The Borda voting rule is a positional scoring rule for $z$ candidates such that in each vote, the first candidate receives $z-1$ points, the second $z-2$ points and so on. The winner in the Borda rule is the candidate with highest total score. We study the manipulation problem of the Borda rule in a setting with two non-manipulators while one of the non-manipulator's vote is weighted. We demonstrate a sharp contrast on computational complexity depending on the weight of the non-manipulator: the problem is NP-hard when the weight is larger than $1$ while there exists an efficient algorithm to find a manipulation when the weight is at most $1$.
In the Possible Winner problem in computational social choice theory, we are given a set of partial preferences and the question is whether a distinguished candidate could be made winner by extending the partial preferences to linear preferences. Previous work has provided, for many common voting rules, fixed parameter tractable algorithms for the Possible Winner problem, with number of candidates as the parameter. However, the corresponding kernelization question is still open and in fact, has been mentioned as a key research challenge. In this paper, we settle this open question for many com…
In this paper, we consider permutation manipulations by any subset of women in the men-proposing version of the Gale-Shapley algorithm. This paper is motivated by the college admissions process in China. Our results also answer an open problem on what can be achieved by permutation manipulations. We present an efficient algorithm to find a strategy profile such that the induced matching is stable and Pareto-optimal (in the set of all achievable stable matchings) while the strategy profile itself is inconspicuous. Surprisingly, we show that such a strategy profile actually forms a Nash equilibr…
Both weighted and unweighted Borda manipulation problems have been proved $\mathcal{NP}$-hard. However, there is no exact combinatorial algorithm known for these problems. In this paper, we initiate the study of exact combinatorial algorithms for both weighted and unweighted Borda manipulation problems. More precisely, we propose $O^*((m\cdot 2^m)^{t+1})$time and $O^*(t^{2m})$time\footnote{$O^*()$ is the $O()$ notation with suppressed factors polynomial in the size of the input.} combinatorial algorithms for weighted and unweighted Borda manipulation problems, respectively, where $t$ is the nu…
The Gibbard-Satterthwaite theorem established that no non-trivial voting rule is strategy-proof, but that does not mean that all voting rules are equally susceptible to strategic manipulation. Over the past fifty years numerous approaches have been proposed to compare the manipulability of voting rules in terms of the probability of manipulation, the domains on which manipulation is possible, the complexity of finding such a manipulation, and others. In the closely related field of matching, Pathak and Sonmez pioneered a notion of manipulability based on case-by-case comparison of manipulable …
UNSW（新南威尔士大学）；NICTA（澳大利亚国家信息与通信技术研究院）；University of Rostock（罗斯托克大学）
Schulze's rule is used in the elections of a large number of organizations including Wikimedia and Debian. Part of the reason for its popularity is the large number of axiomatic properties, like monotonicity and Condorcet consistency, which it satisfies. We identify a potential shortcoming of Schulze's rule: it is computationally vulnerable to manipulation. In particular, we prove that computing an unweighted coalitional manipulation (UCM) is polynomial for any number of manipulators. This result holds for both the unique winner and the co-winner versions of UCM. This resolves an open question…
In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard. Especially among computational agents, it is reasonable to measure this hardness by computational complexity. Some earlier work has been done in this area, but it was assumed that the number of voters and candidates is unbounded. We deri…
In many coalition formation games the utility of the agents depends on a social network. In such scenarios there might be a manipulative agent that would like to manipulate his connections in the social network in order to increase his utility. We study a model of coalition formation in which a central organizer, who needs to form $k$ coalitions, obtains information about the social network from the agents. The central organizer has her own objective: she might want to maximize the utilitarian social welfare, maximize the egalitarian social welfare, or simply guarantee that every agent will ha…
Manipulation is a problem of fundamental importance in the context of voting in which the voters exercise their votes strategically instead of voting honestly to prevent selection of an alternative that is less preferred. The Gibbard-Satterthwaite theorem shows that there is no strategy-proof voting rule that simultaneously satisfies certain combinations of desirable properties. Researchers have attempted to get around the impossibility results in several ways such as domain restriction and computational hardness of manipulation. However these approaches have been shown to have limitations. Si…
Shortlisting of candidates--selecting a group of "best" candidates--is a special case of multiwinner elections. We provide the first in-depth study of the computational complexity of strategic voting for shortlisting based on the perhaps most basic voting rule in this scenario, l-Bloc (every voter approves l candidates). In particular, we investigate the influence of several tie-breaking mechanisms (e.g., pessimistic versus optimistic) and group evaluation functions (e.g., egalitarian versus utilitarian). Among other things, conclude that in an egalitarian setting strategic voting may indeed b…
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…
Weighted voting games apply to a wide variety of multi-agent settings. They enable the formalization of power indices which quantify the coalitional power of players. We take a novel approach to the study of the power of big vs.~small players in these games. We model small (big) players as having single (multiple) votes. The aggregate relative power of big players is measured w.r.t.~their votes proportion. For this ratio, we show small constant worst-case bounds for the Shapley-Shubik and the Deegan-Packel indices. In sharp contrast, this ratio is unbounded for the Banzhaf index. As an applica…
A negotiating team is a group of two or more agents who join together as a single negotiating party because they share a common goal related to the negotiation. Since a negotiating team is composed of several stakeholders, represented as a single negotiating party, there is need for a voting rule for the team to reach decisions. In this paper, we investigate the problem of strategic voting in the context of negotiating teams. Specifically, we present a polynomial-time algorithm that finds a manipulation for a single voter when using a positional scoring rule. We show that the problem is still …
We prove that the constructive weighted coalitional manipulation problem for the Schulze voting rule can be solved in polynomial time for an unbounded number of candidates and an unbounded number of manipulators.
Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipu…
We investigate the parameterized complexity of strategic behaviors in generalized scoring rules. In particular, we prove that the manipulation, control (all the 22 standard types), and bribery problems are fixed-parameter tractable for most of the generalized scoring rules, with respect to the number of candidates. Our results imply that all these strategic voting problems are fixed-parameter tractable for most of the common voting rules, such as Plurality, r-Approval, Borda, Copeland, Maximin, Bucklin, etc., with respect to the number of candidates.
Classical results in voting theory show that strategic manipulation by voters is inevitable if a voting rule simultaneously satisfy certain desirable properties. Motivated by this, we study the relevant question of how often a voting rule is manipulable. It is well known that elections with a large number of voters are rarely manipulable under impartial culture (IC) assumption. However, the manipulability of voting rules when the number of candidates is large has hardly been addressed in the literature and our paper focuses on this problem. First, we propose two properties (1) asymptotic strat…
Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipu…
In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations. We prove the following theorem: if we fix the number of alternatives, generate $n$ votes i.i.d. according to a distribution $π$, and let $n$ go to infinity, then for any $ε>0$, with probability at least $1-ε$, the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) $Θ(\sqrt n)$, (3) $Θ(n)$, and (4) $\infty$. This theorem holds for any set of vote operations, any indivi…
Control and manipulation are two of the most studied types of attacks on elections. In this paper, we study the complexity of control attacks on elections in which there are manipulators. We study both the case where the "chair" who is seeking to control the election is allied with the manipulators, and the case where the manipulators seek to thwart the chair. In the latter case, we see that the order of play substantially influences the complexity. We prove upper bounds, holding over every election system with a polynomial-time winner problem, for all standard control cases, and some of these…
