Computing and Testing Pareto Optimal Committees

03/18/2018 ∙ by Haris Aziz, et al. ∙ UNSW Université Paris-Dauphine 0

Selecting a set of alternatives based on the preferences of agents is an important problem in committee selection and beyond. Among the various criteria put forth for the desirability of a committee, Pareto optimality is a minimal and important requirement. As asking agents to specify their preferences over exponentially many subsets of alternatives is practically infeasible, we assume that each agent specifies a weak order on single alternatives, from which a preference relation over subsets is derived using some preference extension. We consider five prominent extensions (responsive, downward lexicographic, upward lexicographic, best, and worst). For each of them, we consider the corresponding Pareto optimality notion, and we study the complexity of computing and verifying Pareto optimal outcomes. We also consider strategic issues: for four of the set extensions, we present a linear-time, Pareto optimal and strategyproof algorithm that even works for weak preferences.

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1 Introduction

Pareto optimality is a central concept in economics and has been termed the “single most important tool of normative economic analysis” [Moulin, 2003]. An outcome is Pareto optimal if there does not exist another outcome that all agents like at least as much and at least one agent strictly prefers. Although Pareto optimality has been considered extensively in single-winner voting and other social choice settings such as fair division or hedonic games, it has received only little attention in multiwinner voting, in which the outcomes are sets of alternatives. Multiwinner voting applies to selecting a set of plans or a committee, hiring team members, movie recommendations, and more. For convenience, we use the terminology “committee” even if our results have an impact far beyond committee elections [Faliszewski et al., 2017, Aziz et al., 2017].

In single-winner voting setting, agents express preferences over alternatives and a single alternative is selected. Pareto optimality in this context is straightforward to define, achieve, and verify. In multiwinner voting, a well-known difficulty is that it is unrealistic to assume that agents will report preferences over all possible committees, since there is an exponential number of them. For this reason, most approaches assume that they only report a small part of their preferences, and that some extension principle is used to induce a preference over all possible subsets from this ‘small input’ over single alternatives [Barberà et al., 2004]. Such preference extensions are also widely used in other social choice settings such as fair division or matching. The most two widely used choices of ‘small inputs’ in multiwinner voting are rankings (linear orders) over alternatives and sets of approved alternatives. In this paper we make a choice that generalizes both of them: agents report weak orders over single alternatives. Then we consider five prominent preference extension principles: the responsive extension, where a set of alternatives is at least as preferred as a set of alternatives if is obtained from by repeated replacements of an alternative by another alternative which is at least as preferred; the optimistic, or ‘best’ (respectively pessimistic, or ‘worst’) extension, which orders subsets of alternatives according to their most (respectively, least) preferred element; the downward lexicographic extension, a lexicographic refinement of the optimistic extension, and the upward lexicographic extension, a lexicographic refinement of the pessimistic (worst) extension.

The responsive extension [Barberà et al., 2004, Roth and Sotomayor, 1990] can be seen as the ordinal counterpart of additivity. The downward lexicographic extension has been considered in various papers [Bossert, 1995, Lang et al., 2012, Klamler et al., 2012]. The ‘best’ set extension has been considered in a number of approaches such as full proportional representation [Chamberlin and Courant, 1983, Monroe, 1995] and other committee voting settings Elkind et al. [2015]. The ‘worst’ set extension, also used by Klamler et al. [2012] and Skowron et al. [2015b], captures settings where the impact of a bad alternative in the selection overwhelms the benefits of good alternatives: for instance, when the decision about a crucial issue will be made by one of the members of the committee but the agent ignores which one and is risk-averse; or the case of a parent’s preferences over a set of movies to be watched by a child. The ‘best’ and ‘worst’ set extensions have been used in coalition formation [Aziz and Savani, 2016, Cechlárová, 2008].

Although set extensions have been implicitly or explicitly considered in multiwinner voting, most of the computational work has dealt with specific voting rules (see the related work section). Instead, we concentrate on Pareto optimality, consider the computation and verification of Pareto optimal committees, as well as the existence of a polynomial-time and strategyproof algorithm that returns Pareto optimal outcomes.

Contributions

We consider Pareto optimality with respect to the five aforementioned preference set extensions. We present various connections between the Pareto optimality notions. For each of the notions, we undertake a detailed study of complexity of computing and verifying Pareto optimal outcomes. Table 1 summarizes the complexity results.

An important take-home message of the results is that testing Pareto optimality or obtaining Pareto improvements over status-quo committees is computationally hard even though computing some Pareto optimal committee is easy. For responsive and downward lexicographic extensions we give a complete characterization of the complexity of testing Pareto optimality when preferences are dichotomous and the size of top equivalence class is two: unless P = NP, Pareto optimality can be tested in polynomial time if and only if the size of the first equivalence classes is at most two. For the ‘best’ extension, we show that even computing a Pareto optimal outcome is NP-hard. Another interesting contrast with the responsive set extension is that even when preferences are dichotomous and the size of top equivalence class is two, testing Pareto optimality is coNP-complete. In contrast to the other extensions, for the ‘worst’ extension, both problems of computing and verifying Pareto optimal outcomes admit polynomial-time algorithms.

We also consider the requirement of strategyproofness on top of Pareto optimality. We show that there exist linear-time Pareto optimal and strategyproof algorithms for committee voting even for weak preferences for four of the five set extensions. The algorithms can be considered as careful adaptations of serial dictatorship for committee voting.

Computation Verification
Set Extension
Responsive (RS) in (Th. 5.5) coNP-C (Th. 5.2), W[2]-hard
in P—dich. prefs and (Th. 5.4)
Downward Lexicographic (DL) in (Th. 7.1) coNP-C (Th. 1), W[2]-hard
Upward Lexicographic (UL) in (Th. 9.1) coNP-C (Th. 2), W[2]-hard
Best () NP-hard (Th. 6.2) coNP-C, W[2]-hard (Th. 6.1)
in P for strict prefs
Worst () in (Th. 8.2) in P (Th. 8.1)
Table 1: Complexity of computing and verifying Pareto optimal committees. (coined by Christos Papadimitriou in a seminar at Simons Institute in 2015) indicates a class of problems in which agents provide the input and the problems admit a strategyproof and polynomial-time algorithm.

2 Related Work

A first related stream of work involves studying specific committee elections rules from a computational point of view (generally with little or no focus on Pareto optimality). Our focus on determining whether a committee is Pareto optimal or on finding a Pareto optimal committee, is in some sense orthogonal to the study of committee election rules. The simplest (and most widely used) rules for electing a committee, called best- rules, compute a score for each alternative based on the ranks, and the alternatives with the best scores are elected Elkind et al. [2014], Faliszewski et al. [2016]. Scoring-based extension principles have also been used by Darmann [2013]. Note that the output of a best- rule is obviously Pareto-optimal for the preferences induced by this scoring function, but not necessarily with respect to other set extensions.

Klamler et al. Klamler et al. [2012] compute optimal committees under a weight constraint for a single agent (therefore optimality is equivalent to Pareto optimality), using several preference extensions including ‘worst’, ‘best’, and downward lexicographic.

The ‘best’ () extension principle has been used in a number of papers on committee elections by full proportional representation, starting with [Chamberlin and Courant, 1983] and studied from a computational point of view in a long series of papers (e.g., [Procaccia et al., 2008, Lu and Boutilier, 2011, Betzler et al., 2013, Skowron et al., 2015a, Elkind and Ismaili, 2015]. These rules obviously output Pareto optimal committees for , but not necessarily for other extensions.

Some of the set extensions considered in this paper have corresponding analogues when extending preferences over alternatives to preferences over ‘lotteries over alternatives.’ In particular, the RS set extension corresponds to SD (stochastic dominance) lottery extension. Also the DL and UL set extensions considered in this paper correspond to DL and UL lottery extensions considered in works in probabilistic social choice [Brandl, 2013, Aziz et al., 2013b, Cho, 2016].

Some works are based on the Hamming extension. Each agent specifies his ideal committee and he prefers committees with less Hamming distance from the ideal committee. The Hamming distance notion can be used to define specific rules such as minimax approval voting [Brams et al., 2007], which selects the committee minimizing the maximum Hamming distance for the agents. Although the output of minimax approval voting is not always Pareto-optimal for the Hamming extension, there are good Pareto-optimal approximations of it Caragiannis et al. [2010]. Note that for dichotomous preferences, the Hamming extension coincides with the responsive and the downward lexicographic extensions, therefore our computational results for responsive set extension for dichotomous preferences also hold for the Hamming and downward lexicographic extensions.

A second line of work concerns understanding the classes of rules that result in Pareto optimal outcomes. Most works along this line bear on a different type of committee elections, called designated-seat voting, where candidates must declare the seat they contest [Benoît and Kornhauser, 2010].111If there are exactly two candidates per seat, then designated voting is equivalent to multiple referenda, where a decision has to be taken on each of a series of yes-no issues. Results about the existence or non-existence of Pareto optimal rules have been presented Özkal-Sanver and Sanver [2006], Benoît and Kornhauser [2010], Cuhadaroǧlu and Lainé [2012].

3 Setup

We consider a set of agents , a set of alternatives and a preference profile such that each is a complete and transitive relation over . We write  to denote that agent  values at least as much as and use  for the strict part of , i.e.,  iff  but not . Finally, denotes ’s indifference relation, i.e., iff both  and .

The relation results in equivalence classes for some such that if and for some . We will use these equivalence classes to represent the preference relation of an agent as a preference list . For example, we will denote the preferences by the list . An agent ’s preferences are strict if the size of each equivalence class is 1. An agent ’s preferences are dichotomous if he partitions the alternatives into just two equivalence classes, i.e., . Let be the maximum size of the most preferred equivalence class, i.e., . For any , we will denote by and the alternatives in that are maximally and minimally preferred by respectively. Thus, if and are respectively the smallest and the largest indices such that and , then and . For , let .

4 Set Extensions and Pareto Optimality

Set Extensions

Set extensions are used for reasoning about the preferences of an agent over sets of alternatives given their preferences over single alternatives. For fixed-size committee voting, the responsive extension is very natural and has been applied in various matching settings as well [Barberà et al., 2004, Roth and Sotomayor, 1990]. For all , we say that if and only if there is an injection from to such that for each , agent weakly prefers to , i.e. .

We define the ‘best’ set extension and the ‘worst’ set extension which are denoted and respectively. For all , if and only if for and . On the other side, if and only if for and .

In the downward lexicographic (DL) extension, an agent prefers a committee that selects more alternatives from his most preferred equivalence class, in case of equality, the one with more alternatives for the second most preferred equivalence class, and so on. Formally, iff for the smallest (if any) with we have .

In the upward lexicographic (UL) extension, an agent prefers a committee that selects less alternatives from his least preferred equivalence class, in case of equality, the one with less alternatives for the second least preferred equivalence class, and so on. Formally, iff for the largest (if any) with we have .

Remark 1.

Consider an agent with preferences over . Let such that . Then,

The relations follow from the definitions.

Efficiency based on Set Extensions

With each set extension , we can define Pareto optimality with respect to . A committee is Pareto optimal with respect to , or simply -efficient, if there exists no committee such that for all and for some . Note that for each of our set extensions, -efficiency coincides with standard Pareto optimality when . An outcome is a Pareto improvement over another if each agent weakly improves and at least one agent strictly improves.

Example 1.

Consider the preference profile:

Suppose . Then,

  • The unique -efficient committee is

  • The unique -efficient committee is .

  • The DL-efficient committees are , , and .

  • The UL-efficient committees are , , and .

  • The RS-efficient committees are , , , and ,

Remark 2.

Consider a committee .

  • If is -efficient, then is -efficient

  • If is -efficient, then is -efficient

The argument is as follows. Suppose is not -efficient, then there exists some other outcome such that for all and for some . In that case for all and for some . Also for all and for some . Hence is neither -efficient not -efficient.

Remark 3.

There always exists a -efficient committee that is also DL-efficient: DL Pareto improvements over a -efficient does not harm any agent with respect to the relation.

Remark 4.

There always exists a -efficient committee that is also UL-efficient: UL Pareto improvements over a -efficient does not harm any agent with respect to the relation.

In Figure 1, we illustrate the relations between the different efficiency notions. Later on in the paper we will present an algorithm that returns a committee that is -efficient and -efficient, and hence -efficient.

-efficiency

-efficiency

-efficiency

-efficiency

-efficiency
Figure 1: Relations between the five notions of efficiency. An arrow from -efficiency to -efficiency means that -efficiency implies -efficiency; a dashed line means there always exists a committee that is both - and -efficient; absence of arrow or line means that the sets of - and -efficient committees can be disjoint.

We also make the following general observation.

Lemma 1.

If there is a polynomial-time algorithm to compute a Pareto improvement over a committee, then there exists a polynomial-time algorithm to compute an -efficient committee under set extensions .

Proof.

Here, we start from any committee and we recursively apply Pareto improvement until we reach a Pareto optimal committee. For the ‘best’ and ‘worst’ extensions, there can be at most Pareto improvements because for one agent there can be at most improvements. Since an -improvement implies an -improvement, let us bound the number of Pareto-improvements with respect to . In each Pareto-improvement, for the agent who strictly improves, the most preferred equivalence class that has different number of alternative in the outcome increases by at least one. Therefore the most preferred equivalence class can be the improving class in at most of the Pareto improvements. Similarly, the number of Pareto improvements in the subsequent less preferred equivalence class improves in a Pareto improvement can be at most of the Pareto improvements. Therefore the total number of DL Pareto-improvements is bounded by . A similar argument holds for UL as well. ∎

We end this section by observing that, under any of the set extensions we consider, a set of Pareto optimal alternatives may be Pareto dominated. Consider the following example.

Example 2.

The set consists of Pareto optimal alternatives but is Pareto dominated by under any of our set extensions.

5 Responsive Set Extension

There is a trivial way to achieve Pareto optimality under the responsive set extension by taking any decreasing scoring vector consistent with the ordinal preferences, finding the total score of each alternative and returning the set of

alternatives with the maximum scores. For instance, on Example 2, the outcome of the rule that outputs the alternatives with the best Borda scores is .

Theorem 5.1.

A Pareto optimal committee under the responsive set extension committee can be computed in linear time.

In many situations, one may already have a status-quo committee and one may want to find a Pareto improvement over it. This problem of testing Pareto optimality and finding a Pareto improvement under the responsive set extension turns out to be a much harder task. Note that if there exists a polynomial-time algorithm to compute a Pareto improvement, then it means that testing Pareto optimality is also polynomial-time solvable.

Theorem 5.2.

Checking whether a committee is Pareto optimal under the responsive set extension is coNP-complete, even for dichotomous preferences and , or for strict preferences.

Proof.

We only present the case where . The reduction is from the NP-complete problem vertex cover Garey and Johnson [1979]. Given a simple graph , the minimum vertex cover problem consists in finding a subset of minimum size such that every edge is incident to some node of . Its decision version vertex cover takes as input a simple graph and an integer and problem is deciding if there exists a vertex cover of with .

Let be an instance of vertex cover, with being one arbitrary edge in . We build the following instance of Pareto optimality under :

  • , where for each edge , is a set of agents, and is a special agent.

  • , where .

  • For each , the preferences of agent , for , and of agent , are

The reduction is clearly done within polynomial time and preferences are dichotomous. We can check easily that committee (of size ) is not Pareto optimal under if and only if there exists a vertex cover of of size at most .

For strict preferences, in the previous reduction we replace by in the preferences of . It is easy to see that the proof is similar. ∎

Using a similar reduction from the Hitting Set problem, we can also prove Theorem 5.3 that concerns a parametrized complexity intractability result Downey and Fellows [2013]. Hitting Set is defined as follows: given a ground set of elements, and a collection of subsets of , does there exist a such that and for all ?

Theorem 5.3.

Checking whether a committee is Pareto optimal under the responsive set extension is W[2]-complete under parameter , even for dichotomous preferences.

For dichotomous preferences we present a complete characterization of the complexity according to the parameter. If , then in any Pareto improvement over committee , any alternative in that is most preferred by some agent needs to be kept selected, and therefore the problem of checking -efficiency is easy. If , from Theorem 5.2, the problem is hard. Remains the case .

Theorem 5.4.

For dichotomous preferences, a Pareto improvement over a committee with respect to the responsive set extension can be computed in polynomial time when .

Proof.

Consider a preference profile where each is dichotomous and verifies , and let . For each , let be the partition associated with .

First, if for all , , then is obviously -efficient. Assume it is not the case, that is, (1) for some , . Let

  • , (by construction, ), and .

  • and .

Now, we build a graph with isomorphic to , and iff for some : each edge of corresponds to the top two alternatives of some agent, provided one of them is in . Let be the size of an optimal vertex cover of .

We first claim that there is a Pareto improvement over if and only if one of follows two conditions is satisfied:

  • , or

  • , and there is an optimal vertex cover of containing either at least an element of for some , or two elements of for some .

We first show that (i) and (ii) are sufficient. If (i) holds then take a committee corresponding to a minimum vertex cover of , add to it the alternatives of , and add alternatives, with at least one in ; this is possible because of (1). If (ii) holds, then take a committee corresponding to a minimum vertex cover of , and add to it the alternatives of . In both cases, the obtained committee contains for all , contains at least one element of for all , and contains either two elements of for some , or an element of for some . Therefore it is a Pareto-improvement over .

Now, we show that (i) and (ii) are necessary. Let be a Pareto improvement of containing a maximum number of alternatives from . We have the following two properties: and is a vertex cover of . holds, since otherwise there would be an such that does not hold. For similar reasons, is a vertex cover of . If , then by adding to it any set of of size we obtain a set of size which constitutes a Pareto improvement of because now, for some . If , then and necessarily either for some or for some .

It remains to be shown that (i) and (ii) can be checked in polynomial time. (i) can be done in polynomial-time because is bipartite: indeed, by construction, is two-colorable with color sets and , and by König’s theorem, for bipartite graphs, the problem of finding the minimum vertex cover is equivalent to computing a maximum matching, hence solvable in polynomial time. As for (ii), if , we have to check whether for some optimal vertex cover of , either (ii.1) holds for some , or (ii.2) for some . In order to check (ii.1), for each such that there exists , we transform into a new bipartite graph where we add a new vertex and an edge . In order to check (ii.2), for each , let ; we transform into a new bipartite graph where we add two new vertices and , and two edges and . Finally, we test if or if for one of these graphs, because all optimal vertex covers of (respectively ) must contain (respectively ). ∎

Example 3.

We illustrate the algorithm in the proof of Theorem 5.4. Let and consider the dichotomous profile, where we specify only the top equivalence class of each agent:

Let . We have , , , , and . We construct the graph : and . We have . Now we consider the four graphs , resulting from the addition to of a new vertex and edge , and , and : results from the addition to of two new vertices and edges and , etc. Two of these graphs have an optimal cover of size 2: , with optimal cover , and , with optimal cover . Therefore, and are -Pareto-improvements over , and is not -efficient.

Figure 2: Graphs corresponding to Example 3

Note that finding an algorithm that computes a Pareto improvement over a committee can be used to decide whether a given a committee of size , is Pareto optimal under the responsive set extension.

Pareto optimality and Strategyproofness

We now try to achieve both -efficiency and strategyproofness simultaneously. A mechanism is strategyproof if reporting truthful preferences is a dominant strategy with respect to the responsive set extension: for all preference profiles and . Note that defining strategyproofness in this way with respect to the RS extension is stronger than defining it for any of the other four extensions considered in this paper. Nonetheless, we will present some positive results with respect to strategyproofness.

A naive way of achieving -efficiency and Pareto optimality is to enumerate the list of possible winning sets and implement serial dictatorship over the possible outcomes as is done in voting [Aziz et al., 2013b]. However, the number of possible outcomes is exponential and responsive preferences result in a partial order over the possible winning sets and not a complete and transitive order. This problem is solved by Algorithm 1 which can be viewed as a computationally efficient serial dictatorship.

Input:
Output: .

1    (last set to be refined)
2    (number of alternatives yet to be fixed) ;
3    (index of the permutation )
4   while  or  do
5      Agent selects first equivalence classes such that and .
6       (we say agent fixes the alternatives in );
7      
8       ;
9      Increment by one
10   end while
11   if  then
12      pick any alternatives from and add them to
13   end if
14   return  
Algorithm 1 Committee Voting Serial Dictatorship
Theorem 5.5.

There exists a linear-time and strategyproof algorithm that returns a committee that is Pareto optimal under the responsive set extension.

Proof.

Consider Algorithm 1. We show that at each stage , agent , implicitly refines the set of feasible committees to the maximal set of most preferred outcomes from the set by providing additional constraints. This is true for the base case . Now assume it holds from to . Note that contains all those alternatives that are strictly less preferred by agents in than the ones they respectively fixed. Moreover, each agent in is indifferent between the alternatives in . As for , he fixes the best alternatives in where is the value such that and . For , the agent only requires that alternatives are selected from his equivalence class which is ensured by the definition of the algorithm. It follows from the argument that the returned set is Pareto optimal under the responsive set extension. For strategyproofness, when an agent turn comes, it only has a choice over fixing the alternatives in and requiring alternatives from his equivalence class . In this case the algorithm already chooses one of the best possible committees for the agent. ∎

Note that for , the algorithm is equivalent to serial dictatorship as formalized by Aziz et al. [2013a]. Note that a committee that is Pareto optimal under the responsive set extension may not be a result of serial dictatorship. This holds even for and the basic voting setting.

The problem with the serial dictatorship algorithm formalized is that it overly favours the agent that is the first in the permutation. One way to limit his power is to let him choose only alternatives. We note that this attempt at having a fairer extension of serial dictatorship comes at an expense because strategyproofness is compromised. Consider the profile in which 1 has preferences and 2 has preferences . For , and permutation , the outcome is . But if agent 1 reports , then the outcome is .

6 ‘Best’ Set Extension

Next, we consider Pareto optimality with respect to , which has been used for defining many rules (see Section 2).

Theorem 6.1.

Unless PNP, there is no polynomial-time algorithm to compute a Pareto improvement over a committee with respect to , even for dichotomous preferences and .

Proof.

We show that if it is not the case, then we can solve polynomially the vertex cover decision problem. Consider an instance of vertex cover given by a simple graph with and , and an integer . Assume the existence of a polynomial-time algorithm Algo that computes a Pareto improvement over a committee with respect to when : given a profile and a set of alternatives , Algo returns, in time polynomial in , Yes if is Pareto optimal with respect to , and otherwise returns a -set of alternatives which Pareto dominates . We will now prove by applying at most times Algo with different inputs that we can decide in polynomial if has a vertex cover of with . We construct the following profile :

  • The set of agents is , where agent corresponds to edge .

  • The set of alternatives is where .

  • Let be an edge of ; the preferences of agent for are:

    The preferences of the last set of agents are given by: for

The reduction is clearly done within polynomial time and the set of preferences given by are dichotomous.

Consider the following inductive procedure: and for , Algo if is not Pareto optimal with respect to , otherwise we return . Let be the solution output after calls. Because Algo is polynomial, the whole procedure is polynomial.

We claim that has a vertex cover of size iff . We will first prove by induction that at each step , is a vertex cover of . For the initial step, it is valid because is a vertex cover of . Assume that it is true for and let us prove that is a vertex cover of . If it is not the case, some edge is not covered. By assumption, is covered by . This implies , which is a contradiction. Hence, . From this hypothesis, we deduce for with a strict preference for some agent. Equivalently, . In conclusion, after recursive calls, if and only if . ∎

Theorem 6.2.

Computing a -efficient committee is NP-hard, even for dichotomous preferences.

Proof.

We give a reduction from Hitting Set. Let , and for each , ’s dichotomous preferences are If there exists a polynomial-time algorithm to compute a -efficient committee, it will return a committee in which each agent gets a most preferred alternative if such a committee exists. But such a committee corresponds to a hitting set of size . ∎

7 Downward Lexicographic Set Extension

We point out that for dichotomous preferences, the responsive set extension coincides with the downward lexicographic set extension. Hence we get a corollary of our results for responsive preferences:

Corollary 1.

Checking whether a committee is -efficient is coNP-complete, even for dichotomous preferences and .

Note that Algorithm 1 returns a -efficient committee. The reason is that each agent in her turn refines the set of possible outcomes to her most preferred subset of outcomes. Each committee is the refined set is at least as preferred with respect to RS (and hence with respect to DL) to all committees in the set of possible outcomes.

Theorem 7.1.

There exists a linear-time and strategyproof algorithm that returns a -efficient committee.

8 ‘Worst’ Set Extension

In contrast to all the other set extensions considered in the paper, Pareto optimality with respect to the ‘worst’ set extension can be checked in polynomial time.

Theorem 8.1.

There exists a polynomial-time algorithm that checks whether a committee is -efficient and computes a Pareto improvement over it if possible.

Proof.

Let . For each , let be the least preferred equivalence class such that . We want to check whether there is a -set of alternatives in which at least some agent gets a strictly better outcome and all the other agents get at least as preferred an outcome. We check this as follows. For , let . We check whether or not. If , we know that there exists a subset of , that is strictly more preferred by and at least as preferred by each agent. The reason is that contains a more preferred worst alternative for agent than and contains at least as preferred worst alternative for other agents than . If , then this means that a Pareto improvement with strictly improving is only possible if the size of the winning set is less than which is not feasible. ∎

We now consider strategyproofness together with -efficiency. We first note that Algorithm 1 may not return a -efficient outcome. However, we construct a suitable strategyproof and -efficient by formalising an appropriate serial dictatorship algorithm for the worst set extension.

Theorem 8.2.

There exists a linear-time and strategyproof algorithm that returns a -efficient committee.

Proof.

Consider the agents in a permutation . The set of alternatives is initialized to . We reduce the set while ensuring that it of size at least . The next agent in the permutation comes and deletes the maximum number of least preferred equivalence classes from his preferences and the corresponding alternatives in while ensuring that . Each successive agent in the permutation gets a most preferred outcome while ensuring that agents before him in the permutation get at least as preferred an outcome as before. Thus the algorithm is strategyproof and Pareto optimal with respect to the ‘worst’ set extension. ∎

9 Upward Lexicographic Set Extension

We point out that for dichotomous preferences, the responsive set extension coincides with the upward lexicographic set extension. Hence we get a corollary of our results for responsive preferences:

Corollary 2.

Checking whether a committee is -efficient is coNP-complete, even for dichotomous preferences and .

Note that Algorithm 1 returns a -efficient committee. The reason is that each agent in her turn refines the set of possible outcomes to her most preferred subset of outcomes. Each committee is the refined set is at least as preferred with respect to RS (and hence with respect to UL) to all committees in the set of possible outcomes.

Theorem 9.1.

There exists a linear-time and strategyproof algorithm that returns a -efficient committee.

10 Conclusions

We considered Pareto optimality in multi-winner voting with respect to a number of prominent set extensions. We presented results on the relations between the notions as well as complexity of computing and verifying Pareto optimal outcomes. Further directions of future work include considering Pareto optimality with respect to other set extensions [Brandt and Brill, 2011]. Another direction is consider the compatibility of Pareto optimality concepts with other axioms. Finally, we remark that our serial dictatorship algorithm can be used to define a multiwinner generalization of random serial dictatorship, which is worth investigating and raises interesting computational problems.

Acknowledgments

This is the extended version of the IJCAI conference paper [Aziz et al., 2016]. The authors thank Felix Brandt for useful pointers and comments. They also thanks the reviewers and attendees of IJCAI 2016 and COMSOC 2016 for useful comments. Jérôme Lang and Jérôme Monnot thank the ANR project CoCoRICo-CoDec.

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