How Hard is it to Bribe the Judges? A Study of the Complexity of Bribery in Judgment Aggregation

Transcription

1 How Hard is it to Bribe the Judges? A Study of the Complexity of Bribery in Judgment Aggregation Dorothea Baumeister 1, Gábor Erdélyi 2, and Jörg Rothe 1 1 Institut für Informatik, Universität Düsseldorf, Düsseldorf, Germany 2 SPMS, Nanyang Technological University, Singapore Abstract. Endriss et al. [1, 2] initiated the complexity-theoretic study of problems related to judgment aggregation. We extend their results for manipulating two specific judgment aggregation procedures to a whole class of such procedures, and we obtain stronger results by considering not only the classical complexity (NP-hardness) but the parameterized complexity (W[2]-hardness) of these problems with respect to natural parameters. Furthermore, we introduce and study the closely related issue of bribery in judgment aggregation, inspired by work on bribery in voting (see, e.g., [3 5]). In manipulation scenarios one of the judges seeks to influence the outcome of the judgment aggregation procedure used by reporting an insincere judgment set. In bribery scenarios, however, an external actor, the briber, seeks to influence the outcome of the judgment aggregation procedure used by bribing some of the judges without exceeding his or her budget. We study three variants of bribery and show W[2]-hardness of the corresponding problems for natural parameters and for one specific judgment aggregation procedure. We also show that in certain special cases one can determine in polynomial time whether there is a successful bribery action. 1 Introduction In judgment aggregation (see, e.g., [6, 7]), the judges have to provide their judgments of a given set of possibly interconnected propositions, and if the simple majority rule is used to aggregate the individual judgments, the famous doctrinal paradox may occur (see [8] for the original formulation and [9] for a generalization). The study of different ways of influencing a judgment aggregation process is important, since the aggregation of different yes/no opinions about possibly interconnected propositions is often used in practice. To avoid the doctrinal paradox and, in general, inconsistencies in the aggregated judgment set, it is common to use a premise-based approach as we do here. In this approach, the individual judgments are given only over the premises, and the outcome for the conclusion is derived from the outcome for the premises. A simple example for such a premise-based judgment aggregation procedure under the majority rule is given in Table 1. In this example, which is due to Bovens and Rabinowicz [10] (see also [11]), the three judges of a tenure committee have to decide whether a candidate deserves tenure, based on their judgments of two issues: first, This work was supported in part by DFG grant RO 1202/12-1 and the European Science Foundation s EUROCORES program LogICCC. The second author was supported by National Research Foundation (Singapore) under grant NRF-RF

2 whether the candidate is good enough in research and, second, whether the candidate is good enough in teaching. The candidate should get tenure if and only if both requirements are satisfactorily fulfilled, which gives the decision of each individual judge in the right column of the table. To aggregate their individual judgments by the majority rule, both of the requirements (teaching and research) are evaluated by yes if and only if a strict majority of judges says yes. The result for the conclusion (whether or not the candidate deserves tenure) is then derived logically from the result of the premises. Note that this premise-based judgment procedure preserves consistency and thus circumvents the doctrinal paradox (which would occur if also the aggregated conclusion were obtained by applying the majority rule to the individual conclusions, leading to the contradiction (yes and yes) implies no ). Table 1. Example illustrating the premise-based procedure for the majority rule [10, 11] teaching research tenure judge 1 yes yes yes judge 2 yes no no judge 3 no yes no majority yes yes yes On the basis of the above example, List [11] concludes that in a premise-based procedure the judges might have an incentive to report insincere judgments. Suppose that in the above example all judges are absolutely sure that they are right, so they all want the aggregated outcome to be identical to their own conclusions. In this case, judge 3 knows that insincerely changing his or her judgment on the candidate s research capabilities from yes to no would aggregate with the other individual judgments on this issue to a no and thus would deny the candidate tenure. For the same reason, judge 2 might have an incentive to give an insincere judgment of the teaching question. This is a classical manipulation scenario, which has been studied in depth in the context of voting (see, e.g., the surveys by Conitzer [12] and Faliszewski et al. [13, 14] and the references cited therein). Strategic judging (i.e., changing one s individual judgments for the purpose of manipulating the collective outcome) was previously considered by List [11] and by Dietrich and List [15]. Endriss et al. [2] were the first to study the computational aspects of manipulation for judgment aggregation scenarios. Returning to the above example, suppose that the judgments of judges 2 and 3 in Table 1 were no for both premises. Then the candidate (who, of course, would like to get tenure by any means necessary) might try to make some deals with some of the judges (for example, offering to apply for joint research grants with judge 3, and offering to take some of the teaching load off judge 2 s shoulders, or just simply bribe the judges with money not exceeding his or her budget) in order to reach a positive evaluation. This is a classical bribery scenario which has been studied in depth in the context of voting (first by Faliszewski et al. [3], see also, e.g., [4, 5]) and in the context of optimal lobbying (first by Christian et al. [16], see also [17] and Section 4 for more

3 details). Manipulation, bribery, and lobbying are usually considered to be undesirable, and most of the recent literature on these topics is devoted to exploring the barriers to prevent such actions in terms of the computational complexity of the corresponding decision problems. We extend the results obtained by Endriss et al. [2] on the complexity of manipulation in judgment aggregation from two specific judgment aggregation procedures to a whole class of such procedures. We study the corresponding manipulation problems not only in terms of their classical complexity but in terms of their parameterized complexity with respect to two natural parameters, one being the total number of judges and the other one being the maximum number of changes in the premises needed in the manipulator s judgment set. The W[2]-hardness results we obtain in particular imply the NP-hardness results Endriss et al. [2] obtained for the unparameterized problem. Finally, inspired by bribery in voting [3], we introduce the concept of bribery in judgment aggregation. We consider three types of bribery (exact bribery, bribery, and microbribery) and define and motivate the corresponding bribery problems for judgment aggregation, building on the related but simpler model of optimal lobbying (see [16, 17]). We show that, for one specific judgment aggregation procedure, each of the three types of bribery is W[2]-hard with respect to natural parameters; again, note that NP-completeness follows for the corresponding unparameterized problems. One natural parameter we study here is again the total number of judges. Showing W[2]- hardness for this parameter implies that the problem remains hard even if the number of judges is bounded by a constant. As this is often the case in judgment aggregation, it is natural to study this parameter. By contrast, we also show that in certain cases one can determine in polynomial time whether there exists a successful bribery action. Both manipulation and bribery were first defined and studied for preference aggregation, especially in voting scenarios. By the above examples we have argued that it makes sense to study these issues also in the context of judgment aggregation. There is, however, one major difference between the aggregation of preferences via voting systems and judgment aggregation. Both fields are closely related but consider different settings (for further details, see [7, 18]). In voting, the individuals report their subjective personal preference over some given alternatives. For example, one voter may prefer alternative a to alternative b, and another voter may prefer b to a. This does not contradict, and even if both voters may not understand the other voter s preferences on a and b, they should accept them. In judgment aggregation, however, the judges report their individual judgment of some given proposition ϕ. If there are two judges, one reporting ϕ is true and the other reporting ϕ is false, they have contradicting individual judgments regarding ϕ. These two judges with opposing judgments for the same proposition will simply believe the other one is wrong. In certain cases it might even be possible to objectively determine the truth value of the proposition and decide who of the judges is right and who is wrong. This would be impossible to say for an individual preference. 2 Preliminaries The formal definition of the judgment aggregation framework follows the work of Endriss et al. [2]. The set of all propositional variables is denoted by PS, and the set of

4 propositional formulas built from PS is denoted by L PS. As connectives in propositional formulas, we allow disjunction ( ), conjunction ( ), implication ( ), and equivalence ( ) in their usual meaning, and the two boolean constants 1 and 0 representing true and false, respectively. Since double negations are undesirable, let α denote the complement of α. This means that if α is not negated then α = α, and if α = β then α = β. The set of formulas to be judged by the judges is called the agenda. Formally, the agenda is a finite, nonempty subset Φ of L PS. As mentioned above, the agenda does not contain doubly negated formulas, and it also holds that α Φ for all α Φ, that is, Φ is required to be closed under complementation. The judgment provided by a single judge is called his or her individual judgment set and corresponds to the propositions in the agenda accepted by this judge. The set of propositions accepted by all judges is called their collective judgment set. An individual or collective judgment set J on an agenda Φ is a subset J Φ. We consider three basic properties of judgment sets, completeness, complementfreeness, and consistency. A judgment set J is said to be complete if it contains α or α for each α Φ. We say J is complement-free if there is no α J with α J. Finally, J is consistent if there is an assignment that satisfies all formulas in J. We denote the set of all complete and consistent subsets of Φ by J(Φ). Obviously, all sets in J(Φ) are also complement-free. We let N={1,...,n} denote the set of judges taking part in a judgment aggregation scenario, and we will always assume that there are at least two judges, so n 2. The individual judgment set of judge i N is denoted by J i, and the profile of all n individual judgment sets is denoted by J=(J 1,...,J n ). To obtain a collective judgment set from a given profile J J(Φ) n, an aggregation procedure F is needed. This is a function F : J(Φ) n 2 Φ, mapping a profile of n complete and consistent judgment sets to a subset of the agenda Φ, the collective judgment set. We consider the same three basic properties for judgment aggregation procedures as for judgment sets. A judgment aggregation procedure F is said to be complete/complement-free/consistent if F(J) is complete/complement-free/consistent for all profiles J J(Φ) n. One particular judgment aggregation procedure studied by Endriss et al. [2] is the premise-based procedure. Definition 1 (Premise-based Procedure [2]). Let the agenda Φ be divided into two disjoint sets, Φ = Φ p Φ c, where Φ p is the set of premises and Φ c is the set of conclusions, and both Φ p and Φ c are closed under complementation. The premise-based procedure is a function PBP : J(Φ) n 2 Φ mapping, for Φ = Φ p Φ c, each profile J=(J 1,...,J n ) to the following judgment set: PBP(J)= {ϕ Φ c = ϕ} with ={ϕ Φ p {i ϕ J i } >n/2}, where S denotes the cardinality of set S and = denotes the satisfaction relation. According to this definition, the majority procedure is applied only to the premises of the agenda, and the collective outcome for the conclusions is derived from the collective outcome of the premises. However, this is not sufficient to obtain a complete and consistent procedure. To achieve this, it is furthermore required that the agenda is

5 closed under propositional variables (i.e., every variable that occurs in a formula of Φ is contained in Φ), that the set of premises is the set of all literals in the agenda, and that the number of judges is odd. Endriss et al. [2] argue that this definition is appropriate, since the problem of determining whether an agenda guarantees a complete and consistent outcome for the majority procedure is an intractable problem. We extend this approach to the class of uniform quota rules as defined by Dietrich and List [19]. We allow an arbitrary quota and do not restrict our scenarios to an odd number of judges. Definition 2 (Premise-based Quota Rule). Let the agenda Φ be divided into two disjoint sets, Φ = Φ p Φ c, where Φ p is the set of premises and Φ c is the set of conclusions, and both Φ p and Φ c are closed under complementation. Divide the set of premises Φ p into two disjoint subsets, Φ 1 and Φ 2, such that for each ϕ Φ p, either ϕ Φ 1 and ϕ Φ 2 or ϕ Φ 2 and ϕ Φ 1. Define a quota q ϕ Q with 0 q ϕ < 1 for every ϕ Φ 1. The quota for every ϕ Φ 2 is then defined as q ϕ = 1 q ϕ. The premise-based quota rule is a function PQR : J(Φ) n 2 Φ mapping, for Φ = Φ p Φ c, each profile J=(J 1,...,J n ) to the following judgment set: where PQR(J)= q {ϕ Φ c q = ϕ}, q ={ϕ Φ 1 {i ϕ J i } >n q ϕ } {ϕ Φ 2 {i ϕ J i } > n q ϕ 1 }. To obtain complete and consistent collective judgment sets, we again require that the agenda Φ is closed under propositional variables, and that Φ p consists of all literals. The number of affirmations needed to be in the collective judgment set may differ for the variables in Φ 1 and in Φ 2. For ϕ Φ 1, at least n q ϕ + 1 affirmations from the judges are needed, and for ϕ Φ 2, n q ϕ affirmations are needed. Clearly, since n q ϕ n q ϕ =n+1, it is ensured that for every ϕ Φ, either ϕ PQR(J) or ϕ PQR(J). Observe that the quota q ϕ = 1 for a literal ϕ Φ 1 is not considered here, since then n+1 affirmations were needed for ϕ Φ 1 to be in the collective judgment set, which is not possible. Hence, the outcome does not depend on the individual judgment sets. By contrast, considering q ϕ = 0 leads to the case that ϕ Φ 1 needs at least one affirmation, and ϕ Φ 2 needs n affirmations, which may be a reasonable choice. If the quota q ϕ is identical for all literals in Φ 1 and hence also the quota q ϕ for all literals in Φ 2, we obtain the special case of uniform premise-based quota rules. The quotas will then be q for all ϕ Φ 1 and q for all ϕ Φ 2. In this paper, we focus on this class of rules, and denote it by UPQR q. For the case of q=1/2 and an odd number of judges, we obtain exactly the premise-based procedure defined by Endriss et al. [2] (see Definition 1). We assume that the reader is familiar with the basic concepts of complexity theory and with complexity classes such as P and NP; see, e.g., [20]. Downey and Fellows [21] introduced parameterized complexity theory; in their framework it is possible to do a more fine-grained multi-dimensional complexity analysis. In particular, NP-complete problems may be easy (i.e., fixed-parameter tractable) with respect to certain parameters confining the seemingly unavoidable combinatorial explosion. If this parameter

6 is reasonably small, a fixed-parameter tractable problem can be solved efficiently in practice, despite its NP-hardness. Formally, a parameterized decision problem is a set L Σ N, and we say it is fixed-parameter tractable (FPT) if there is a constant c such that for each input(x,k) of size n= (x,k) we can determine in time O( f(k) n c ) whether (x,k) is in L, where f is a function depending only on the parameter k. The main hierarchy of parameterized complexity classes is: FPT=W[0] W[1] W[2] W[l] XP. In our results, we will focus on only the class W[2], which refers to problems that are considered to be fixed-parameter intractable. In order to show that a parameterized problem is W[2]-hard, we will give a parameterized reduction from the W[2]-complete problem k-dominating SET (see [21]). We say that a parameterized problem A parameterized reduces to a parameterized problem B if each instance (x,k) of A can be transformed in time O(g(k) x c ) (for some function g and some constant c) into an instance(x,k ) of B such that(x,k) Aif and only if(x,k ) B, where k = g(k). 3 Problem Definitions Bribery problems in voting theory, as introduced by Faliszewski et al. [3] (see also, e.g., [4, 5]), model scenarios in which an external actor seeks to bribe some of the voters to change their votes such that a distinguished candidate becomes the winner of the election. In judgment aggregation it is not the case that one single candidate wins, but there is a decision for every formula in the agenda. So the external actor might seek to obtain exactly his or her desired collective outcome by bribing the judges, or he or she might be interested only in the desired outcome of some formulas in Φ. The exact bribery problem is then defined as follows for a given aggregation procedure F. EXACT-F-BRIBERY Given: Question: An agenda Φ, a profile T J(Φ) n, a consistent and complement-free judgment set J (not necessarily complete) desired by the briber, and a positive integer k. Is it possible to change up to k individual judgment sets in T such that for the resulting new profile T it holds that J F(T )? Note that if J is a complete judgment set then the question is whether J = F(T ). Since in the case of judgment aggregation there is no winner, we also adopt the approach Endriss et al. [2] used to define the manipulation problem in judgment aggregation. In their definition, an outcome (i.e., a collective judgment set) is more desirable for the manipulator if its Hamming distance to the manipulator s desired judgment set is smaller, where for an agenda Φ the Hamming distance H(J,J ) between two complete and consistent judgment sets J,J J(Φ) is defined as the number of positive formulas in Φ on which J and J differ. The formal definition of the manipulation problem in judgment aggregation is as follows, for a given aggregation procedure F.

7 F-MANIPULATION Given: An agenda Φ, a profile T J(Φ) n 1, and a consistent and complete judgment set J desired by the manipulator. Question: Does there exist a judgment set J J(Φ) such that H(J,F(T,J )) < H(J,F(T,J))? Now, we can give the formal definition of bribery in judgment aggregation, where the briber seeks to obtain a collective judgment set having a smaller Hamming distance to the desired judgment set, then the original outcome has. In bribery scenarios, we extend the above approach of Endriss et al. [2] by allowing that the desired outcome for the briber may be an incomplete (albeit consistent and complement-free) judgment set. This reflects a scenario where the briber may be interested only in some part of the agenda. The definition of Hamming distance is extended accordingly as follows. Let Φ be an agenda, J J(Φ) be a complete and consistent judgment set, and J Φ be a consistent and complement-free judgment set. The Hamming distance H(J,J ) between J and J is defined as the number of formulas from J on which J does not agree: H(J,J )= {ϕ ϕ J ϕ J}. Observe that if J is also complete, this extended notion of Hamming distance coincides with the notion Endriss et al. [2] use. F-BRIBERY Given: Question: An agenda Φ, a profile T J(Φ) n, a consistent and complement-free judgment set J (not necessarily complete) desired by the briber, and a positive integer k. Is it possible to change up to k individual judgment sets in T such that for the resulting new profile T it holds that H(F(T ),J)<H(F(T),J)? Faliszewski et al. [5] introduced microbribery for voting systems. We adopt their notion so as to apply to judgment aggregation. In microbribery for judgment aggregation, if the briber s budget is k, he or she is not allowed to change up to k entire judgment sets but instead can change up to k premise entries in the given profile (the conclusions change automatically if necessary). F-MICROBRIBERY Given: Question: An agenda Φ, a profile T J(Φ) n, a consistent and complement-free judgment set J (not necessarily complete) desired by the briber, and a positive integer k. Is it possible to change up to k entries among the premises in the individual judgment sets in T such that for the resulting profile T it holds that H(F(T ),J)<H(F(T),J)? EXACT-F-MICROBRIBERY is defined analogously to the corresponding bribery problem with the difference that the briber is allowed to change only up to k entries in T rather than to change k complete individual judgment sets.

8 In our proofs we will make use of the following two problems. First, we will use DOMINATING SET, a classical problem from graph theory. Given a graph G =(V, E), a dominating set is a subset V V such that for each v V \V there is an edge{v,v } in E with v V. The size of a dominating set V is the number V of its vertices. DOMINATING SET Given: A graph G=(V,E), with the set V of vertices and the set E of edges, and a positive integer k V. Question: Does G have a dominating set of size at most k? DOMINATING SET is NP-complete (see [22]) and, when parameterized by the upper bound k on the size of the dominating set, its parameterized variant (denoted by k- DOMINATING SET, to be explicit) is W[2]-complete [21]. Second, we will use the following problem: OPTIMAL LOBBYING Given: Question: An m n 0-1 matrix L (whose rows represent the voters, whose columns represent the referenda, and whose 0-1 entries represent No/Yes votes), a positive integer k m, and a target vector x {0,1} n. Is there a choice of k rows in L such that by changing the entries of these rows the resulting matrix has the property that, for each j, 1 j n, the jth column has a strict majority of ones (respectively, zeros) if and only if the jth entry of the target vector x of The Lobby is one (respectively, zero)? OPTIMAL LOBBYING has been introduced and, parameterized by the number k of rows The Lobby can change, shown to be W[2]-complete by Christian et al. [16] (see also [17] for a more general framework and more W[2]-hardness results). Note that a multiple referendum as in OPTIMAL LOBBYING can be seen as the special case of a judgment aggregation scenario where the agenda is closed under complementation and propositional variables and contains only premises and where the majority rule is used for aggregation. For illustration, consider the following simple example of a multiple referendum. Suppose the citizens of a town are asked to decide by a referendum whether two projects, A and B (e.g., a new hospital and a new bridge), are to be realized. Suppose the building contractor (who, of course, is interested in being awarded a contract for both projects) sets some money aside to attempt to influence the outcome of the referenda, by bribing some of the citizens without exceeding this budget. Observe that an EXACT-PBP-BRIBERY instance with only premises in the agenda and with a complete desired judgment set J is nothing other than an OPTIMAL LOBBY- ING instance, where J corresponds to The Lobby s target vector. 1 Requiring the citizens to give their opinion only for the premises A and B of the referendum and not for the conclusion (whether both projects are to be realized) again avoids the doctrinal paradox. 1 Although exact bribery in judgment aggregation thus generalizes lobbying in the sense of Christian et al. [16] (which is different from bribery in voting, as defined by Faliszewski et al. [3]), we will use the term bribery rather than lobbying in the context of judgment aggregation.

9 Again, the citizens might also vote strategically in these referenda. Both projects will cost money, and if both projects are realized, the amount available for both must be reduced. Some citizens may wish to support some project, say A, but they are not satisfied if the amount for A would be reduced when both projects are realized. For them it is natural to consider the possibility of reporting insincere votes (provided they know how the others will vote); this may turn out to be more advantageous for them, as then they possibly can prevent that both projects are realized. 4 Results 4.1 Manipulation in Judgment Aggregation We start by extending the result of Endriss et al. [2] that PBP-MANIPULATION is NPcomplete. We study two parameterized versions of the manipulation problem and establish W[2]-hardness results for them with respect to the uniform premise-based quota rule. Theorem 1. For each rational quota q, 0 q<1, UPQR q -MANIPULATION is W[2]- hard when parameterized either by the total number of judges, or by the maximum number of changes in the premises needed in the manipulator s judgment set. Proof. We start by giving the details for q = 1/2, and later explain how this proof can be extended to capture any other rational quota values q with 0 q < 1. The proof for both parameters will be by one reduction from the W[2]-complete problem k-dominating SET. Given a graph G =(V,E) with the set of vertices V = {v 1,...,v n }, define N(v i ) as the closed neighborhood of vertex v i, i.e., the union of the set of vertices adjacent to v i and the vertex v i itself. Then, V is a dominating set for G if and only if N(v i ) V /0 for each 1 i n. We will now describe how to construct a bribery instance for judgment aggregation. Let the agenda Φ contain the variables 2 v 1,...,v n,y and their negations, the formula ϕ i = (v 1 i v j i ) y and its negation, where {v 1 i,...,vj i }=N(v i) for each i, 1 i n, and n 1 syntactic variations of each of these formulas and its negation. This can be seen as giving each formula ϕ i a weight of n. A syntactic variation of a formula can, for example, be obtained by an additional conjunction with the constant 1. Furthermore, Φ contains the formula v 1 v n, its negation, and n 2 k 2 syntactic variations of this formula and its negation; this can be seen as giving this formula a weight of n 2 k 1. The set of judges is N ={1,2,3}, with the individual judgment sets J 1, J 2, and J 3 (where J 3 is the judgment set of the manipulative judge), and the collective judgment set as shown in Table 2. Note that the Hamming distance between J 3 and the collective judgment set is 1+n 2. We claim that there is an alternative judgment set for J 3 that yields a smaller Hamming distance to the collective outcome if and only if there is a dominating set of size at most k for G. ( ) Assume that there is a dominating set V of G with V =k. (If V <k, we simply add any k V vertices to obtain a dominating set of size exactly k.) Regarding 2 We use the same identifiers v 1,...,v n for the vertices of G and the variables in Φ, specifying the intended meaning only if it is not clear from the context.

10 Table 2. Construction for the proof of Theorem 1 Judgment Set v 1 v n y ϕ 1 ϕ n v 1 v n J J J UPQR 1/2 (J) the premises, the judgment set of the manipulator contains the variables v i V and also the literal y. Then the collective outcome also contains the variables v i V, and since V is a dominating set, each ϕ i, 1 i n, evaluates to true and the formula v 1 v n is also evaluated to true. The Hamming distance to the original judgment set of the manipulator is then k+ 1+(n 2 k 1)=n 2. Hence the manipulation was successful, and the number of entries changed in the judgment set of the manipulator is exactly k. ( ) Now assume that there is a successful manipulation with judgment set J. The manipulator can change only the premises in the agenda to achieve a better outcome for him or her. A change for the literal y changes nothing in the collective outcome, hence the changes must be within the set {v 1,...,v n }. Including j of the v i into J has the effect that these v i are included in the collective judgment set, and that all variations of the formula v 1 v n and of those ϕ i that are evaluated to true are also included in the collective judgment set. Iflformulas ϕ i are evaluated to true in the collective judgment set, the Hamming distance is j+ 1+(n 2 nl)+(n 2 k 1). Since the manipulation was successful, the Hamming distance can be at most n 2. If l < n, it must hold that j k n, which is not possible given that k n and j>0. Hence,l=n and j= k. Then exactly k literals v i are set to true, and since this satisfies all ϕ i, they must correspond to a dominating set of size k, concluding the proof for the quota q=1/2 and three judges. This proof can be adapted to work for any fixed number m 3 of judgment sets S 1,...,S m and for any rational value of q, with 1 m q < m. The agenda remains the same, but S 1,...,S mq are each equal to the judgment set J 1 and S mq +1,...,S m 1 are each equal to the judgment set J 2. The judgment set S m of the manipulative judge equals the judgment set J 3, and the quota is q for every positive variable and 1 q for every negative variable. The number of affirmations every positive formula needs to be in the collective judgment set is then mq +1. Then the same argumentation as above holds. The remaining case, where 0 mq < 1, can be handled by a slightly modified construction. Since the number of judges is fixed for any fixed value of m and q, and the number of premises changed by the manipulator depends only on the size k of the dominating set, W[2]-hardness for UPQR q -MANIPULATION holds for both parameters. Since DOMINATING SET is an NP-complete problem, NP-completeness of UPQR q - MANIPULATION follows immediately from the proof of Theorem 1 for any fixed number n 3 of judges. Note that NP-hardness of UPQR q -MANIPULATION could have also been shown by a modification of the proof of Theorem 2 in [2], but this reduction would

11 not be appropriate to establish W[2]-hardness, since the corresponding parameterized version of SAT is not known to be W[2]-hard. As mentioned above, studying the parameterized complexity for the parameter total number of judges is very natural. The second parameter we have considered for the manipulation problem in Theorem 1 is the maximum number of changes in the premises needed in the manipulator s judgment set. Hence this theorem shows that the problem remains hard even if the number of premises the manipulator can change is bounded by a fixed constant. This is also very natural, since the manipulator may wish to report a judgment set that is as close as possible to his or her sincere judgment set, because for a completely different judgment set it might be discovered too easily that he was judging strategically. In contrast to the hardness results stated in Theorem 1, the following proposition shows that, depending on the agenda, there are cases in which UPQR q -MANIPULATION is solvable in polynomial time. Proposition 1. If the agenda contains only premises then UPQR q -MANIPULATION is in P. Proof. Assume that the agenda Φ contains only premises. Then every variable is considered independently. Let n be the number of judges. If ϕ is contained in the judgment set J of the manipulator, and ϕ does not have n q+1 (respectively, n(1 q) ) affirmations without considering J, it cannot reach the required number of affirmations if the manipulator switches from ϕ to ϕ in his or her judgment set. The W[2]-hardness result for UPQR q -MANIPULATION, parameterized by the number of judges, stated in Theorem 1 implies that there is little hope to find a polynomialtime algorithm for the general problem even when the number of judges participating is fixed. However, Proposition 1 tells us that if the agenda is simple and contains no conclusions, the problem can be solved efficiently even when the number of judges participating is not fixed. 4.2 Bribery in Judgment Aggregation In this section we will study the complexity of several bribery problems for the premisebased procedure PBP, i.e., UPQR 1/2 for an odd number of judges. We will again establish even W[2]-hardness results for two natural parameters for these bribery problems. Theorem 2. PBP-BRIBERY is W[2]-hard when parameterized either by the total number of judges, or by the number of judges that can be bribed. Proof. We will show W[2]-hardness by a slightly modified construction from Theorem 1. We start by considering the case, where the briber is allowed to bribe exactly one judge. The notation and the agenda from that proof remain unchanged, but the individual judgment sets are slightly different. The first two judges remain unchanged, but the third judge has the same judgment set as the second one, and the desired judgment set J is equal to J 3. Since the quota is 1/2, two affirmations are needed to be in the collective judgment set. Again the briber cannot benefit from bribing one judge to switch from y to y in his or her individual judgment set. Hence the change must be in the set of

12 variables {v 1,...,v n } from the second or the third judge. By a similar argument as in the proof of Theorem 1, there is a successful bribery action if and only if there is a dominating set of size at most k for the given graph. Now we consider the case that the briber is allowed to bribe more than one judge. If the briber is allowed to bribe k judges, we construct an instance with 2k+1 judges, where one judgement set is equal to J 1 and the remaining 2k individual judgment sets are equal to J 2. It is again not possible for the briber to change the entry for y, and the briber must change the entry for any v i in the judgment sets from k judges to obtain a different collective outcome. This construction works by similar arguments as above. Since the total number of judges and the number of judges that can be bribed depends only on k, W[2]-hardness follows for both parameters. As in the case of manipulation, the proof of Theorem 2 immediately implies an NP-completeness result for PBP-BRIBERY. Next, we turn to microbribery. Here the briber can change only up to a fixed number of entries in the individual judgment sets. We again start by proving W[2]-hardness for the parameters number of judges and number of microbribes allowed. Theorem 3. PBP-MICROBRIBERY is W[2]-hard when parameterized either by the total number of judges, or by the number of microbribes allowed. Proof. The proof that PBP-MICROBRIBERY is W[2]-hard is similar to the proof of Theorem 2. The given instance for the k-dominating SET Problem is the graph G=(V,E) and the positive integer k. The agenda Φ is defined as in the proof of Theorem 1. The number of judges is 2k+ 1, where the individual judgment sets of k judges are of type J 1 and the remaining k+ 1 individual judgment sets are of type J 2. The desired outcome of the briber is the judgment set J 3. The number of affirmations needed to be in the collective judgment set is at least k+ 1, and the number of entries the briber is allowed to change is at most k. Since none of the judges have y in their individual judgment sets, the briber cannot change the collective outcome for y to 1. Hence all entries that can be changed are for the variables v 1,...,v n. Obviously, setting the value for one v i in one of the judges of type J 2 to 1 causes v i to be in the collective judgment set and all other changes have no effect on the collective judgment set. By similar arguments as in the proof of Theorem 1, there is a successful microbribery action if and only if the given graph has a dominating set of size at most k. Since both the total number of judges and the number of entries the briber is allowed to change depend only on k, W[2]-hardness follows directly for both parameters. Again, NP-hardness of PBP-MICROBRIBERY follows immediately from that of DOMINATING SET. Theorem 4. EXACT-PBP-BRIBERY is W[2]-hard when parameterized by the number of judges that can be bribed. Proof. Observe that an exact bribery instance with only premises in the agenda and with a complete desired judgment set J is exactly the OPTIMAL LOBBYING problem. Since this problem is W[2]-complete for the parameter number of rows that can be changed, EXACT-PBP-BRIBERY inherits the W[2]-hardness lower bound, where the parameter is the number of judges that can be bribed.

13 Note that W[2]-hardness with respect to any parameter directly implies NP-hardness for the corresponding unparameterized problem, so EXACT-PBP-BRIBERY is also NPcomplete (all unparameterized problems considered here are easily seen to be in NP). Theorem 5. EXACT-PBP-MICROBRIBERY is W[2]-hard when parameterized either by the number of judges, or by the number of microbribes. Proof. Consider the construction in the proof of Theorem 3, and change the agenda such that there are only n 2 2 (instead of n 2 k 2) syntactic variations of the formula v 1 v n (i.e., this can be seen as giving a weight of n 2 1 to this formula), and that the desired judgment set J is incomplete and contains all conclusions. By similar arguments as above, a successful microbribery of k entries is possible if and only if there is a dominating set for G of size at most k. As for the manipulation problem, we studied in Theorems 2 through 5 the bribery problems for the natural parameter total number of judges. It turned out that for that parameter BRIBERY, MICROBRIBERY, and their exact variants are W[2]-hard for the premise-based procedure for the majority rule. Hence these four problems remain hard even if the total number of judges is fixed. Furthermore we considered the parameter number of judges allowed to bribe for PBP-BRIBERY and its exact variant and the parameter number of microbribes allowed for PBP-MICROBRIBERY and its exact variant. Both parameters concern the budget of the briber. Since the briber aims at spending as little money as possible, it is also natural to consider this parameter. But again W[2]-hardness was shown in all cases, which means that bounding the budget by a fixed constant does not help to solve the problem easily (i.e., it is unlikely to be fixed-parameter tractable). Although the exact microbribery problem is computationally hard in general for the aggregation procedure PBP, there are some interesting naturally restricted instances where it is computationally easy. Theorem 6. If the desired judgment set J is complete or if the desired judgment set is incomplete but contains all of the premises or only premises, then EXACT-PBP- MICROBRIBERY is in P. Proof. We give only an informal description of the algorithm that computes a successful microbribery. Input: Our algorithm takes as an input a complete profile T, a consistent judgment set J, and a positive integer k. Step 1: For each premise present in J, compute the minimum number of entries that have to be flipped in order to make the collective judgment on that premise equal to the desired judgment set s entry on that premise. Note that this can be done in linear time, since it is a simple counting. Let d i denote the number of entries needed to flip for premise i. Step 2: Check if i d i k. Output: If i d i k, output the entries which have to be flipped and halt. Otherwise, output bribery impossible and halt.

14 Clearly, this algorithm works in polynomial time. The output is correct, since if we need less than k flips in the premises, the premises are evaluated exactly as they are in J, and the conclusions follow automatically, since we are using a premise-based procedure. 5 Conclusions Following up a line of research initiated by Endriss et al. [1, 2], we have studied the computational complexity of problems related to manipulation and bribery in judgment aggregation. In particular, the complexity of bribery though deeply investigated in the context of voting [3 5] has not been studied before in the context of judgment aggregation. For three natural scenarios modelling different ways of bribery, we have shown that the corresponding problems are computationally hard even with respect to their parameterized complexity (namely, W[2]-hard) for natural parametrizations. In addition, extending the results of Endriss et al. [2] on the (classical) complexity of manipulation in judgment aggregation, we have obtained W[2]-hardness for the class of uniform premise-based quota rules, for each reasonable quota. From all W[2]-hardness results we immediately obtain the corresponding NP-hardness results, and since all problems considered are easily seen to be in NP, we have NP-completeness results. It remains open, however, whether one can also obtain matching upper bounds in terms of parameterized complexity. We suspect that all W[2]-hardness results in this paper in fact can be strengthened to W[2]-completeness results. Faliszewski et al. [3] introduced and studied also the priced and weighted versions of bribery in voting. These notions can be reasonably applied to bribery in judgment aggregation: The priced variant means that judges may request different amounts of money to be willing to change their judgments according to the briber s will, and the weighted variant means that the judgments of some judges may be heavier than those of others. Although we have not defined this in a formal setting here, note that our hardness results carry over to more general problem variants as well. A more interesting task for future research is to try to complement our parameterized worstcase hardness results by studying the typical-case behavior for these problems, as is currently done intensely in the context of voting. Another interesting task is to study these problems for other natural parameters and for other natural judgment aggregation procedures. Acknowledgments: We thank the anonymous reviewers for their helpful reviews and literature pointers. References 1. Endriss, U., Grandi, U., Porello, D.: Complexity of judgment aggregation: Safety of the agenda. In: Proceedings of the 9th International Joint Conference on Autonomous Agents and Multiagent Systems, IFAAMAS (May 2010)

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