Complexity of and Algorithms for Borda Manipulation

We prove that it is NP-hard for a coalition of two manipulators to compute how to manipulate the Borda voting rule. This resolves one of the last open problems in the computational complexity of manipulating common voting rules. Because of this NP-hardness, we treat computing a manipulation as an approximation problem where we try to minimize the number of manipulators. Based on ideas from bin packing and multiprocessor scheduling, we propose two new approximation methods to compute manipulations of the Borda rule. Experiments show that these methods significantly outperform the previous best known %existing approximation method. We are able to find optimal manipulations in almost all the randomly generated elections tested. Our results suggest that, whilst computing a manipulation of the Borda rule by a coalition is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice

Complexity of and Algorithms for the Manipulation of Borda, Nanson and Baldwin’s Voting Rules

Abstract We investigate manipulation of the Borda voting rule, as well as two elimination style voting rules, Nanson's and Baldwin's voting rules, which are based on Borda voting. We argue that these rules have a number of desirable computational properties. For unweighted Borda voting, we prove that it is NP-hard for a coalition of two manipulators to compute a manipulation. This resolves a long-standing open problem in the computational complexity of manipulating common voting rules. We prove that manipulation of Baldwin's and Nanson's rules is computationally more difficult than manipulation of Borda, as it is NP-hard for a single manipulator to compute a manipulation. In addition, for Baldwin's and Nanson's rules with weighted votes, we prove that it is NP-hard for a coalition of manipulators to compute a manipulation with a small number of candidates. Because of these NP-hardness results, we compute manipulations using heuristic algorithms that attempt to minimise the number of manipulators. We propose several new heuristic methods. Experiments show that these methods significantly outperform the previously best known heuristic method for the Borda rule. Our results suggest that, whilst computing a manipulation of the Borda rule is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice. In contrast to the Borda rule, our experiments with Baldwin's and Nanson's rules demonstrate that both of them are often more difficult to manipulate in practice. These results suggest that elimination style voting rules deserve further study

Combining Voting Rules Together

We propose a simple method for combining together voting rules that performs a run-off between the different winners of each voting rule. We prove that this combinator has several good properties. For instance, even if just one of the base voting rules has a desirable property like Condorcet consistency, the combination inherits this property. In addition, we prove that combining voting rules together in this way can make finding a manipulation more computationally difficult. Finally, we study the impact of this combinator on approximation methods that find close to optimal manipulations

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