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High-level description of the algorithm. The algorithm solving [MATH] [MATH] [MATH] CM with consistent manipulators iteratively looks for an optimal canonical solution for every possible (non-negative) number [MATH] of kept candidates (alternatively the algorithm checks all feasible possibilities of choosing the droppe... |
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 [MATH] time. Similarly, computing [MATH] takes [MATH] ... |
[MATH] to find an optimal canonical solution for some fixed number [MATH] of kept candidates. Finally, we have at most [MATH] possible values of [MATH] . Summing the times up, together with the fact that [MATH] , we obtain a running time |
[MATH] What remains to be done. Procedure describes how to look for an optimal canonical solution for a fixed number [MATH] of kept candidates. First, partition the candidate set in the following way. By [MATH] we denote the kept candidates (which are the top [MATH] candidates according to [MATH] ). Consequently, the [... |
[EQUATION] be the set of distinguished candidates . Each distinguished candidate, if supported, is preferred over [MATH] to be selected into the winning [MATH] -egroup. Consequently, the distinguished candidates are all candidates who can potentially be part of the winning [MATH] -egroup. We remark that to fulfill our ... |
[MATH] is not necessarily equal to [MATH] . The remaining candidates cannot be part of the winning [MATH] -egroup under any circumstances assuming [MATH] kept candidates. Also, set [MATH] might consist of less than [MATH] required candidates (which is the case when there are too few candidates that, after supported, wo... |
Input: Election [MATH] ; number [MATH] of approvals in [MATH] Bloc rule; size [MATH] of the winning [MATH] -egroup; a partition of [MATH] into kept candidates |
[MATH] (such that [MATH] and [MATH] ), a dropped candidate [MATH] , and distinguished candidates [MATH] (such that [MATH] ). Output : Optimal supported candidates set [MATH] |
[MATH] {the [MATH] most valuable candidates from [MATH] [MATH] [MATH] arbitrary candidates from [MATH] if [MATH] then [MATH] the [MATH] weakest candidates from [MATH] |
[MATH] top [MATH] strongest candidates from [MATH] [MATH] [MATH] the [MATH] weakest candidates from [MATH] 10 [MATH] the [MATH] most valuable candidates from [MATH] |
11 12 end if 13 return [MATH] Procedure 1 A procedure of finding an optimal set of supported candidates. Detailed description of the algorithm. |
Before studying Procedure in detail, consider Figure 1 illustrating the procedure on example data. In line , the procedure builds set [MATH] of supported candidates using the [MATH] best valued distinguished candidates. Since only the distinguished candidates might be a part of the winning [MATH] -egroup besides the ke... |
[MATH] approvals were used; that is, whether [MATH] . If not, then there are exactly [MATH] remaining approvals to use. Note that at this stage set [MATH] |
contains [MATH] supported candidates who correspond to the best possible [MATH] -egroup, however, without spending all approvals. Let us call this |
[MATH] -egroup [MATH] . It is possible that there is no way to spend the remaining [MATH] approvals without changing the winning [MATH] -egroup [MATH] . Then substitutions of candidates occur. The new candidates in the [MATH] -egroup can be only those that are distinguished and so far unsupported whereas the exchanged ... |
previously unsupported candidates were supported. The procedure makes such a simulation and computes the number [MATH] of necessary substitutions, in lines |
. Supporting the [MATH] weakest unsupported candidates and then the [MATH] most valuable so far unsupported distinguished candidates gives the optimal [MATH] -egroup for [MATH] kept candidates (when all approvals are spent). Note that the number [MATH] of approvals is strictly lower than the number of candidates, so on... |
[MATH] The algorithm we presented can be applied also for pessimistic and optimistic evaluation because of the possibility of simulating these evaluations by a lexicographic order in time [MATH] (see |
Proposition 1 ). For Bloc , we will show that manipulators can always vote identically to achieve an optimal [MATH] -egroup. In a nutshell, for every egroup the manipulators can only increase the scores of its members by voting exactly for them. This fact leads to the next corollary. |
Corollary 2 Let [MATH] denote the number of candidates, [MATH] denote the number of voters, and [MATH] denote the number of manipulators. One can solve |
Bloc [MATH] [MATH] Coalitional Manipulation in [MATH] time for any [MATH] and [MATH] Proof. We show that for Bloc [MATH] [MATH] Coalitional Manipulation the manipulators have no incentive to deviate from one optimal profile (i.e., they vote in the same manner). Let us fix an optimal [MATH] -egroup [MATH] . If there exi... |
[MATH] candidates from [MATH] ). Observe that in the Bloc voting rule by shifting a candidate up in a preference order we only increase the candidate’s score; as a result, we cannot prevent the candidate from winning by doing such a shift. Using this observation, we can exchange candidate [MATH] with candidate [MATH] i... |
5.2 Egalitarian: Hard Even for Simple Tie-Breaking In Subsection 4.2 , we showed that already breaking ties might be computationally intractable. These intractability results only hold with respect to the egalitarian evaluation and optimistic manipulators. We now show that this intractability transfers to coalitional m... |
Proposition 4 For any tie-breaking rule [MATH] , there is a polynomial-time many-one reduction from [MATH] -Tie-Breaking to [MATH] Bloc [MATH] [MATH] Coalitional Manipulation |
Proof. We reduce an instance of [MATH] -Tie-Breaking to [MATH] -Bloc [MATH] [MATH] Coalitional Manipulation ; however, before we describe the actual reduction, we present a useful observation concerning |
[MATH] -Tie-Breaking in the next paragraph. Let us fix an instance [MATH] of [MATH] -Tie-Breaking with a confirmed set [MATH] , a pending set [MATH] , a size [MATH] of an egroup, a threshold [MATH] , and a set of manipulators represented by a family [MATH] of utility functions. We can construct a new equivalent instanc... |
[MATH] -egroups and their values of egalitarian utility do not change. Now we can phrase our reduction from [MATH] -Tie-Breaking to |
[MATH] -Bloc [MATH] [MATH] Coalitional Manipulation . Let us fix an instance [MATH] of [MATH] -Tie-Breaking with a confirmed set [MATH] , a pending set [MATH] , a size [MATH] |
of an egroup, a threshold [MATH] , and a set [MATH] of [MATH] utility functions. Because of the observation about “pumping” instances of [MATH] -Tie-Breaking , we can assume, without loss of generality, that [MATH] holds. In the constructed instance of [MATH] -Bloc [MATH] [MATH] CM equivalent to [MATH] we build an elec... |
[MATH] . The construction starts from ensuring the scores of the confirmed candidates. Observe, that in this step we add at most [MATH] voters (in case [MATH] ). If [MATH] , then we have to add some dummy candidates in this step. We can upper-bound the number of the added dummy candidates by [MATH] (this bound is not t... |
[MATH] -egroup by approving candidates in this subset exactly once. However, to be sure that we are able to distribute all approvals such that there is no tie, we ensure that the remaining [MATH] approvals can be distributed to some candidates without changing the outcome. To achieve this goal we add exactly [MATH] dum... |
By our construction, we are always able to approve enough pending candidates to form a [MATH] -egroup without considering ties, and we cannot make a dummy candidate a winner under any circumstances. Thus, if [MATH] -Tie-Breaking has a solution |
[MATH] , then we approve every candidate [MATH] such that [MATH] was in the pending set [MATH] before, and we obtain a solution to the reduced instance. In the opposite case, if there is no such a [MATH] -egroup whose egalitarian utility value is at least [MATH] , then the corresponding instance of |
[MATH] -Bloc [MATH] [MATH] Coalitional Manipulation also has no solution since the possible [MATH] -egroups are exactly the same. The reduction runs in polynomial time. |
Observe that the reduction proving Proposition 4 does not change the egroup size [MATH] . Additionally, the increase of the number of manipulators in resulting instances is polynomially bounded in the egroup size [MATH] of input instances. This is due to the fact that even if we need to “pump” an initial instance to ac... |
[MATH] manipulators. Thus, together with Theorem 1 and Theorem 2 Proposition 4 leads to the following corollary. Corollary 3 Let [MATH] be an arbitrary tie-breaking rule. Then, |
[MATH] Bloc [MATH] [MATH] Coalitional Manipulation is [MATH] -hard. Let [MATH] denote the number of manipulators, [MATH] denote the evaluation threshold and [MATH] denote the size of an egroup. Then, parameterized by [MATH] |
[MATH] Bloc [MATH] [MATH] CM is [MATH] -hard. Parameterized by [MATH] [MATH] Bloc [MATH] [MATH] CM is [MATH] -hard even if [MATH] and every manipulator only gives either utility one or zero to each candidate. |
Finally, by using ideas from Theorem 4 and an adaptation of the ILP from Theorem 3 as a subroutine, we show that, for the combined parameter “the number of manipulators and the number of different utility values”, fixed-parameter tractability of [MATH] -Tie-Breaking transfers to coalitional manipulation for both optimi... |
Theorem 5 Let [MATH] denote the number of manipulators and [MATH] denote the number of different utility values. Parameterized by [MATH] [MATH] Bloc [MATH] [MATH] Coalitional Manipulation with |
[MATH] is fixed-parameter tractable. Proof. In a nutshell, we divide [MATH] Bloc [MATH] [MATH] CM and [MATH] Bloc [MATH] [MATH] CM into subproblems solvable in |
[MATH] time with respect to the combined parameter “number of manipulators and number of different utility values.” We show that solving polynomially many subproblems is enough to solve the problems. |
The main idea. We split the proof into two parts. In the first part, we define subproblems and show how to find a solution assuming that the subproblems are solvable in [MATH] time with respect to the parameter. In the second part, we show that, indeed, the subproblems are fixed-parameter tractable using their ILP form... |
[MATH] Bloc [MATH] [MATH] CM and [MATH] Bloc [MATH] [MATH] CM are the same, so let us consider an arbitrary input with an election [MATH] where [MATH] [MATH] , a size |
[MATH] of an excellence-group, and [MATH] manipulators represented by a set [MATH] of their utility functions. Let [MATH] be the number of different utility values. |
An election resulting from a manipulation and a corresponding [MATH] -egroup emerging from the manipulation can be described by three non-negative integer parameters: |
1. the lowest final score [MATH] of any member of the [MATH] -egroup; 2. the number [MATH] of promoted candidates from the [MATH] -egroup with a score higher than [MATH] that, at the same time, have score at most [MATH] without taking manipulative votes into consideration; |
3. the number [MATH] of border candidates with score [MATH] Observe that if as a result of a manipulation the lowest final score of members in a final [MATH] -egroup is [MATH] , then the promoted candidates are part of the [MATH] -egroup regardless of the tie-breaking method used. For border candidates, however, it mig... |
[MATH] -egroup; thus, the term border candidates. From now on, we refer to the election situation characterized by parameters [MATH] as a (input) state . Additionally, we call a set of manipulators’ votes a manipulation |
Part 1: High-level description of the algorithm. For now, we assume that there is a procedure [MATH] which runs in [MATH] time with respect to the combined parameter “number of manipulators and number of different utility values.” Procedure [MATH] , takes values |
[MATH] [MATH] [MATH] and an instance of the problem, and finds a manipulation which leads to a [MATH] -egroup maximizing the egalitarian utility under either egalitarian optimistic or egalitarian pessimistic tie-breaking with respect to the input state. If such a manipulation does not exist, then procedure [MATH] retur... |
[MATH] Bloc [MATH] [MATH] CM and [MATH] Bloc [MATH] [MATH] CM runs [MATH] for all possible combinations of values [MATH] [MATH] , and [MATH] Eventually, it chooses the best manipulation returned by [MATH] or returns “no” if [MATH] always returned so. Since the value of |
[MATH] is at most [MATH] and [MATH] together with [MATH] are both upper-bounded by the number of candidates, we run [MATH] at most [MATH] times. Because the input size grows polynomially with respect to the growth of values [MATH] [MATH] , and [MATH] , the overall algorithm runs in [MATH] time with respect to the combi... |
Part 2: Basics and preprocessing for the ILP. To complete the proof we describe procedure [MATH] used by the above algorithm. In short, the procedure builds and solves an ILP program that finds a manipulation leading to the state described by the input values. Before we describe the procedure in details, we start with ... |
[MATH] that altogether form the input of [MATH] . For each candidate [MATH] , let a size- [MATH] vector [MATH] , referred to as a type vector , define the |
type of [MATH] . We denote the set of all possible type vectors by [MATH] . Observe that [MATH] . With each type vector [MATH] [MATH] , we associate a set [MATH] containing only candidates of type [MATH] . We also distinguish the candidates with respect to their initial score compared to [MATH] . A candidate of type |
[MATH] [MATH] , with score [MATH] [MATH] , belongs to group [MATH] . We denote all candidates with a score (excluding manipulative votes) higher than [MATH] by |
[MATH] , whereas by [MATH] we denote the candidates with a score (excluding manipulative votes) strictly lower than [MATH] . For each type [MATH] of a candidate, we define function |
[MATH] , which gives the number of candidates of type [MATH] that are obligatory part of the winning [MATH] -egroup. At the beginning, procedure [MATH] tests whether the input values |
[MATH] [MATH] , and [MATH] represent a correct state. From the fact that there has to be at least one candidate with score [MATH] , we get the upper bound [MATH] for value [MATH] . To have enough candidates to complete the [MATH] -egroup, we need at least [MATH] |
candidates with score [MATH] after the manipulation which gives [MATH] . Finally, the state is incorrect if the corresponding set [MATH] contains [MATH] or more candidates. If the input values are incorrect, then [MATH] returns “no.” Otherwise, [MATH] continues with building a corresponding ILP program. We give two sep... |
ILP: Common part. For each group [MATH] [MATH] [MATH] , we introduce variables [MATH] and [MATH] indicating the numbers of, respectively, border and promoted candidates from group [MATH] . Additionally, we introduce variables [MATH] and [MATH] . The former represents the number of approvals used to get the obligatory n... |
[EQUATION] The expressions ensure that exactly [MATH] candidates are selected to be promoted ( ), exactly [MATH] candidates are selected to be border ones ( ), and that, for every group, the sum of border and promoted candidates is not greater that the cardinality of the group ( ). The last two formulae ensure that can... |
[EQUATION] Then, we model spending the [MATH] remaining votes (if any) to use all approvals. [EQUATION] The upper bound on the number of votes one can spend without changing the outcome presented in equation ( 11 ) consists of three summands. The first one indicates the number of approvals which can be spent for candid... |
ILP extension for optimistic egalitarian tie-breaking. In the second part, we find the final [MATH] -egroup by completing it with the border candidates according to the particular tie-breaking mechanism. Let us first focus on the case of the optimistic egalitarian tie-breaking. We introduce constraints allowing us to m... |
[MATH] indicating the number of border candidates of the given group chosen to be in the final [MATH] -egroup. The following constraints ensure that we select exactly [MATH] border candidates to complete the winning egroup and that, for each group [MATH] , we do not select more candidates than available. |
[EQUATION] To complete the description of the ILP, we add the final expression defining the egalitarian utility [MATH] of the final [MATH] -excellence-group. The goal of the ILP program is to maximize [MATH] |
[EQUATION] Since the goal is to maximize [MATH] , our program simulates the egalitarian optimistic tie-breaking. ILP extension for pessimistic egalitarian tie-breaking. |
To solve a subproblem for the case of pessimistic egalitarian tie-breaking, we need a different approach. We start with an additional notation. For each type of candidate [MATH] , let [MATH] denote the number of border candidates of this type. For each type [MATH] and manipulator [MATH] [MATH] we introduce a new intege... |
preferences; we call these candidates the designated candidates of type [MATH] of manipulator [MATH] . For each variable [MATH] , we define a binary variable [MATH] which has value one if at least one candidate of type [MATH] is a designated candidate of manipulator [MATH] . Similarly, we define [MATH] to indicate that... |
[EQUATION] The following forces the semantics of the variables [MATH] ; that is, a variable [MATH] [MATH] [MATH] , has value one if and only if variable [MATH] is at least one. |
[EQUATION] Similarly, for the variables [MATH] , we ensure that [MATH] [MATH] [MATH] is one if and only if manipulator [MATH] designates all available candidates of type [MATH] |
[EQUATION] Since our task is to perform pessimistic tie-breaking, we have to ensure that the designated candidates for each manipulator are the candidates whom the manipulator gives the least utility. We impose it by forcing that the more valuable candidates (for a particular manipulator) are used only when all candida... |
[MATH] variables in the following constraint. [EQUATION] Finally, we give the last expression where [MATH] represents the pessimistic egalitarian [MATH] -egroup’s utility which our ILP program wants to maximize: |
[EQUATION] The ILP programs, for both tie-breaking variants, use at most [MATH] variables so, according to Lenstra ( 1983 , are in [MATH] with respect to the combined parameter [MATH] . Consequently, procedure |
[MATH] is in [MATH] with respect to the same parameter. After presenting the [MATH] result for egalitarian coalitional manipulation with optimistic or pessimistic egalitarian tie-breaking in Theorem 5 , we proceed with an analogous result for egalitarian coalitional manipulation with one of the four remaining tie-break... |
Theorem 6 Let [MATH] denote the number of manipulators and [MATH] denote the number of different utility values. Parameterized by [MATH] [MATH] Bloc [MATH] [MATH] Coalitional Manipulation is fixed-parameter tractable. |
Proof. The general proof idea is to show an algorithm which solves problem [MATH] Bloc [MATH] [MATH] Coalitional Manipulation To solve [MATH] Bloc [MATH] [MATH] Coalitional Manipulation we create an ILP program for all possible value combinations of the following parameters: |
the lowest final score [MATH] of any member of the [MATH] -egroup and the candidate [MATH] which is the least preferred member of the |
[MATH] -egroup with final score [MATH] with respect to the tie-breaking rule [MATH] Having [MATH] fixed, let [MATH] denote the set of candidates which get at least |
[MATH] approvals from the non-manipulative voters or which are preferred to [MATH] with respect to [MATH] and get exactly [MATH] approvals from the non-manipulative voters. Assuming that the combination of parameter values is correct, all candidates from [MATH] must belong to the |
[MATH] -egroup. We check whether [MATH] , that is, whether there is space for candidate [MATH] in the [MATH] -egroup. If the check fails, then we skip the corresponding combination of solution parameter values. Next, we ensure that |
[MATH] obtains final score exactly [MATH] . If [MATH] receives less than [MATH] or more than [MATH] approvals from non-manipulative voters, then we discard this combination of solution parameter values. Otherwise, let [MATH] denote number of additional approvals candidate [MATH] needs in order to get final score [MATH] |
We define the type of some candidate [MATH] to be the size- [MATH] vector [MATH] . We denote by [MATH] the set of all possible types. Observe that [MATH] . With each type vector [MATH] [MATH] , we associate a set |
[MATH] containing only the candidates of type [MATH] Having [MATH] (and [MATH] ) fixed, we distinguish candidates according to types further. For [MATH] , all candidates with score [MATH] that are preferred (resp. not preferred) to candidate [MATH] according to |
[MATH] , fall into group [MATH] (resp. [MATH] ). For each type [MATH] of a candidate, we define function [MATH] which gives the number of candidates of type [MATH] who are obligatory part of the winning |
[MATH] -egroup. We denote by [MATH] candidates which do not fall to any of such groups. We give the following ILP formulation of the problem using |
[MATH] variables. For all groups [MATH] and [MATH] [MATH] [MATH] , we introduce variables [MATH] and [MATH] respectively. The variables indicate, respectively, the number of candidates from groups [MATH] and |
[MATH] whom we push to the winning [MATH] -egroup. Also, we introduce two additional variables [MATH] and [MATH] . The former one represents the minimal value of the total utility achieved by manipulators. The latter one indicates the number of votes which were spent without changing the outcome. To shorten the ILP we ... |
[EQUATION] Intuitively, [MATH] is the number of approvals used to make potential winners the winners. Also, we define [EQUATION] |
[MATH] represents the number of approvals which cannot be used if one wants to avoid pushing candidates outside of the solution (given by values of the variables [MATH] ) to the winning [MATH] -egroup; for example, if some candidate [MATH] needs [MATH] approvals to be part of the winning committee, then we subtract [MA... |
[MATH] -egroup. We define the following constraints to construct our program the goal of which is to maximize [MATH] [EQUATION] Constraint ( 24 ) ensures that the candidates picked into a solution are available and can be part of the solution. Observe that candidates in [MATH] have to be part of the solution and candid... |
[MATH] cannot be part of the solution. These two facts are ensured by Constraints ( 25 ) and ( 26 ). Constraint ( 27 ) forbids spending more votes than possible to push some candidates to the [MATH] -egroup. The same role for “wasted” approvals plays Constraint ( 28 ). The upper bound of wasted approvals is counted in ... |
[MATH] -egroup when [MATH] is maximized. Using our technique we can obtain a solution by making [MATH] ILPs with at most [MATH] variables. According to Lenstra’s famous result (Lenstra, 1983 , the constructed ILPs yield fixed-parameter tractability with respect to the combined parameter [MATH] |
Conclusion We developed a new model for and started the first systematic study of coalitional manipulation for multiwinner elections. Our analysis revealed that multiwinner coalitional manipulation requires models which are significantly more complex than those for single-winner coalitional manipulation or multiwinner ... |
In our work, on the one hand, we generalized tractability results for coalitional manipulation of [MATH] -Approval by Conitzer et al. ( 2007 and Lin ( 2011 |
and for non-coalitional manipulation of Bloc by Meir et al. ( 2008 and Obraztsova et al. ( 2013 to tractability of coalitional manipulation of [MATH] Bloc in case of utilitarian or candidate-wise egalitarian evaluation of egroups. On the other hand, we showed that coalitional manipulation becomes intractable in case of... |
Let us discuss a few findings in more detail (Table surveys all our results). We studied lexicographic, optimistic, and pessimistic tie-breaking and showed that, with the exception of egalitarian group evaluation, winner groups can be determined very efficiently. The intractability (NP-hardness, parameterized hardness ... |
regardless of the specific tie-breaking rule. On the contrary, coalitional manipulation becomes tractable for the other two evaluation strategies—“candidate-wise” egalitarian and utilitarian. Additionally, for few candidates and few different utility values the voters assign to the candidates, manipulation becomes trac... |
In our study, we entirely focused on shortlisting as one of the simplest tasks for multiwinner elections to analyze our evaluation functions. It is interesting and non-trivial to develop models for multiwinner rules that aim for proportional representation or diversity. For shortlisting, extending our studies to non-ap... |
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