A university wants to select the two favorite compositions in classical style to be played during the next graduation ceremony. The students were asked to submit their favorite compositions. Then a jury consisting of seven members (three juniors and four seniors) from the university staff selects from the six most frequently submitted compositions as follows: Each jury member approves two compositions and the two winners are those obtaining most of the approvals. The six options provided by the students are “Beethoven: Piano Concerto No. 5 ()”, “Beethoven: Symphony No. 6 ()”, “Mozart: Clarinet Concerto ()”, “Mozart: Requiem K626 ()”, “Uematsu: Final Fantasy ()”, and “Badelt: Pirates of the Caribbean ()”. The three junior jury members are excited about recent audio-visual presentation arts (both interactive and passive) and approve and . Two of the senior jury members are Mozart enthusiasts, and the other two senior jury members are Beethoven enthusiasts. Hence, when voting truthfully, two of them would approve the two Mozart compositions and the other two would approve the two Beethoven compositions. The winners of the selection process would be and , both receiving three approvals whereas every other composition receives only two approvals.
The senior jury members meet every Friday evening and discuss important academic issues including the graduation ceremony music selection processes, why movie background noise recently counts as classical music,111http://www.classicfm.com/radio/hall-of-fame/ and the influence of video games on the ability of making important decisions. During such a meeting they agreed that a graduation ceremony should always be accompanied by a composition of a first-class composer and by another composition of a second-class composer (two compositions of a first-class composer may overstrain students and junior researchers). Thus, finally all four senior jury members approve and making these two compositions playing during the graduation ceremony.
Already this little example above (which will be the base of our running example throughout the paper) illustrates important aspects of strategic voting in multiwinner elections. In case of coalitional manipulation for single-winner elections (where a coalition of voters casts untruthful votes in order to influence the outcome of an election; a topic which has been intensively studied in the literature (Rothe, 2015; Brandt et al., 2016)) one can always assume that a coalition of manipulators agrees on trying to make a distinguished alternative win the election. In case of multiwinner elections, however, already determining concrete possible goals of a coalition seems to be a non-trivial task: There may be exponentially many different outcomes which can be reached through strategic votes of the coalition members and each member could have its individual evaluation of these outcomes.
Multiwinner voting rules come up very naturally whenever one has to select from a large set of candidates a smaller set of “the best” candidates. Surprisingly, although at least as practically relevant as single-winner voting rules, the multiwinner literature is much less developed than the single-winner literature. In recent years (see a survey of Faliszewski et al. (2017a)), however, research into multiwinner voting rules, their properties, and algorithmic complexity grew significantly (Aziz et al., 2017a; Betzler et al., 2013; Aziz et al., 2017b, 2015; Barrot et al., 2013; Barberà and Coelho, 2008, 2010; Elkind et al., 2017; Faliszewski et al., 2016, 2017b; Meir et al., 2008; Obraztsova et al., 2013; Skowron, 2015; Skowron et al., 2015). When selecting a group of winning candidates, various goals can be interesting; namely, proportional representation, diversity, or a short list (see Elkind et al. (2017)). We focus on the last scenario. Here the goal is to select the best (say highest-scoring) group of candidates.
Shortlisting comes very naturally in the context of selection committees; for instance, for human resources departments that need to select, for a fixed number of positions, the best qualified applicants. A standard way of candidate selection in the context of shortlisting is to use scoring-based voting rules. We focus on the two most natural ones: SNTV (single non-transferable vote—each voter gives one point to one candidate) and -Bloc (each voter gives one point to each of different candidates, so SNTV is the same as -Bloc). Obviously, for such voting rules it is trivial to determine the score of each individual candidate. The main goal of our work is to model and understand coalitional manipulation in a computational sense—that is, to introduce a formal description of how a group of manipulators can influence the election outcome by casting strategic votes and whether it is possible to find an effective strategy for the manipulators to change the outcome in some desired way. In this fashion, we complement well-known work: manipulation for single-winner rules initiated by Bartholdi III et al. (1989), coalitional manipulation for single-winner rules initiated by Conitzer et al. (2007), and (non-coalitional) manipulation for multiwinner rules initiated by Meir et al. (2008). In coalitional manipulation scenarios, given full knowledge about other voters’ preferences, one has a set of manipulative voters who want to influence the election outcome in a favorable way by casting their votes strategically.
To come up with a useful framework for coalitional manipulation for multiwinner elections, we first have to identify the exact mathematical model and questions to be asked. The straightforward extensions of coalitional manipulation for single-winner elections or (non-coalitional) manipulation for multiwinner elections do not fit. Extending the single-winner variant directly, one would probably assume that the coalition agrees on making a distinguished candidate part of the winners or that the coalition agrees on making a distinguished candidate group part of the winners. The former is unrealistic because in multiwinner settings one typically cares about more than just one candidate—especially in shortlisting it is natural that one wants rather some group of “similarly good” candidates to become part of winners instead of only one representative of such a group. Agreeing on a distinguished candidate group to be part of the winners is also problematic since there may be exponentially many “equally good” candidate groups for the coalition. Notably, this was not a problem in the single-winner case; there, one can test for a successful manipulation towards each possible candidate avoiding an exponential increase of the running time (compared to the running time of such a test for a single candidate). The single-manipulator model for multiwinner rules ofMeir et al. (2008) is a helpful first step: the manipulator specifies the utility of each candidate; the utility for a candidate group is obtained by adding up the utilities of each group member. Aggregating utilities, however, becomes non-trivial for a coalition of manipulators which may have totally different utilities for single candidates but still have strong incentives to work together (e.g., as we have seen in our introductory example). Besides formalizing this either in a utilitarian or egalitarian way, our modeling also aims to distinguish between optimistic and pessimistic manipulators. In case of a tie, optimistic manipulators assume that the tie is broken in favor of them while pessimistic ones assume the opposite scenario. It turns out that this difference in manipulators’ behavior leads to significant differences in terms of computational complexity. Technically, analyzing the issues discussed above requires to study tie-breaking mechanisms and (winning) group evaluation functions.
Related Work. To the best of our knowledge, there is no previous work on coalitional manipulation in the context of multiwinner elections. We refer to recent textbooks for an overview of the huge literature on single-winner (coalitional) manipulation (Rothe, 2015; Brandt et al., 2016). Most relevant to this paper, Lin (2011) showed that coalitional manipulation in single-winner elections under -Approval is solvable in linear time by a greedy algorithm. Meir et al. (2008) introduced (non-coalitional) manipulation for multiwinner elections. While identifying manipulation for several voting rules as NP-hard problems, they showed that manipulation remains polynomial-time solvable for Bloc (which can be interpreted as a multiwinner equivalent of -Approval). Obraztsova et al. (2013) extended the latter result for different tie-breaking strategies and identified further tractable special cases of multiwinner scoring rules but conjectured manipulation to be hard in general for (other) scoring rules. Summarizing, Bloc is simple but comparably well-studied and was, hence, selected as a showcase for our study of the presumably computationally harder coalitional manipulation.
Organization. Section 2 introduces basic notations and formal concepts. In Section 3, we develop our model for coalitional manipulation in multiwinner elections. Its variants respect different ways of evaluating candidate groups (utilitarian vs. egalitarian) and two kinds of manipulators behavior (optimistic vs. pessimistic). In Section 4, we present algorithms and complexity results for computing the output of several tie-breaking rules that allow to model optimistic and pessimistic manipulators. In Section 5, we formally define the coalitional manipulation problem and explore its computational complexity using -Bloc as a showcase. We refer to our conclusion and Table 1 (Section 6) for a detailed overview of our findings.
For a positive integer , let . We use the toolbox of parameterized complexity (Cygan et al., 2015; Downey and Fellows, 2013; Flum and Grohe, 2006; Niedermeier, 2006) to analyze the computational complexity of our problems in a fine-grained way. To this end, we always identify a parameter that is typically a positive integer. We call a problem parameterized by fixed-parameter tractable (in ) if it is solvable in time, where is the size of a given instance encoding, is the value of the parameter, and is an arbitrary computable (typically super-polynomial) function. To preclude fixed-parameter tractability, we use an established complexity hierarchy of classes of parameterized problems, . It is widely believed that all inclusions are proper. The notions of hardness for parameterized classes are defined through parameterized reductions similar to classical polynomial-time many-one reductions—in this work, it suffices to ensure that the value of the parameter in the problem we reduce to depends only on the value of the parameter of the problem we reduce from. Occasionally, we use a combined parameter which is a more explicit way of expressing a parameter .
An election consists of a set of candidates and a multiset of votes. Votes are linear orders over —for example, for we write to express that candidate is the most preferred and candidate is the least preferred according to vote . We write if the corresponding vote is clear from the context.
A multiwinner voting rule222Some literature use the name multiwinner voting correspondence for what we called multiwinner voting rule. In that case, a multiwinner voting rule is required to return exactly one set of the desired size. This is usually achieved by combining a multiwinner voting correspondence with a tie-breaking rule. is a function that, given an election and an integer , outputs a family of co-winning size- subsets of representing the co-winning -excellence-groups. We use -egroup as an abbreviation for -excellence-group. The reason we do not use the established term “committee” is that in shortlisting applications “committee” traditionally rather refers to voters and not to candidates.
We consider scoring rules which assign points to candidates based on their positions in the votes. By , we denote the total number of points that candidate obtains, and we use when restricting the election to a subset of voters. A (multiwinner) scoring rule selects a family of co-winning -egroups with the maximum total sum of scores. It holds that if and only if . We focus on the family of -Bloc multiwinner voting rules which assign, for each vote, one point to each of the top candidates.333The case where coincides with the size of the egroup is typically referred to as Bloc; -Bloc corresponds to SNTV (Meir et al., 2008). The case where is also referred to as Limited Vote (or Limited Voting).
Referring back to our introductory example, we have a set of candidates and a set of voters . The voters , , and represent the three junior jury members, whereas , and , represent, respectively, the Beethoven and Mozart enthusiasts among the senior jury members. In the example, we described a way of manipulating the election by the senior jury members which leads to selecting two music pieces. There are several ways to illustrate this manipulation using our model. Below we present one of the possible sets of casted votes that represents the manipulated election:
Following the introductory example, we are choosing an egroup of size . Using the Bloc multiwinner voting rules (which coincides with our introductory example), the winning -egroup consist of candidates and . However, under the SNTV voting rule the situation would change, and the winners would be and . SNTV and Bloc alike output a single winning egroup in this example, and thus tie-breaking is negligible.
To select a single -egroup from the set of co-winning -egroups one has to consider tie-breaking rules. A multiwinner tie-breaking rule is a mapping that, given an election and a family of co-winning -egroups, outputs a single -egroup. Among them, there is a set of natural rules that is of particular interest in order to model the behavior of manipulative voters. Indeed, in case of a single manipulator both pessimistic tie-breaking as well as optimistic tie-breaking have been considered in addition to lexicographic and randomized tie-breaking (Meir et al., 2008; Obraztsova et al., 2013). To model optimistic and pessimistic tie-breaking in a meaningful manner444We cannot simply use ordinal preferences: Obraztsova et al. (2013) observed that already in case of a single manipulator one cannot simply set the fixed lexicographic order of the manipulators’ preferences (resp. the reverse of it) over candidates to model optimistic (resp. pessimistic) tie-breaking. For example, it is a strong restriction to assume that a manipulator would always prefer its first choice together with its fourth choice towards its second choice together with its third choice. It might be that only its first choice is really acceptable (in which case the assumption is reasonable) or that the first three choices are comparably good but the fourth choice is absolutely unacceptable (in which case the assumption is wrong)., we use the model introduced by Obraztsova et al. (2013) in which a manipulative voter is described not only by the preference order of the candidates but also by a utility function . To cover this in the tie-breaking process, coalition-specific tie-breaking rules get—in addition to the original election, the manipulators’ votes, and the co-winning excellence-groups—the manipulators’ utility functions in the input. The formal implementations of these rules and their properties are discussed in Subsection 3.2.
3 Model for Coalitional Manipulation
In this section, we formally define and explain our model and the respective variants. To this end, we discuss how we evaluate a -egroup in terms of utility for a coalition of manipulators and introduce tie-breaking rules that model optimistic or pessimistic viewpoints of the manipulators.
3.1 Evaluating -egroups
As already discussed in the introduction, one should not extend the model of coalitional manipulation for single-winner elections to multiwinner elections in the simplest way (e.g., by assuming that the manipulators agree on some distinguished candidate or on some distinguished egroup). Instead, we follow Meir et al. (2008) and assume that we are given a utility function over the candidates for each manipulator and a utility level which, if achieved, indicates a successful manipulation. Meir et al. (2008) compute the utility of an egroup by summing up the utility values the manipulator assigns to each member of the egroup.
At first glance, summing up the utility values assigned by each manipulator to each member of an egroup seems to be the most natural extension for a coalition of manipulators. However, this utilitarian variant does not guarantee single manipulators to gain non-zero utility. In extreme cases it could even happen that some manipulator is worse off compared to voting sincerely, as demonstrated in Example 2.
Consider the election where is a set of candidates and is the following multiset of three votes:
Additionally, consider two manipulators, and , that report utilities to the candidates as depicted in the table below.
Let us analyze the winning -egroup under the SNTV voting rule. Observe that if the manipulators vote sincerely, then together they give one point to and one to (one point from each manipulator). Combining the manipulators’ votes with the non-manipulative ones, the winning -egroup consists of candidates and that both have score two; no other candidate has greater or equal score, so tie-breaking is unnecessary. The value of such a group is equal to seven according to the utilitarian evaluation variant. Manipulator ’s utility is seven. However, both manipulators can do better by giving their points to candidate . Then, the winners are candidates and , giving the total utility of (according to the utilitarian variant). Observe that in spite of growth of the total utility, the utility value gained by , which is one, is lower than in the case of sincere voting.
In Example 2 manipulator devotes its satisfaction to the utilitarian satisfaction of the group of the manipulators; that is, is worse off voting strategically compared to voting sincerely. Despite this issue, however, the utilitarian viewpoint can be justified if the manipulators are able to compensate such losses of utility of some manipulators, for example, by paying money to each other. For cases where manipulators cannot do that, we introduce two egalitarian evaluation variants. The (egroup-wise) egalitarian variant aims at maximizing the minimum satisfaction of the manipulators with the whole -egroup. The candidate-wise egalitarian variant aims at maximizing the manipulators’ satisfaction resulting from the summation of the minimum satisfactions every single candidate contributes. We do not distinguish “candidate-wise utilitarian” variant since this variant would be equivalent to the (regular) utilitarian variant.
We formalize the described variants of -egroup evaluation (for manipulators) with Definition 1.
Given a set of candidates , a -egroup , , and a family of manipulators’ utility functions where , we consider the following functions:
Intuitively, these functions determine the utility of a -egroup according to, respectively, the utilitarian and the egalitarian variants of evaluating by a group of manipulators (identifying manipulators with their utility functions). We omit subscript when is clear from the context. To illustrate Definition 1 we apply it in Example 3.
Consider our example set of candidates and two manipulators , whose utility functions over the candidates are depicted in the table below.
Then, evaluating the utility of -egroup applying the three different evaluation variants gives:
Analyzing Example 3, we observe that we can compute the utilitarian value of an egroup by summing up the overall utilities each candidate contributes to all manipulators. Analogously, we can deal with the candidate-wise egalitarian variant by taking the minimum utility associated to each candidate as the overall utility of this candidate. In both variants we obtain a single utility function.
Without loss of generality, one can assume that there is a single non-zero valued utility function over the candidates under the utilitarian or candidate-wise egalitarian evaluation.
3.2 Breaking Ties
Before formally defining our tie-breaking rules, we briefly discuss some necessary notation and central concepts. Consider an election , a size for the egroup to be chosen, and a scoring-based multiwinner voting rule . We can partition the set of candidates into three sets , , and as follows: The set contains the confirmed candidates, that is, candidates that are in all co-winning -egroups. The set contains the pending candidates, that is, candidates that are only in some co-winning -egroups. The set contains the rejected candidates, that is, candidates that are in no co-winning -egroup. Observe that , , and that every candidate from receives fewer points than every candidate from . Additionally, all candidates in receive the same number of points.
We define the following families of tie-breaking rules which are considered in this work. In order to define optimistic and pessimistic rules, we assume that in addition to , , and , we are given a family of utility functions which are used to evaluate the -egroups as discussed in Subsection 3.1.
Lexicographic . A tie-breaking belongs to if and only if ties are broken lexicographically with respect to some predefined order of the candidates from . That is, selects all candidates from and the top candidates from with respect to .
Optimistic , . A tie-breaking belongs to if and only if it always selects some -egroup such that and there is no other -egroup with and .
Pessimistic , . A tie-breaking belongs to if and only if it always selects some -egroup such that and there is no other -egroup with and .
3.3 Limits of Lexicographic Tie-Breaking
From the above discussion, we can conclude that lexicographic tie-breaking is straightforward in the case of scoring-based multiwinner voting rules. Basically any subset of the desired cardinality from the set of pending candidates can be chosen. In particular, the best pending candidates with respect to the given order can be chosen. We remark that applying lexicographic tie-breaking may be more complicated for general multiwinner voting rules.
It remains to be clarified whether one can find a reasonable order of the pending candidates in order to model optimistic or pessimistic tie-breaking rules in a simple way. We show that this is possible for every , , , using the fact that in these cases we can safely assume that there is only one non-zero valued utility function (see Observation 1). On the contrary, there is a counterexample for and . On the way to prove these claims we need to formally define what it means that one family of tie-breaking rules can be used to simulate another family of tie-breaking rules.
Let be a fixed set of candidates. Let be a set of confirmed and be a set of pending candidates. Let be a family of utility functions and be a size of an egroup. Consider some subset of the set . For two families and of tie-breaking rules we say that can -simulate if there are some rules and such that and have the same output for all possible elections assuming all elements from together with set are fixed. We call rule a -simulator.
At first glance Definition 2 might seem overcomplicated. However, it is tailored to grasp different degrees of simulation possibilities. On the one hand, one can always find a lexicographic order and use it for breaking ties if all: confirmed candidates, pending candidates, utility functions, and the size of an egroup are known. Thus, one needs some flexibility in the definition of simulation for it to be non-trivial. On the other hand, it is somewhat obvious that without fixing the utility functions, one cannot simulate optimistic or pessimistic tie-breaking rules. In other words, we have:
Let and . The family of lexicographic tie-breaking rules does not -simulate .
Next, we show that for some cases it is sufficient to fix just the utility functions in order to simulate optimistic or pessimistic tie-breaking rules (see Proposition 1). For other cases, however, one has to fix all: confirmed candidates, pending candidates, utility functions, and the size of an egroup (see Proposition 2).
Let be a set of candidates, be a family of utility functions, , and . Let and . Then the family of lexicographic tie-breaking rules can -simulate , and a -simulator can be found in time.
Recall from Observation 1 that if , then there is always a set of utility functions with just one non-zero valued utility function that is equivalent to . Hence, we compute such a function in time as follows: In the utilitarian case, function assigns every candidate the sum of utilities the manipulators give to the candidate. Considering the candidate-wise egalitarian evaluation, function assigns every candidate the minimum utility value among utilities given to the candidate over all manipulators. We say an order of the candidates is consistent with some utility function if implies for optimistic tie-breaking and implies for pessimistic tie-breaking. Any lexicographic tie-breaking rule defined by an order that is consistent with the utility function simulates . We compute a consistent order by sorting the candidates according to in time. ∎
Proposition 1 describes a strong feature of optimistic utilitarian and candidate-wise egalitarian tie-breaking and their pessimistic variants. Intuitively, the proposition says that for these tie-breaking mechanisms one can compute a respective linear order of candidates. Then one can forget all the details of the initial tie-breaking mechanism and use the order to determine winners. The order can be computed even without knowing the details of an election. Unfortunately, the simulation of pessimistic and optimistic egalitarian tie-breaking turns out to be more complicated.
Let be a set of candidates, be a family of utility functions, be a set of confirmed candidates, be a set of pending candidates, and be a size of an egroup. For each , , the lexicographic tie-breaking family of rules does not -simulate assuming .
From Observation 2 we already know that the family of lexicographic tie-breaking rules cannot -simulate the family of egalitarian pessimistic tie-breaking rules or the family of egalitarian optimistic tie-breaking rules.
Next, we build one counterexample for each of the remaining size-three subsets of to show our theorem. To this end, let us fix a set of candidates (compatible with our running example) and a family of utility functions as depicted in the table below.
First, we prove that the family cannot -simulate (i.e., is unfixed) for . Let us fix , . We consider the optimistic variant of egalitarian tie-breaking, so we are searching for the best possible -egroup. Looking at the values of , we see that candidate gives the best possible egalitarian evaluation value which is four. This means that a -simulator has to use an order where precedes both and . However, it turns out that if we set , then the best -egroup consists exactly of candidates and . This leads to a contradiction because now candidates and should precede in ’s lexicographic order. Consequently, family does not -simulate . Using the same values of utility functions and the same sequence of the values of we get a proof for the pessimistic variant of egalitarian evaluation.
Second, we prove that the family cannot -simulate (i.e., set is unfixed) for . This time, we fix , . We construct the first case by setting . Using the fact that in both functions candidate has utility zero, we choose exactly the same candidate as in the proof of -simulation for the case ; that is, for the optimistic variant, the winning -egroup is and . Consequently, this leads to the fact that precedes and in the potential -simulator’s lexicographic order. Towards a contradiction, we set . The situation is exactly the same as in the proof of the -simulation case. Now, the winning -egroup consists of and which ends the proof for the optimistic case. By almost the same argument, the result holds for the pessimistic variant.
Finally, we prove that the family cannot -simulate (i.e., set is unfixed) for . We fix , . For the first case we pick . The best egalitarian evaluation happens for the -egroup consisting of and . This imposes that, in the potential -simulator’s order, and precede the remaining candidates (in particular, precedes ). However, for the best -egroup changes to that consisting of and which gives a contradiction ( precedes ). As in the previous cases, the same argument provides a proof for the pessimistic variant. ∎
Proposition 2 implies that pessimistic and optimistic egalitarian tie-breaking cannot be simulated without having full knowledge about an election. In terms of computational complexity, however, pessimistic egalitarian tie-breaking remains tractable whereas optimistic egalitarian tie-breaking is intractable. In the next section, among other things, we show that the egalitarian optimistic tie-breaking significantly differs from the egalitarian pessimistic tie-breaking with respect of hardness of computing winners.
4 Complexity of Tie-Breaking
It is natural to ask whether the tie-breaking rules proposed in Subsection 3.2 are practical in terms of their computational complexity. If not, then there is little hope for coalitional manipulation because tie-breaking is a subtask to be solved by the manipulators.
Clearly, we can apply every lexicographic tie-breaking rule that is defined through some predefined order of the candidates in linear time. Hence, we focus on the rules that model optimistic or pessimistic manipulators. To this end, we analyze the following computational problem.
-Tie-Breaking (-TB), ,
Input: A set of candidates partitioned into a set of pending candidates and a set of confirmed candidates, the size of the excellence-group such that , a family of manipulators’ utility functions where , and a non-negative, integral evaluation threshold .
Question: Is there a size- set such that is selected according to , , and ?
Naturally, we may assume that the number of candidates and the number of utility functions are polynomially bounded in the size of the input. However, both the evaluation threshold and the utility function values are encoded in binary.
Note that an analogous problem has not been considered for single-winner elections since, for single-winner elections, optimistic and pessimistic tie-breaking rules can be easily simulated by straightforward lexicographic tie-breaking rules.
4.1 Utilitarian and Candidate-Wise Egalitarian: Tie-Breaking Is Easy
As a warm-up, we observe that tie-breaking can be applied and evaluated efficiently if the -egroups are evaluated according to the utilitarian or candidate-wise egalitarian variant. The corresponding result follows almost directly from Proposition 1.
Let denote the number of candidates and denote the number of manipulators. Then one can solve -Tie-Breaking in time for , .
The algorithm works in two steps. First, compute a lexicographic tie-breaking rule that simulates in time as described in Proposition 1. Second, apply tie-breaking rule , and evaluate the resulting -egroup in time. The running time of applying a lexicographic tie-breaking rule is linear with respect to the input length (see Subsection 3.3). ∎
4.2 Egalitarian: Being Optimistic Is Hard
In this subsection, we consider the optimistic and pessimistic tie-breaking rules when applied for searching a -egroup evaluated according to the egalitarian variant. First, we show that applying and evaluating egalitarian tie-breaking is computationally easy for pessimistic manipulators but computationally intractable for optimistic manipulators even if the size of the egroup is small. Being pessimistic, the main idea is to “guess” the manipulator that is least satisfied and select the candidates appropriately. We show the computational hardness of the optimistic case via a reduction from the -complete Set Cover problem parameterized by solution size (Downey and Fellows, 2013).
Let denote the number of candidates, denote the number of manipulators, denote the evaluation threshold, and denote the size of an egroup. Then one can solve -Tie-Breaking in time, but -Tie-Breaking is -hard and -hard when parameterized by even if and every manipulator only gives either utility one or zero to each candidate.
For the pessimistic case, it is sufficient to “guess” the least satisfied manipulator by iterating through possibilities. Then, select the pending candidates with the smallest total utility for this manipulator in time. Finally, comparing the -egroup with the worst minimum satisfaction over all manipulators with the lower bound on satisfaction level given in the input solves the problem.
We prove the hardness for the optimistic case reducing from the -hard Set Cover problem which, given a collection of subsets of universe and an integer , asks whether there exists a family of size at most such that . Let us fix an instance of Set Cover. To construct an -Tie-Breaking instance, we introduce pending candidates representing subsets in and manipulators representing elements of the universe. Note that there are no confirmed and rejected candidates. Each manipulator gives utility one to candidate if set contains element and zero otherwise. We set the excellence-group size and the threshold to be .
We observe that if there is a size- subset such that , then there exists a family —consisting of the sets represented by candidates in —such that each element of the universe belongs to the set . On the contrary, if we cannot pick a group of candidates of size for which every manipulator’s utility is at least one, then instance is a ‘no’ instance. This follows from the fact that for each size- subset there exists at least one manipulator for whom . This translates to the claim that there exists no size- subset such that all elements in belong to the union of the sets in .
Since Set Cover is -hard and -hard with respect to parameter , we obtain that our problem is also -hard and -hard when parameterized by the size of an excellence-group. ∎
Inspecting the -hardness proof of Theorem 1, we learn that a small egroup size (alone) does not make -Tie-Breaking computationally tractable even for very simple utility functions. Next, using a parameterized reduction from the -complete Multicolored Clique problem (Fellows et al., 2009), we show that there is still no hope for fixed-parameter tractability (under standard assumptions) even for the combined parameter “number of manipulators and egroup size”; intuitively, this parameter covers situations where few manipulators are going to influence an election for a small egroup.
Let denote the size of an egroup and denote the number of manipulators. Then, parameterized by , -Tie-Breaking is -hard.
We describe a parameterized reduction from the -hard Multicolored Clique problem (Fellows et al., 2009). In this problem, given an undirected graph , a non-negative integer , and a vertex coloring , we ask whether graph admits a colorful -clique, that is, a size- vertex subset such that the vertices in are pairwise adjacent and have pairwise distinct colors. Without loss of generality, we assume that the number of vertices of each color is the same; to be referred as in the following. Let , , be a Multicolored Clique instance. Let denote the set of vertices of color , and let , defined for , , denote the set of edges that connect a vertex of color to a vertex of color .
Candidates. We create one confirmed candidate and pending candidates. More precisely: for each , we create one vertex candidate for each vertex , and for each such that we create one edge candidate for each edge , . We set the size of the egroup to and set the evaluation threshold . Next, we describe the manipulators and explain the high-level idea of the construction.
Manipulators and main idea. Our construction will ensure that there is a -egroup with and if and only if contains vertex candidates and edge candidates that encode a colorful -clique. To this end, we introduce the following manipulators.
For each color , there is a color manipulator ensuring that the -egroup contains a vertex candidate corresponding to a vertex of color . Herein, variable denotes the id of the vertex candidate (resp. vertex) that is selected for color .
For each such that , there is one color pair manipulator ensuring that the -egroup contains an edge candidate corresponding to an edge connecting vertices of colors and . Herein, variable denotes the id of the edge candidate (resp. edge) that is selected for color pair , .
For each such that , there are two verification manipulators , ensuring that vertex is incident to edge if or incident to edge otherwise.
If there exists a -egroup in agreement with the description in the previous three points, then this -egroup must encode a colorful -clique.
Color manipulator , , has utility for the confirmed candidate , utility one for each candidate corresponding to a vertex of color , and utility zero for the remaining candidates.
Color pair manipulator , , , has utility for the confirmed candidate , utility one for each candidate corresponding to an edge connecting a vertices of colors and , and utility zero for the remaining candidates.
Verification manipulator , , , has utility for candidate , , utility for each candidate corresponding to an edge that connects vertex to a vertex of color , and utility zero for the remaining candidates.
Verification manipulator , , , has utility for candidate , , utility for each candidate corresponding to an edge that connects vertex , to a vertex of color , and utility zero for the remaining candidates.
Correctness. We argue that the graph admits a colorful clique of size if and only if there is a -egroup with and .
Suppose that there exists a colorful clique of size . Create the -egroup as follows. Start with and add every vertex candidate that corresponds to some vertex of and every edge candidate that corresponds to some edge of . Each color manipulator and color pair manipulator receives total utility , because contains, by definition, one vertex of each color and one edge connecting two vertices for each color pair. It is easy to verify that the verification manipulator must receive utility from a vertex candidate and utility from an edge candidate and that the verification manipulator must receive utility from a vertex candidate and utility from an edge candidate. Thus, .
Suppose that there exists a -egroup such that . Since each color manipulator cannot achieve utility unless belongs to the winning -egroup, it follows that . Because each color manipulator receives total utility at least , must contain some vertex candidate corresponding to a vertex of color for some . We say that selects vertex . Since each color pair manipulator receives total utility at least , must contain some edge candidate corresponding to an edge connecting a vertex of color and a vertex of color for some . We say that selects edge . We implicitly assumed that each color manipulator and color pair manipulator contributes exactly one selected candidate to . This assumption is true because there are exactly such manipulators and each needs to select at least one candidate; hence, is exactly of the desired size. In order to show that the corresponding vertices and edges encode a colorful -clique, it remains to show that no selected edge is incident to a vertex that is not selected. Assume towards a contradiction that selects an edge and some vertex . However, either verification manipulator or verification manipulator receives the total utility at most ; a contradiction. ∎
Finally, devising an ILP formulation, we show that -Tie-Breaking becomes fixed-parameter tractable when parameterized by the combined parameter “number of manipulators and number of different utility values”. This parameter covers situations with few manipulators that have simple utility functions; in particular, when few voters have utility functions. Together with Theorem 1 and Theorem 2, following Theorem 3 shows that neither few manipulators nor few utility functions make -TB fixed-parameter tractable, but only combining these two parameters allows us to deal with the problem in time.
Let denote the number of different utility values and denote the number of manipulators. Then, parameterized by , -Tie-Breaking is fixed-parameter tractable.
We define the type of any candidate to be the size-vector . Let be the set of all possible types. Naturally, the size of is upper-bounded by . We denote the set of candidates of type by . Now, the ILP formulation of the problem using exactly variables reads as follows. For each type , we introduce variable indicating the number of candidates of type in an optimal -egroup. We use variable to represent the minimal value of the total utility achieved by manipulators. We define the following ILP with the goal to maximize (indicating the utility gained by the least satisfied manipulator) subject to:
Constraint set (1) ensures that the solution is achievable with given candidates. Constraint (2) guarantees a choice of an egroup of size . The last set of constraints imposes that holds at most the minimal value of the total utility gained by manipulators. By a famous result of Lenstra (1983), this ILP formulation with the number of variables bounded by yields that -Tie-Breaking is fixed-parameter tractable when parameterized by the combined parameter . ∎
5 Complexity of Coalitional Manipulation
In the previous section, we have seen that breaking ties optimistically or pessimistically—an essential subtask to be solved by the manipulators—can be computationally challenging; in most cases, however, this problem turned out to be computationally easy. In this section, we move on to our full framework and analyze the computational difficulty of voting strategically for a coalition of manipulators. To this end, we formalize our central computational problem. Let be a multiwinner voting rule and let be a multiwinner tie-breaking rule.
---Coalitional Manipulation (---CM),
Input: An election , a searched egroup size , manipulators represented by their utility functions such that , and a non-negative, integral evaluation threshold .
Question: Is there a size- multiset of manipulative votes over such that -egroup wins the election under and , and ?
The ---CM problem is defined very generally; namely, one can consider any multiwinner voting rule (in particular, any single-winner voting rule is a multiwinner voting rule with ). In our paper, however, we focus on -Bloc; hence, from now on, we narrow down our analysis of ---CM to the ---CM problem.
As a step on the way to show our results we also use a restricted version of ---Coalitional Manipulation that we call ---Coalitional Manipulation with consistent manipulators. In this variant, the input stays the same, but all manipulators cast exactly the same vote to achieve the objective.
To increase readability, we decided to represent manipulators by their utility functions. As a consequence, we frequently use, for example, referring to the manipulator itself, even if we do not care about the values of utility function
at the moment of usage. In the paper, we also stick to the term “voters” meaning the setof voters of an input election. We never call manipulators “voters”; however, we speak about the manipulative votes they cast.
As for the encoding of the input of ---CM, we use a standard assumption; namely, that the number of candidates, the number of voters, and the number manipulators are polynomially upper-bounded in the size of the input. Analogously to -Tie-Breaking, both the evaluation threshold and the utility function values are encoded in binary.
5.1 Utilitarian and Candidate-Wise Egalitarian: Manipulation Is Tractable
We show that ---Coalitional Manipulation can be solved in polynomial time for any constant , any , and any .
For Bloc (i.e., ), we give a quadratic-time algorithm with respect to the size of the input. We start with an algorithm for ---CM with consistent manipulators and show that this algorithm solves also Bloc---CM. The algorithm “guesses” the minimum score among all members of the winning egroup and then carefully (with respect to the tie-breaking method) selects the best candidates that can reach this score.
In several proofs in Subsection 5.1 we use the value of a candidate for manipulators (coalition) and say that a candidate is more valuable or less valuable than another candidate. Although we cannot directly measure the value of a candidate for the whole manipulators’ coalition in general, thanks to Observation 1, we can assume a single non-zero utility function when discussing the utilitarian and candidate-wise egalitarian variants. Thus, assigning a single value to each candidate is justified.
Let denote the number of candidates, denote the number of voters, and denote the number of manipulators. Then one can solve ---Coalitional Manipulation with consistent manipulators in time for any and .
Consider an instance of ---CM with consistent manipulators with an election where is a candidate set and is a multiset of non-manipulative votes, manipulators, an egroup size , and a lexicographic order used by to break ties. In essence, we introduce a constrained solution form called a canonical solution and argue that it is sufficient to analyze only this type of solutions. Then we provide an algorithm that efficiently seeks for an optimal canonical solution.
At the beginning, we observe that when manipulators vote consistently, then we can arrange the top candidates of a manipulative vote in any order. Hence, the solution to our problem is a size- subset (instead of an order) of candidates which we call a set of supported candidates; we call each member of this set a supported candidate.
Strength order of the candidates. Additionally, we introduce a new order of the candidates. It sorts them descendingly with respect to the score they receive from voters and, as a second criterion, according to the position in . Intuitively, the easier it is for some candidate to be a part of a winning -egroup, the higher is the candidate’s position in . As a consequence, we state Claim 1.
Let us fix an instance of ---CM with consistent manipulators and a solution which leads to a winning -egroup . For every supported (resp. unsupported) candidate , the following holds:
If is part of the winning -egroup, then every supported (resp. unsupported) predecessor of , according to , belongs to .
If is not part of the winning -egroup, then every supported (resp. unsupported) successor of , according to , does not belong to .
Claim 1 justifies thinking about as a “strength order”; hence, in the proof we use the terms stronger and weaker candidate. Using Claim 1, we can fix some candidate as the weakest in the winning -egroup and then infer candidates that have to be and that cannot be part of this -egroup. To formalize this idea, we introduce the concept of a canonical solution.
Canonical solutions. Assuming the case where , we call a solution leading to a winning -egroup canonical if all candidates of the winning egroup are supported; that is, . In the opposite case, , solution is canonical if and is a set of the weakest candidates in . For the latter case, the formulation describes the solution which favors supporting weaker candidates first and ensures that no approval is given to a candidate outside the winning -egroup.
Canonical solutions are achievable from every solution without changing the outcome. Observe that one cannot prevent a candidate from winning by supporting the candidate more because this only increases the candidate’s score. Consequently, we can always transfer approvals to all candidates from the winning -egroup. For the case of , we then have to rearrange the approvals in such a way that only the weakest members of the -egroup are supported. However, such a rearrangement cannot change the outcome because, according to Claim 1, we can transfer an approval from some stronger candidate to weaker keeping both of them in the winning -egroup.
Dropped and kept candidates. Observe that for every solution (including canonical solutions), we can always find the strongest candidate who is not part of the winning egroup. We call this candidate the dropped candidate. Note that we use the strength order in the definition of the dropped candidate; this order does not take manipulative votes into account. Moreover, without loss of generality, we can assume that the dropped candidate is not a supported candidate. This is because if the dropped candidate is not in the winning -egroup even if supported, then we can support any other candidate outside of the winning -egroup without changing the winning -egroup (see Claim 1). There always exists some candidate to whom we can transfer our support because . Naturally, by definition of the dropped candidate, all candidates stronger than the dropped candidate are members of the winning -egroup. We call these candidates kept candidates.
High-level description of the algorithm. The algorithm solving ---CM with consistent manipulators iteratively looks for an optimal canonical solution for every possible (non-negative) number of kept candidates (alternatively the algorithm checks all feasible possibilities of choosing the dropped candidate). Observe that . The upper bound is the consequence of the fact that each kept candidate is (by definition) in the winning -egroup. Since all candidates except for kept candidates have to be supported to be part of the winning egroup, we need at least kept candidates, in order to be able to complete the -egroup.
Running time. To analyze the running time of the algorithm described in the previous paragraph, several steps need to be considered. At the beginning we have to compute values of candidates and then sort the candidates with respect to their value. This step runs in time. Similarly, computing takes time. Having both orders, Procedure 1 (described in detail later in this proof) needs to find an optimal canonical solution for some fixed number of kept candidates. Finally, we have at most possible values of . Summing the times up, together with the fact that , we obtain a running time .
What remains to be done. Procedure 1 describes how to look for an optimal canonical solution for a fixed number of kept candidates. First, partition the candidate set in the following way. By we denote the kept candidates (which are the top candidates according to ). Consequently, the -st strongest candidate is the dropped candidate; say . For every value of , the corresponding dropped candidate, by definition, is not allowed to be part of the winning egroup. Let
be the set of distinguished candidates. Each distinguished candidate, if supported, is preferred over to be selected into the winning -egroup. Consequently, the distinguished candidates are all candidates who can potentially be part of the winning -egroup. We remark that to fulfil our assumption that the dropped candidate is not part of a winning egroup, it is obligatory to support at least distinguished candidates. Note that is not necessarily equal to . The remaining candidates cannot be a part of the winning -egroup under any circumstances assuming kept candidates. Making use of the described division into , , and , Procedure 1 incrementally builds set of supported candidates associated with an optimal solution until all possible approvals are used.
Detailed description of the algorithm. Before studying Procedure 1 in detail, consider Figure 1 illustrating the procedure on example data. In line 1, the procedure builds set of supported candidates using the best valued distinguished candidates. Since only the distinguished candidates might be a part of the winning -egroup besides the kept candidates, there is no better outcome achievable. Then, in line 1, the remaining approvals, if they exist, are used to support secured candidates. This operation does not change the resulting -egroup. Then Procedure 1 checks whether all approvals were used; that is, whether . If not, then there are exactly remaining approvals to use. Note that at this stage set contains supported candidates who correspond to the best possible -egroup, however, without spending all approvals. Let us call this -egroup . It is possible that there is no way to spend the remaining approvals without changing the winning -egroup . Then substitutions of candidates occur. The new candidates in the -egroup can be only those that are distinguished and so far unsupported whereas the exchanged ones can be only so far supported distinguished candidates. This means that each substitution lowers the overall value of the winning -egroup. So, the best what can be achieved is to find the minimal number of substitutions and then pick the most valuable remaining candidates from to be substituted. The minimal number of substitutions can be found by analyzing how many candidates would be exchanged in the winning -egroup if the weakest previously unsupported candidates were supported. The procedure makes such a simulation and computes the number of necessary substitutions, in lines 1-1. Supporting the weakest unsupported candidates and then the most valuable so far unsupported distinguished candidates gives the optimal -egroup for kept candidates (when all approvals are spent). Note that the number of approvals is strictly lower than the number of candidates, so one always avoids supporting .