4 0 0 1 1 1 come; this result improves upon the previously best known al- gorithm that has an approximation ratio of 3. We are further- more interested in approximation algorithms that are resistant Recently, a new voting rule, the minimax solution, was in- to manipulation by (coalitions of) voters, i.e., algorithms that troduced as a means to achieve a compromise between the do not motivate voters to misreport their true preferences in voters’ preferences (Brams et al. 2007). The minimax solu- order to improve their distance from the outcome. We com- tion picks the k candidates for which the maximum (Ham- plement previous results in the literature with new upper and ming) distance of any voter from the outcome is minimized. lower bounds on strategyproof and group-strategyproof algo- Since this rule minimizes the disagreement with the least rithms. satisfied voter, it tends to result in outcomes that are more widely acceptable than the minisum solution. On the nega- Introduction tive side the minimax solution has two main drawbacks that prevent its applicability: (i) the problem of computing the Approval voting is a very popular voting protocol mainly minimax solution is NP-hard, and (ii) voters may have in- used for committee elections (Brams & Fishburn 2007). In centives to misreport their preference in order to improve such a protocol, the voters are allowed to vote for, or ap- the distance of their true preference from the outcome. Our prove of, as many candidates as they like. In the last three main goal in this paper is to tackle these issues by resorting decades, many scientific societies and organizations have to approximation algorithms. adopted approval voting for their council elections. The so- Approximation algorithms tackle the computational lution concept that has been used in almost all such elections hardness of an optimization problem by producing (in in practice is the minisum solution, i.e., output the commit- polynomial-time) solutions provably close to optimal ones tee which, when seen as a -vector, minimizes the sum of 0/1 for any problem instance; see (Vazirani 2001) for a cover- the Hamming distances to the ballots. We assume through- age of early work in the field. We refer to the optimization out the paper that the committee should be of some prede- problem of computing the minimax solution as k-minimax Copyright c 2010, Association for the Advancement of Artificial approval. (LeGrand et al. 2007) present a 3-approximation Intelligence (www.aaai.org). All rights reserved. algorithm for the problem; given an instance, the algorithm 737 2 produces a solution (i.e., a set of k candidates) so that its than 3− k1+ and infinity. Our lower boundsare not based on distance from any voter’s preference is at most 3 times the any computational complexity assumption and, hence, hold maximum distance of the voters from the minimax solution. for exponential-time algorithms as well. The algorithm is very simple to describe and we will refer to it here as the k-completion algorithm: it arbitrarily picks Notation and Definitions a voter and computes a set of k candidates which has min- We fix some notation used in the following. We typically imum distance from this voter. An immediate question is use n to denote the number of voters and m for the number whether algorithms with better approximation ratios exist. of candidates. We denote the set of candidates by A.A Another interesting question is whether we can have good preference is simply a subset of A. A profile P is a tuple approximations by non-dictatorial algorithms. Note that the P = (P1, ..., Pn) where Pi denotes the preference of voter k-completion algorithm is dictatorial as it is based only on i (i.e., the set of candidates she approves). Throughout the one voter’s preferences. paper we make the reasonable assumption that n > k. When The issue of resistance to manipulation is the very subject this is not explicitly mentioned (e.g., in some lower bound of Mechanism Design; see (Nisan 2007) for an introduc- proofs), we can complete the profile by adding indifferent tion to the field. In our context, it translates to algorithms voters (that approve no candidate). for k-minimax approval which, given a profile, compute an We extend the notion of (Hamming) distance to subsets of approximate solution in such a way that no single voter or A as follows. We say that the distance of two sets Q and T a coalition of voters have any incentive to misreport their is the total number of candidates in which they differ, i.e., preferences in order to decrease their distance from the out- d(Q, T ) = |Q \ T | + |T \ Q| = |Q| + |T | − 2|Q ∩ T |. come. The corresponding properties of resistance to manip- ulation by single voters and coalitions of voters are known as Note that this is precisely the Hamming distance of the sets, strategyproofness and group-stratefyproofness, respectively. when seen as binary vectors where the ith coordinate of each (LeGrand et al. 2007) prove that the minimax solution is vector equals 1 if the ith candidate belongs to the set. not resistant to manipulation while the k-completion algo- rithm is. They also pose the question of computing the Approximation Algorithms best possible bound on the approximationratio of algorithms We begin by establishing a connection between Pareto- that are resistant to manipulation. This question falls within efficiency and low approximation ratio. the line of research on mechanisms without monetary trans- Definition 1. Given a profile P , a size-k set K ⊆ A is called fers (Schummer & Vohra 2007) and, in particular, approxi- Pareto-efficient with respect to P if there is no other size-k mate mechanism design without money (Procaccia & Ten- ′ ′ set K ⊆ A such that d(K ,Pi∗ ) < d(K, Pi∗ ) for some nenholtz 2009). ∗ ′ voter i and d(K ,Pi) ≤ d(K, Pi) for any other voter i. We make progress in both directions. Concerning the ap- An algorithm for k-minimax approval is Pareto-efficient if, proximability of k-minimax approval by polynomial-time on input any profile P , its outcome is Pareto-efficient with algorithms, we first establish a connection between the prop- respect to P . erty of Pareto-efficiency and approximability. As a corol- The next lemma significantly extends the class of 3- lary, we obtain that Minisum (i.e., the algorithm that re- approximation algorithms for minimax-approval and will be turns a minisum solution) has approximation ratio at most proved very useful later. Interestingly, Minisum is Pareto- 3 − 2 for k-minimax approval. Our strongest result in k1+ efficient; the proof follows by the definition of Pareto- this direction is an algorithm based on linear programming efficiency and the fact that Minisum minimizes the sum of that achieves an improved approximation ratio of 2; this is the distances of the outcome from the voters. a significant improvement compared to the previously best Lemma 2. Any Pareto-efficient algorithm for k-minimax known algorithms. The algorithm is based on rounding the 2 fractional solution of a natural linear programming relax- approval has approximation ratio at most 3 − k1+ .