Detecting Possible Manipulators in Elections

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. Since prevention of manipulation seems to be elusive, an interesting research direction therefore is detection of manipulation. Motivated by this, we initiate the study of detection of possible manipulators in an election. We formulate two pertinent computational problems - Coalitional Possible Manipulators (CPM) and Coalitional Possible Manipulators given Winner (CPMW), where a suspect group of voters is provided as input to compute whether they can be a potential coalition of possible manipulators. In the absence of any suspect group, we formulate two more computational problems namely Coalitional Possible Manipulators Search (CPMS), and Coalitional Possible Manipulators Search given Winner (CPMSW). We provide polynomial time algorithms for these problems, for several popular voting rules. For a few other voting rules, we show that these problems are in NP-complete. We observe that detecting manipulation maybe easy even when manipulation is hard, as seen for example, in the case of the Borda voting rule.Comment: Accepted in AAMAS 201

Schulze and Ranked-Pairs Voting are Fixed-Parameter Tractable to Bribe, Manipulate, and Control

Schulze and ranked-pairs elections have received much attention recently, and the former has quickly become a quite widely used election system. For many cases these systems have been proven resistant to bribery, control, or manipulation, with ranked pairs being particularly praised for being NP-hard for all three of those. Nonetheless, the present paper shows that with respect to the number of candidates, Schulze and ranked-pairs elections are fixed-parameter tractable to bribe, control, and manipulate: we obtain uniform, polynomial-time algorithms whose degree does not depend on the number of candidates. We also provide such algorithms for some weighted variants of these problems

A Smooth Transition from Powerlessness to Absolute Power

We study the phase transition of the coalitional manipulation problem for generalized scoring rules. Previously it has been shown that, under some conditions on the distribution of votes, if the number of manipulators is $o(\sqrt{n})$, where $n$ is the number of voters, then the probability that a random profile is manipulable by the coalition goes to zero as the number of voters goes to infinity, whereas if the number of manipulators is $\omega(\sqrt{n})$, then the probability that a random profile is manipulable goes to one. Here we consider the critical window, where a coalition has size $c\sqrt{n}$, and we show that as $c$ goes from zero to infinity, the limiting probability that a random profile is manipulable goes from zero to one in a smooth fashion, i.e., there is a smooth phase transition between the two regimes. This result analytically validates recent empirical results, and suggests that deciding the coalitional manipulation problem may be of limited computational hardness in practice.Comment: 22 pages; v2 contains minor changes and corrections; v3 contains minor changes after comments of reviewer

The Complexity of Online Manipulation of Sequential Elections

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 manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. We obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model. And we obtain what to the best of our knowledge are the first P^NP[1]-completeness and P^NP-completeness results in the field of computational social choice, in particular proving such completeness for, respectively, the complexity of 3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition manipulation of veto elections.Comment: 24 page