Complexity Results for Manipulation, Bribery and Control of the Kemeny Procedure in Judgment Aggregation

08/08/2016 ∙ by Ronald de Haan, et al. ∙ TU Wien 0

We study the computational complexity of several scenarios of strategic behavior for the Kemeny procedure in the setting of judgment aggregation. In particular, we investigate (1) manipulation, where an individual aims to achieve a better group outcome by reporting an insincere individual opinion, (2) bribery, where an external agent aims to achieve an outcome with certain properties by bribing a number of individuals, and (3) control (by adding or deleting issues), where an external agent aims to achieve an outcome with certain properties by influencing the set of issues in the judgment aggregation situation. We show that determining whether these types of strategic behavior are possible (and if so, computing a policy for successful strategic behavior) is complete for the second level of the Polynomial Hierarchy. That is, we show that these problems are Σ^p_2-complete.

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1 Introduction

An important topic in the research field of computational social choice is the (im)possibility of strategic behavior in collective decision making. This is epitomized by the eminence of results such as the Gibbard-Satterthwaite Theorem [16, 22], that identifies various conditions under which strategic voting (or manipulation) is, in principle, unavoidable. Manipulation in voting is a typical example of strategic behavior, and involves individuals reporting insincere preferences with the aim of obtaining a group outcome that is preferable for them.

Since strategic behavior in collective decision making is generally considered to be (socially) undesirable, a lot of research effort has been invested in diagnosing what social choice procedures are resistant to strategic behavior, and under what conditions. An important research direction along these lines investigates how computational complexity can be used to establish that various social choice procedures are (in many cases) practically immune to strategic behavior [2, 5]. For example, in many cases, it is in principle possible to manipulate voting rules (by reporting insincere preferences), but determining what insincere preference leads to a better outcome is computationally so demanding that it prevents manipulative behavior from being a useful policy.

Contributions

In this paper, we use the framework of computational complexity theory to study several scenarios of strategic behavior in the setting of judgment aggregation. Judgment aggregation studies collective decision making on a set of issues that are logically related [12]. In particular, we study three scenarios of strategic behavior for the Kemeny judgment aggregation procedure—which is one of the most prominent judgment aggregation procedures known from the literature. We investigate:

  1. manipulation, where an individual reports an insincere individual judgment in an attempt to enforce a preferable group judgment (from their point of view);

  2. bribery, where an external party bribes several individuals that are involved in the group decision process (that is, the briber stipulates their individual judgments) in order to obtain a group judgment with certain properties; and

  3. control, where an external party controls the set of issues that are involved in the judgment aggregation setting, with the aim of achieving a group judgment with certain properties.

Concretely, we study various different decision problems that formalize the computational tasks involved in the strategic behavior in each of these scenarios. We show that all the computational problems that we consider in this paper are -complete. That is, we show that:

  • Manipulation for the Kemeny rule in judgment aggregation is -complete.

  • Bribery for the Kemeny rule in judgment aggregation is -complete.

  • Control (by adding or removing issues) for the Kemeny rule in judgment aggregation is -complete.

(Completeness for the complexity class indicates that a problem is computationally intractable. Even the easier problem of checking whether a given candidate solution is in fact a solution is not efficiently solvable—it requires solving an NP-complete problem.)

Various different frameworks have been used in the literature to formalize the setting of judgment aggregation (see, e.g., [13]). The computational complexity results that we develop hold for two commonly considered judgment aggregation frameworks: formula-based judgment aggregation and constraint-based judgment aggregation. We discuss these judgment aggregation frameworks in more detail in Section 2.2. (In order to capture the scenario of control naturally in the constraint-based judgment aggregation framework, we consider a slightly extended variant of this framework. For more details, see Section 2.2.3.)

Most of the various forms of strategic behavior that we consider in this paper involve the incentive of achieving a preferable group outcome. There are various ways to define preference relations over (individual and group) judgments. The preferences that we study are based on weighted Hamming distances. That is, we consider weight functions that assign to each issue a weight that indicates how important it is for an individual (or for an external party) that the group judgment agree with their judgment on this issue. Such weight functions naturally induce preference relations over judgments.

In addition, we study variants of the strategic behavior scenarios where the objective is to obtain a group judgment that includes a given set of conclusions. This can be seen as an all-or-nothing variants (the group outcome either includes the required set of conclusions or it does not), whereas the variants involving preferences based on weighted Hamming distances offer a more gradual view (maybe the optimal outcome is not possible, but the current outcome can still be improved slightly by behaving strategically).

Worst-case Complexity

The computational intractability results that we provide in this paper can be seen as positive results, since they show that various kinds of undesirable strategic behavior cannot be used efficiently across the board due to computational complexity obstructions. However, it is important to emphasize that the computational complexity results that we provide in this paper are worst-case complexity results. Worst-case intractability results indicate that there is no algorithm that works efficiently in all possible cases. However, it might well be the case that there are restricted settings where several forms of strategic behavior are efficiently possible.

In order to consolidate the conclusion that strategic behavior for the Kemeny procedure in judgment aggregation is computationally intractable, further research is needed. Such further research would have to establish that the various forms of strategic behavior remain computationally intractable in many restricted settings. A key tool for establishing computational complexity results for restricted settings is the paradigm of parameterized complexity [10, 11, 15, 20]

—this is a framework where the complexity of computational problems is measured in a multi-dimensional way, in contrast to the classical theory of computational complexity, where the complexity of problems is measured only in terms of the input size in bits.

Related Work

The concept of manipulation in judgment aggregation has been studied before in the literature, both from an axiomatic point of view [4, 6, 8] and from a computational complexity point of view [3, 14]. The complexity analysis of manipulation in judgment aggregation that has been done in the literature is restricted to uniform premise-based quota rules. Additionally, bribery in judgment aggregation has been studied from a computational complexity point of view for uniform premise-based quota rules [3].

Outline

We begin in Section 2 by considering relevant notions from computational complexity and judgment aggregation that we use in this paper. Then, in Section 3, we develop the intractability results for the scenario of manipulation. In Section 4, we turn to the scenario of bribery, and in Section 5, we consider the scenario of control (by adding or deleting issues). Finally, we conclude in Section 6.

2 Preliminaries

Before we turn to the complexity results that we develop in this paper, we review several relevant concepts from computational complexity theory and judgment aggregation.

2.1 Complexity Theory

We begin with reviewing some basic notions from computational complexity. We assume the reader to be familiar with the complexity classes P and NP, and with basic notions such as polynomial-time reductions. For more details, we refer to textbooks on computational complexity theory (see, e.g., [1]).

We briefly review the classes of the Polynomial Hierarchy (PH) [19, 21, 23, 24]. In order to do so, we consider quantified Boolean formulas. A (fully) quantified Boolean formula (in prenex form) is a formula of the form , where all  are propositional variables, each  is either an existential or a universal quantifier, and  is a (quantifier-free) propositional formula over the variables . Truth for such formulas is defined in the usual way.

To consider the complexity classes of the PH, we restrict the number of quantifier alternations occurring in quantified Boolean formulas, i.e., the number of times where . We consider the complexity classes , for each . Let  be an arbitrary, fixed constant. The complexity class  consists of all decision problems for which there exists a polynomial-time reduction to the problem , that is defined as follows. Instances of the problem are quantified Boolean formulas of the form  , where  if 

is odd and 

if  is even, where , and where  is quantifier-free. The problem is to decide if the quantified Boolean formula is true. The complementary class  consists of all decision problems for which there exists a polynomial-time reduction to the problem co-, that is complementary to the problem . The Polynomial Hierarchy (PH) consists of the classes and , for all .

Alternatively, one can characterize the class

using nondeterministic polynomial-time algorithms with access to an oracle for an NP-complete problem. Let 

be a decision problem. A Turing machine 

with access to an  oracle is a Turing machine with a dedicated oracle tape and dedicated states  and . Whenever  is in the state , it does not proceed according to the transition relation, but instead it transitions into the state  if the oracle tape contains a string  that is a yes-instance for the problem , i.e., if , and it transitions into the state  if . Intuitively, the oracle solves arbitrary instances of  in a single time step. The class consists of all decision problems that can be solved in polynomial time by a nondeterministic Turing machine that has access to an -oracle, for some .

2.2 Judgment Aggregation

Next, we introduce the two formal judgment aggregation frameworks that we use in this paper: formula-based judgment aggregation (as used by, e.g., [7, 14, 18]) and constraint-based judgment aggregation (as used by, e.g., [17]). Moreover, we briefly discuss an extended variant of the constraint-based judgment aggregation framework (as considered in, e.g., [13]).

2.2.1 Formula-Based Judgment Aggregation

We begin with the framework of formula-based judgment aggregation.

An agenda is a finite, nonempty set  of formulas that does not contain any doubly-negated formulas and that is closed under complementation. Moreover, if  is an agenda, then we let  denote the pre-agenda associated to the agenda . We denote the bitsize of the agenda  by . A judgment set  for an agenda  is a subset . We call a judgment set  complete if  or  for all ; and we call it consistent if there exists an assignment that makes all formulas in  true. Intuitively, the consistent and complete judgment sets are the opinions that individuals and the group can have.

We associate with each agenda  an integrity constraint , that can be used to further restrict the set of feasible opinions. Such an integrity constraint consists of a single propositional formula. We say that a judgment set  is -consistent if there exists a truth assignment that simultaneously makes all formulas in  and  true. Let  denote the set of all complete and -consistent subsets of . We say that finite sequences  of complete and -consistent judgment sets are profiles, and where convenient we equate a profile  with the (multi)set . Moreover, for , we let  denote the profile .

A judgment aggregation procedure (or rule) for the agenda  and the integrity constraint  is a function  that takes as input a profile , and that produces a non-empty set of non-empty judgment sets. We call a judgment aggregation procedure  resolute if for any profile  it returns a singleton, i.e., ; otherwise, we call  irresolute. We call a judgment aggregation procedure  anonymous if for every profile  and for every permutation  it holds that , where . An example of a resolute, anonymous judgment aggregation procedure is the strict majority rule Majority, where , where  if and only if  occurs in the strict majority of judgment sets in , for all , and where  if and only if , for all . We call a judgment aggregation procedure  complete and -consistent, if  is complete and -consistent, respectively, for every  and every . The procedure Majority is not consistent. Consider the agenda  with , and the profile , where , and . The unique outcome  in  is inconsistent.

The Kemeny aggregation procedure is based on a notion of distance. This distance is based on the Hamming distance  between two complete judgment sets . Intuitively, the Hamming distance  counts the number of issues on which two judgment sets disagree. Let  be a single -consistent and complete judgment set, and let  be a profile. We define the distance between  and  to be . Then, we let the outcome  of the Kemeny rule be the set of those  for which there is no  such that . (If  and  are clear from the context, we often write  to denote .) Intuitively, the Kemeny rule selects those complete and -consistent judgment sets that minimize the cumulative Hamming distance to the judgment sets in the profile. The Kemeny rule is irresolute, complete, -consistent and anonymous.

2.2.2 Constraint-Based Judgment Aggregation

We continue with the framework of constraint-based judgment aggregation.

Let  be a finite set of issues, in the form of propositional variables. Intuitively, these issues are the topics about which the individuals want to combine their judgments. A truth assignment  is called a ballot, and represents an opinion that individuals and the group can have. We will also denote ballots 

by a binary vector 

, where  for each . Moreover, we say that  is a partial ballot, and that  agrees with a ballot  if  whenever , for all . As in the case for formula-based judgment aggregation, we introduce an integrity constraint , that can be used to restrict the set of feasible opinions (for both the individuals and the group). The integrity constraint  is a propositional formula on the variables . We define the set  of rational ballots to be the ballots (for ) that satisfy the integrity constraint . Rational ballots in the constraint-based judgment aggregation framework correspond to complete and -consistent judgment sets in the formula-based judgment aggregation framework. We say that finite sequences  of rational ballots are profiles, and where convenient we equate a profile  with the (multi)set . Moreover, for , we let  denote the profile .

A judgment aggregation procedure (or rule), for the set  of issues and the integrity constraint , is a function  that takes as input a profile , and that produces a non-empty set of ballots. We call a judgment aggregation procedure  resolute if for any profile  it returns a singleton, i.e., ; otherwise, we call  irresolute. We call a judgment aggregation procedure  anonymous if for every profile  and for every permutation  it holds that , where . We call a judgment aggregation procedure  rational (or consistent), if  is rational for every  and every .

As an example of a judgment aggregation procedure we consider the strict majority rule Majority, where  and where each  agrees with the majority of the -th bits in the ballots in  (in case of a tie, we arbitrarily let ). To see that Majority is not rational, consider the set  of issues, the integrity constraint , and the profile , where , and . The unique outcome  in  is not rational.

The Kemeny aggregation procedure is defined for the constraint-based judgment aggregation framework as follows. Similarly to the case for formula-based judgment aggregation, the Kemeny rule is based on the Hamming distance , between two rational ballots  and  for the set  of issues and the integrity constraint . Let  be a single ballot, and let  be a profile. We define the distance between  and  to be . Then, we let the outcome  of the Kemeny rule be the set of those ballots  for which there is no  such that . (If  and  are clear from the context, we often write  to denote .) The Kemeny rule is irresolute, anonymous and rational.

2.2.3 Extended Constraint-Based Judgment Aggregation

Finally, we consider an extended variant of the constraint-based judgment aggregation framework. In the constraint-based judgment aggregation framework, we consider a set  of issues and an integrity constraint in the form of a propositional formula  that satisfies the constraint that . However, in some situations it is more convenient to allow the integrity constraint  to contain additional variables. In the extended constraint-based framework, we relax the condition that , and we allow arbitrary propositional formulas as integrity constraints. This modification requires us to adapt the notion of rationality accordingly. A ballot  is said to be rational if  is satisfiable—that is, if there is some truth assignment  such that  satisfies .

2.2.4 Preferences over Opinions

Strategic behavior for judgment aggregation (such as the problems of manipulation, bribery and control) involves the incentive to obtain a “better” outcome. Therefore, in order to study strategic behavior, it is essential to define a notion of preference over opinions—that is, when is one opinion “better than” (or preferred over) another opinion.

In the worst case, the number of possible opinions that play a role is exponential in the number of issues—e.g., for  issues there could be up to  possible opinions. As a result, it is unreasonable to expect agents to explicitly specify a preference relation over all (feasible) opinions. Instead it makes more sense to use a compact specification language to represent a preference relation over opinions. In this paper, we will use one such specification method that can be used to capture a wide range of preferences. Various preference relations over opinions have been studied in the literature [3, 6, 8].

We consider preferences based on a weighted Hamming distance. We define this weighted Hamming distance for the setting of formula-based judgment aggregation. Definitions for the setting of constraint-based judgment aggregation are entirely similar. Take an agenda  together with an integrity constraint . An agent can specify their preference relation over complete and -consistent judgment sets  by providing a weight function  that produces a weight  for each formula . Intuitively, for each , the weight  indicates how important it is for the agent that the outcome agrees with their truthful opinion on the issue . (Alternatively, one could consider weight functions that produce rational or real weights.) Then, for two complete judgment sets  and  the weighted Hamming distance  is defined as follows:

That is, for each formula  that  and  disagree on, the weighted Hamming distance is increased by .

Using this notion of weighted Hamming distance, we can define a preference relation for an agent. Suppose that the agent’s truthful opinion is given by a complete and -consistent judgment set . Moreover, suppose that the agent’s view on the relative importance of the separate issues is given by a weight function . Then the preference relation  for this agent is defined as follows. For any two complete and -consistent judgment sets , it holds:

Correspondingly, a judgment set  is (strictly) preferred over another judgment set  if and only if the weighted Hamming distance from  to  is (strictly) smaller than the weighted Hamming distance from  to .

A particular case of the weighted Hamming distance is the unweighted Hamming distance. That is, the case where  for all . Whenever the weight function  is the constant function that always returns , we drop the “” from the notation—that is, the unweighted Hamming distance between two judgment sets  and  is denoted by .

Other preference relations

In the literature, there have been various proposals for notions of preference over opinions. For example, the phenomenon of manipulation in judgment aggregation has been studied in the settings (1) where one judgment set is preferred over another if it agrees with a fixed optimal judgment set on at least one issue where the other judgment set disagrees [8], and (2) where one judgment set is preferred over a second judgment set if it agrees with a fixed optimal judgment set on at least one issue where the second judgment set disagrees, and for all issues it holds that if the second judgment set agrees with the optimal judgment set then the first judgment set also agrees with the optimal [8]. Other preference relations that have been investigated are top-respecting preferences and closeness-respecting preferences. The class of top-respecting preferences contains all preferences that prefer a single most preferred judgment set over all other judgment sets (and the preference between the other judgment sets is arbitrary) [3, 6]. The class of closeness-respecting preferences contains preferences that additionally satisfy the condition of closeness: if one judgment set agrees with the most preferred judgment on a superset of issues compared to another judgment set, then the one judgment is preferred over the other [3, 6].

3 Manipulation

The first form of strategic behavior in judgment aggregation that we consider is manipulation. This concerns cases where individuals aim to influence the outcome of the aggregation procedure in their favor by reporting an insincere judgment, that is, by reporting a judgment that differs from their beliefs.

For irresolute judgment aggregation procedures such as the Kemeny procedure, one can consider various requirements on the strategically reported insincere judgments. For instance, one could require that every outcome for the insincere judgment is preferred over every outcome for the sincere judgment. Alternatively, one could require that there is at least one outcome for the insincere judgment that is preferred over every outcome for the sincere judgment. Correspondingly, we consider the following decision problems. (We formalize these problems for the setting of formula-based judgment aggregation. For the setting of constraint-based judgment aggregation, these problems are defined entirely similarly.)

Cautious-Manipulation(Kemeny) Instance: An agenda  with an integrity constraint , a weight function , and a profile . Question: Is there a complete and consistent judgment set such that for all and for all it holds that ?

Optimistic-Manipulation(Kemeny) Instance: An agenda  with an integrity constraint , a weight function , and a profile . Question: Is there a complete and consistent judgment set and some such that for all it holds that ?

Pessimistic-Manipulation(Kemeny) Instance: An agenda  with an integrity constraint , a weight function , and a profile . Question: Is there a complete and consistent judgment set such that for all there is some such that ?

Superoptimistic-Manipulation(Kemeny) Instance: An agenda  with an integrity constraint , a weight function , and a profile . Question: Is there a complete and consistent judgment set , some and some such that ?

Safe-Superoptimistic-Manipulation(Kemeny) Instance: An agenda  with an integrity constraint , a weight function , and a profile . Question: Is there a complete and consistent judgment set such that (1) for all and for all it holds that , and such that (2) there exists some and some such that ?

3.1 Complexity Results

In this section, we prove the following result.

Theorem 1.

The following problems are -complete:

  • Cautious-Manipulation(Kemeny),

  • Optimistic-Manipulation(Kemeny),

  • Pessimistic-Manipulation(Kemeny),

  • Superoptimistic-Manipulation(Kemeny), and

  • Safe-Superoptimistic-Manipulation(Kemeny).

Moreover, -hardness holds already for the case where the manipulator’s preferences are based on the unweighted Hamming distance.

This result follows from Propositions 25 and 7, and Corollaries 6 and 8, that we establish below.

Proposition 2.

Cautious-Manipulation(Kemeny) is in .

Proof.

We describe a nondeterministic polynomial-time algorithm with access to an NP oracle that solves the problem. Let  specify an instance of Cautious-Manipulation(Kemeny). The algorithm proceeds in several steps.

Firstly, (1) the algorithm determines the minimum distance  from  to any complete and consistent judgment set . That is,  is the cumulative unweighted Hamming distance from the judgments in  to the judgment sets . This can be done in (deterministic) polynomial time using  queries to an NP oracle.

Then, (2) the algorithm determines the minimum distance  (weighted by ) from  to any judgment set , that is, from  to any complete and consistent judgment set  that has cumulative unweighted Hamming distance  to the profile . This can also be done in (deterministic) polynomial time using an NP oracle.

Next, (3a) the algorithm guesses a complete judgment set  together with a truth assignment , and it checks whether  satisfies both  and . This can be done in nondeterministic polynomial time. Moreover, (3b) the algorithm determines the minimum distance  from  to any complete and consistent judgment set . Finally, (3c) the algorithm determines by using a single query to an NP oracle whether there exists some complete and consistent judgment set  such that  and . If this is the case, the algorithm rejects; otherwise, the algorithm accepts.

It is straightforward to verify that the algorithm runs in nondeterministic polynomial time. Moreover, the algorithm accepts the input (for some sequence of nondeterministic choices) if and only if there is some complete and consistent judgment set  such that for all  and for all  it holds that . ∎

Proposition 3.

Optimistic-Manipulation(Kemeny) is in .

Proof.

We describe a nondeterministic polynomial-time algorithm with access to an NP oracle that solves the problem. Let  specify an instance of Optimistic-Manipulation(Kemeny). The algorithm proceeds in several steps. For the first two steps, the algorithm proceeds exactly as the algorithm described in the proof of Proposition 2. That is, (1) the algorithm computes  and (2) it computes , both in deterministic polynomial time using an NP oracle.

For the third step, the algorithm proceeds in a similar fashion as the algorithm described in the proof of Proposition 2. That is, (3a) the algorithm guesses a complete judgment set  together with a truth assignment , and it checks whether  satisfies both  and . Also, (3b) the algorithm determines the minimum unweighted Hamming distance  from  to any complete and consistent judgment set . This can be done using  queries to an NP oracle. Then, (3c) the algorithm guesses some complete judgment set  together with a truth assignment , and it checks whether  satisfies both  and . Moreover, the algorithm checks whether —that is, —and , and accepts if and only if this is the case.

It is straightforward to verify that the algorithm runs in nondeterministic polynomial time. Moreover, the algorithm accepts the input (for some sequence of nondeterministic choices) if and only if there is some complete and consistent judgment set  and some  such that for all  it holds that . ∎

Proposition 4.

Pessimistic-Manipulation(Kemeny) is in .

Proof.

We describe a nondeterministic polynomial-time algorithm with access to an NP oracle that solves the problem. Let  specify an instance of Pessimistic-Manipulation(Kemeny). The algorithm proceeds in several steps. For the first step, the algorithm proceeds exactly as the algorithm described in the proof of Proposition 2. That is, (1) the algorithm computes  in deterministic polynomial time using  queries to an NP oracle.

Then, (2) the algorithm determines the maximum distance  (weighted by ) from  to any judgment set , that is, from  to any complete and consistent judgment set  that has cumulative unweighted Hamming distance  to the profile . This can also be done in (deterministic) polynomial time using an NP oracle.

Next, (3a) the algorithm guesses a complete judgment set  together with a truth assignment , and it checks whether  satisfies both  and . This can be done in nondeterministic polynomial time. Moreover, (3b) the algorithm determines the minimum unweighted Hamming distance  from  to any complete and consistent judgment set . Finally, (3c) the algorithm determines by using a single query to an NP oracle whether there exists some complete and consistent judgment set  such that  and . If this is the case, the algorithm rejects; otherwise, the algorithm accepts.

It is straightforward to verify that the algorithm runs in nondeterministic polynomial time. Moreover, the algorithm accepts the input (for some sequence of nondeterministic choices) if and only if there is some complete and consistent judgment set  such that for all  there is some  such that . ∎

Proposition 5.

Superoptimistic-Manipulation(Kemeny) is in .

Proof.

We describe a nondeterministic polynomial-time algorithm with access to an NP oracle that solves the problem. Let  specify an instance of Superoptimistic-Manipulation(Kemeny). The algorithm proceeds in several steps. For the first steps, the algorithm proceeds exactly as the algorithm described in the proof of Proposition 2. That is, (1) the algorithm computes , in deterministic polynomial time using  queries to an NP oracle.

Then, (2) the algorithm guesses a complete judgment set  together with a truth assignment , and it checks whether  satisfies both  and , and whether .

For the third step, the algorithm proceeds in a similar fashion as the algorithm described in the proof of Proposition 2. That is, (3a) the algorithm guesses a complete judgment set  together with a truth assignment , and it checks whether  satisfies both  and . Also, (3b) the algorithm determines the minimum distance  from  to any complete and consistent judgment set . This can be done using  queries to an NP oracle. Then, (3c) the algorithm guesses some complete judgment set  together with a truth assignment , and it checks whether  satisfies both  and . Moreover, the algorithm checks whether —that is, —and , and accepts if and only if this is the case.

It is straightforward to verify that the algorithm runs in nondeterministic polynomial time. Moreover, the algorithm accepts the input (for some sequence of nondeterministic choices) if and only if there is some complete and consistent judgment set , some  and some  such that . ∎

Corollary 6.

Safe-Superoptimistic-Manipulation(Kemeny) is in .

Proof (sketch).

The algorithms described in the proofs of Proposition 2 and 5 can straightforwardly be modified and combined to form a nondeterministic polynomial-time algorithm with access to an NP oracle that solves Safe-Superoptimistic-Manipulation(Kemeny). ∎

Proposition 7.

Cautious-Manipulation(Kemeny) is -hard.

Proof.

We show -hardness by giving a reduction from the satisfiability problem for quantified Boolean formulas of the form . Let  be a quantified Boolean formula. Let  and . Without loss of generality, we assume that  is a multiple of —that is, that  for some . We construct an agenda , an integrity constraint , a weight function , and a profile  as follows.

We consider fresh variables  and fresh variables . Moreover, we consider fresh variables , fresh variables , fresh variables  for , fresh variables  for  and , and fresh variables  for  and , where . We then let the agenda  consist of the variables  and  and the fresh variables we introduced above, together with their negations. That is, we let .

We then define the integrity constraint  as follows. We let

and

(Here  denotes exclusive disjunction.)

As a result of the definition of , each complete and consistent judgment set  satisfies (at least) one of the following four conditions.

  1. For some , the judgment set  includes each formula  for .

  2. The judgment set  includes exactly one of  and  for each , it includes none of the formulas , it includes exactly one of  and  for each , it includes none of the formulas , it includes either all formulas  or all formulas  for , and  does not satisfy .

  3. The judgment set  includes exactly one of  and  for each , it includes none of the formulas , it includes all formulas , for each  it includes either none or both of  and , and it includes all formulas  for .

  4. The judgment set  includes all formulas , for each  it includes either none or both of  and , it includes all formulas , for each  it includes either none or both of  and , and it includes either all formulas  or all formulas  for .

We define  by letting  for each . In other words, we consider the unweighted Hamming distance.

Finally, we let , where  are defined as described in Table 1. In this table, the indices  range over all possible values, and for each  we write a  if  and a  if .

Table 1: The profile  that we use in the proof of Proposition 7.

In the remainder, we will argue that there is some truth assignment  such that for all truth assignments  it holds that  is true if and only if there is some complete and consistent judgment set  such that for all  and for all  it holds that .

Firstly, we observe that , where  and  are defined as described in Table 2. Both  and  are complete and consistent, and have a cumulative Hamming distance of  to the profile . It is straightforward to verify that no complete and consistent judgment set has a smaller cumulative Hamming distance to the profile . (For instance, the judgment sets of the form , as described below in Table 3, have a cumulative Hamming distance of