Computational aspects of strategic behaviour in elections with top-truncated ballots

Understanding when and how computational complexity can be used to protect elections against different manipulative actions has been a highly active research area over the past two decades. Much of this literature, however, makes the assumption that the voters or agents specify a complete preference ordering over the set of candidates. There are many multiagent systems applications, and even real-world elections, where this assumption is not warranted, and this in turn raises a series of questions on the impact of partial voting on the complexity of manipulative actions. In this paper, we focus on two of these questions. First, we address the question of how hard it is to manipulate elections when the agents specify only top-truncated ballots. Here, in particular, we look at the weighted manipulation problem—both constructive and destructive manipulation—when the voters are allowed to specify top-truncated ballots, and we provide general results for all scoring rules, for elimination versions of all scoring rules, for the plurality with runoff rule, for a family of election systems known as Copeland\(^{\alpha }\), and for the maximin protocol. The second question we address is the impact of top-truncated voting on the complexity of manipulative actions in electorates with structured preference profiles. In particular, we consider electorates that are single-peaked and we show how, for many voting protocols, allowing top-truncated voting reimposes the \(\mathcal {NP}\)-hardness shields that normally vanish in such electorates.     

The complexity of priced control in elections

We study the complexity of priced control in elections. Naturally, if a given control type is NP-hard for a given voting system ? then its priced variant is NP-hard for this rule as well. It is, however, interesting what effect introducing prices has on the complexity of those control problems that without prices are tractable. We show that for four prominent voting rules (plurality, approval, Condorcet, and Copeland) introducing prices does not increase the complexity of control by adding/deleting candidates/voters. However, we do show an example of a scoring rule for which such an effect takes place.     

Complexity of control by partitioning veto elections and of control by adding candidates to plurality elections

Control by partition refers to situations where an election chair seeks to influence the outcome of an election by partitioning either the candidates or the voters into two groups, thus creating two first-round subelections that determine who will take part in a final round. The model of partition-of-voters control attacks is remotely related to “gerrymandering” (maliciously resizing election districts). While the complexity of control by partition has been studied thoroughly for many voting systems, there are no such results known for the important veto voting system. We settle the complexity of control by partition for veto in a broad variety of models. In addition, by giving a counterexample we observe that a reduction from the literature (Chen et al. 2015) showing the parameterized complexity of control by adding candidates to plurality elections, parameterized by the number of voters, is technically flawed, and we show how this reduction can be adapted to make it correct.     

How hard is control in single-crossing elections?

Election control problems model situations where some entity (traditionally called the election chair) wants to ensure some candidate’s victory by either adding or deleting candidates or voters. The complexity of deciding if such control actions can be successful is well-studied for many typical voting rules and, usually, such control problems are \(\mathrm {NP}\)-complete. However, Faliszewski et al. (Inf Comput 209(2):89–107, 2011) have shown that many control problems become polynomial-time solvable when we consider single-peaked elections. In this paper we show that a similar phenomenon applies to the case of single-crossing elections. Specifically, we consider the complexity of control by adding/deleting candidates/voters under plurality, Condorcet, and approval voting. For each of these control types and each of the rules, we show that if the control type is \(\mathrm {NP}\)-complete in general, it becomes polynomial-time solvable for single-crossing elections.     

Parameterized complexity of candidate control in elections and related digraph problems

There are different ways for an external agent to influence the outcome of an election. We concentrate on “control” by adding or deleting candidates. Our main focus is to investigate the parameterized complexity of various control problems for different voting systems. To this end, we introduce natural digraph problems that may be of independent interest. They help in determining the parameterized complexity of control for different voting systems including Llull, Copeland, and plurality voting. Devising several parameterized reductions, we provide an overview of the parameterized complexity of the digraph and control problems with respect to natural parameters such as adding/deleting only a bounded number of candidates or having only few voters.     

Bribery in voting with CP-nets

We investigate the computational complexity of finding optimal bribery schemes in voting domains where the candidate set is the Cartesian product of a set of variables and voters use CP-nets, an expressive and compact way to represent preferences. To do this, we generalize the traditional bribery problem to take into account several issues over which agents vote, and their inter-dependencies. We consider five voting rules, three kinds of bribery actions, and five cost schemes. For most of the combinations of these parameters, we find that bribery in this setting is computationally easy.     

Anyone but him: The complexity of precluding an alternative

Preference aggregation in a multiagent setting is a central issue in both human and computer contexts. In this paper, we study in terms of complexity the vulnerability of preference aggregation to destructive control. In particular, we study the ability of an election's chair to, through such mechanisms as voter/candidate addition/suppression/partition, ensure that a particular candidate (equivalently, alternative) does not win. And we study the extent to which election systems can make it impossible, or computationally costly (NP-complete), for the chair to execute such control. Among the systems we study—plurality, Condorcet, and approval voting—we find cases where systems immune or computationally resistant to a chair choosing the winner nonetheless are vulnerable to the chair blocking a victory. Beyond that, we see that among our studied systems no one system offers the best protection against destructive control. Rather, the choice of a preference aggregation system will depend closely on which types of control one wishes to be protected against. We also find concrete cases where the complexity of or susceptibility to control varies dramatically based on the choice among natural tie-handling rules.     

Computational complexity of manipulation: A survey

In situations when a group of people has to make a decision based on the set of individual preferences, they use a certain aggregation method, in particular, voting. One of the main problems for any non-dictatorial social choice rule is the possibility for the voters to achieve a more preferable outcome of the voting by misrepresenting their preferences. Such actions on behalf of the voters are called manipulation, or strategic voting. One approach used to compare social choice rules in terms of how hard they are to manipulate is to find the complexity classes of manipulation problems for a given aggregation method. In this work, we present a survey of the studies of complexity classes of manipulation problems under various model assumptions and constraints.     

Exact Complexity of the Winner Problem for Young Elections

In 1977 Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for \p _|| ^NP , the class of problems solvable in polynomial time by parallel access to NP. Analogous results for Lewis Carroll's 1876 voting scheme were recently established by Hemaspaandra et al. In contrast, we prove that the winner and ranking problems in Fishburn's homogeneous variant of Carroll's voting scheme can be solved efficiently by linear programming.     

The learnability of voting rules

Scoring rules and voting trees are two broad and concisely-representable classes of voting rules; scoring rules award points to alternatives according to their position in the preferences of the voters, while voting trees are iterative procedures that select an alternative based on pairwise comparisons. In this paper, we investigate the PAC-learnability of these classes of rules. We demonstrate that the class of scoring rules, as functions from preferences into alternatives, is efficiently learnable in the PAC model. With respect to voting trees, while in general a learning algorithm would require an exponential number of samples, we show that if the number of leaves is polynomial in the size of the set of alternatives, then a polynomial training set suffices. We apply these results in an emerging theory: automated design of voting rules by learning.     

一种优化计算slater投票获胜者的Picat方法

slater投票规则是基于锦标赛的投票规则,主要是通过构造无环锦标赛,找到与原锦标赛差异最小的一个,从中选出获胜者。针对求解难度为NP难的slater投票算法,提出了一种基于相似候选项集的优化求解slater问题的Picat方法。相比于非优化求解slater问题的方法,该方法缩小了slater算法的解空间,有效地减少了求解slater获胜者的计算量,提高了计算速度。实验结果表明,优化求解slater问题的Picat方法的计算速度优于非优化的Picat方法;当候选项人数少于20时,求解slater问题的回答集程序（ASP）方法的计算速度和计算能力优于优化的Picat方法,但当候选项人数超过30时,优化的Picat方法（用可满足问题求解器）的计算速度和计算能力优于ASP方法。     

Is computational complexity a barrier to manipulation?

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by mis-reporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. In this survey article, we summarize the evidence for and against computational complexity being a barrier to manipulation. We look both at techniques identified to increase complexity (for example, hybridizing together two or more voting rules), as well as other features that may change the computational complexity of computing a manipulation (for example, if votes are restricted to be single peaked then some of the complexity barriers fall away). We discuss recent theoretical results that consider the average case, as well as simple greedy and approximate methods. We also describe how computational ??phase transitions??, which have been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems, have provided insight into the hardness of manipulating voting rules in practice. Finally, we consider manipulation of other related problems like stable marriage and tournament problems.     

Winner determination in voting trees with incomplete preferences and weighted votes

In multiagent settings where agents have different preferences, preference aggregation can be an important issue. Voting is a general method to aggregate preferences. We consider the use of voting tree rules to aggregate agents’ preferences. In a voting tree, decisions are taken by performing a sequence of pairwise comparisons in a binary tree where each comparison is a majority vote among the agents. Incompleteness in the agents’ preferences is common in many real-life settings due to privacy issues or an ongoing elicitation process. We study how to determine the winners when preferences may be incomplete, not only for voting tree rules (where the tree is assumed to be fixed), but also for the Schwartz rule (in which the winners are the candidates winning for at least one voting tree). In addition, we study how to determine the winners when only balanced trees are allowed. In each setting, we address the complexity of computing necessary (respectively, possible) winners, which are those candidates winning for all completions (respectively, at least one completion) of the incomplete profile. We show that many such winner determination problems are computationally intractable when the votes are weighted. However, in some cases, the exact complexity remains unknown. Since it is generally computationally difficult to find the exact set of winners for voting trees and the Schwartz rule, we propose several heuristics that find in polynomial time a superset of the possible winners and a subset of the necessary winners which are based on the completions of the (incomplete) majority graph built from the incomplete profiles.     

Manipulation can be hard in tractable voting systems even for constant-sized coalitions

Voting theory has become increasingly integrated with computational social choice and multiagent systems. Computational complexity has been extensively used as a shield against manipulation of voting systems, however for several voting schemes this complexity may cause calculating the winner to be computationally difficult. Of the many voting systems that have been studied with regard to election manipulation, a few have been found to have an unweighted coalitional manipulation problem that is NP-hard for a constant number of manipulators despite having a winner problem that is in P. We survey this interesting class of voting systems and the work that has analyzed their complexity.     

Positional Voting Methods Satisfying the Criteria of Weak Mutual Majority and Condorcet Loser

This paper considers a voting problem in which the individual preferences of electors are defined by the ranked lists of candidates. For single-winner elections, we apply the criterion of weak positional dominance (WPD, PD), which is closely related to the positional scoring rules. Also we formulate the criterion of weak mutual majority (WMM), which is stronger than the majority criterion but weaker than the criterion of mutual majority (MM). Then we construct two modifications for the median voting rule that satisfy the Condorcet loser criterion. As shown below, WPD and WMM are satisfied for the first modification while PD and MM for the second modification. We prove that there is no rule satisfying WPD and MM simultaneously. Finally, we check a list of 37 criteria for the constructed rules.     

Manipulation Complexity of Same-System Runoff Elections

Do runoff elections, using the same voting rule as the initial election but just on the winning candidates, increase or decrease the complexity of manipulation? Does allowing revoting in the runoff increase or decrease the complexity relative to just having a runoff without revoting? For both weighted and unweighted voting, we show that even for election systems with simple winner problems the complexity of manipulation, manipulation with runoffs, and manipulation with revoting runoffs are independent. On the other hand, for some important, well-known election systems we determine what holds for each of these cases. For no such systems do we find runoffs lowering complexity, and for some we find that runoffs raise complexity. Ours is the first paper to show that for natural, unweighted election systems, runoffs can increase the manipulation complexity.     

RLWE同态加密算法的多候选人电子投票协议

使用安全协议保护选民隐私、保证投票公正有效是投票电子信息化的基础,安全协议的复杂度则是电子投票应用的最大阻碍。提出了一种基于RLWE同态加密算法的多候选人电子投票协议,可支持多候选人,也能满足对选民隐私的保护。该协议利用基于RLWE的同态加密算法的加法同态性质在计票环节使用密文计票保护选民的私密,利用中国剩余定理的性质对选票进行批处理,提升计票能力。该投票协议能支持多候选人投票并最终知晓每个候选人最终票数,并设置公示机构公示投票过程中的每个步骤,用于公开验证。     

Parameterized computational complexity of control problems in voting systems

Voting systems are common tools in a variety of areas. This paper studies parameterized computational complexity of control of Plurality, Condorcet and Approval voting systems, respectively. The types of controls considered include adding or deleting candidates or voters, under constructive or destructive setting. We obtain the following results: (1) constructive control by adding candidates in Plurality voting is W[2]-hard with respect to the parameter “number of added candidates”, (2) destructive control by adding candidates in Plurality voting is W[2]-hard with respect to the parameter “number of added candidates”, (3) constructive control by adding voters in Condorcet voting is W[1]-hard with respect to the parameter “number of added voters”, (4) constructive control by deleting voters in Condorcet voting is W[1]-hard with respect to the parameter “number of deleted voters”, (5) constructive control by adding voters in Approval voting is W[1]-hard with respect to the parameter “number of added voters”, and (6) constructive control by deleting voters in Approval voting is W[2]-hard with respect to the parameter “number of deleted voters”.     

The acclamation consensus state and an associated ranking rule

The study of conditions, under which the existence of an “absolute” best winner can be assured, is a hot topic in the field of social choice. Unanimity is an evident example of a condition under which the winner is obvious. However, many more properties weaker than unanimity have been analysed in literature: the presence of a Condorcet winner, strong stochastic transitivity, the presence of a candidate that Borda dominates all other candidates, etc. Unfortunately, one could easily find a prominent ranking rule, for which the outcome does not agree with these relaxed conditions. In this study, we aim to identify a condition weaker than unanimity, but under which the social outcome is still obvious. This condition, defined as the conjunction of three properties already studied by the present authors and hereinafter referred to as acclamation, will be proven to be a meeting point for the most prominent ranking rules in social choice theory, and will be used for introducing an intuitively appealing ranking rule.