We consider the problem of predicting winners in elections, for the case where we are given complete knowledge about all possible candidates, all possible voters (together with their preferences), but where it is uncertain either which candidates exactly register for the election or which voters cast their votes. Under reasonable assumptions, our problems reduce to counting variants of election control problems. We either give polynomial-time algorithms or prove #P-completeness results for counting variants of control by adding deleting candidates voters for Plurality, k-Approval, Approval, Condorcet, and Maximin voting rules. We consider both the general case, where voters' preferences are unrestricted, and the case where voters' preferences are single-peaked.
From a technical standpoint, our research continues the line of work on the complexity of control. This line of work was initiated by Bartholdi, Tovey, and Trick @cite_20 , and then continued by Hemaspaandra, Hemaspaandra, and Rothe (who introduced the destructive cases), by (who considered multiwinner rules and who generalized the idea of the constructive and destructive cases), by Faliszewski, Hemaspaandra, and Hemaspaandra (who introduced multimode model of control), by Faliszewski, Hemaspaandra, and Hemaspaandra @cite_17 (who were first to consider control for weighted elections), by Rothe and Schend @cite_32 (who initiated the empirical study of the complexity of control problems), and by many other researchers, who provided results for specific voting rules and who introduced various other novel means of studying control problems (see, e.g., the following papers @cite_36 @cite_14 @cite_33 @cite_49 @cite_24 @cite_38 @cite_45 ; we also point the readers to the survey @cite_29 ).
@cite_38 @cite_14 @cite_33 @cite_36 @cite_29 @cite_32 @cite_24 @cite_45 @cite_49 @cite_20 @cite_17
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In 1992, Bartholdi, Tovey, and Trick opened the study of control attacks on elections-- attempts to improve the election outcome by such actions as adding deleting candidates or voters. That work has led to many results on how algorithms can be used to find attacks on elections and how complexity-theoretic hardness results can be used as shields against attacks. However, all the work in this line has assumed that the attacker employs just a single type of attack. In this paper, we model and study the case in which the attacker launches a multipronged (i.e., multimode) attack. We do so to more realistically capture the richness of real-life settings. For example, an attacker might simultaneously try to suppress some voters, attract new voters into the election, and introduce a spoiler candidate. Our model provides a unified framework for such varied attacks. By constructing polynomialtime multiprong attack algorithms we prove that for various election systems even such concerted, flexible attacks can be perfectly planned in deterministic polynomial time. Recent work by Procaccia, Rosenschein and Zohar [14] established some results regarding the complexity of manipulation and control in elections with multiple winners, such as elections of an assembly or committee; that work provided an initial understanding of the topic. In this paper, we paint a more complete picture of the topic, investigating four prominent multi-winner voting rules. First, we characterize the complexity of manipulation and control in these voting rules under various kinds of formalizations of the manipulator's goal. Second, we extend the results about complexity of control to various well-known types of control. This work enhances our comprehension of which multi-winner voting rules should be employed in various settings. We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule. An important aspect of mechanism design in social choice protocols and multiagent systems is to discourage insincere and manipulative behaviour. We examine the computational complexity of false-name manipulation in weighted voting games which are an important class of coalitional voting games. Weighted voting games have received increased interest in the multiagent community due to their compact representation and ability to model coalitional formation scenarios. Bachrach and Elkind in their AAMAS 2008 paper examined divide and conquer false-name manipulation in weighted voting games from the point of view of Shapley-Shubik index. We analyse the corresponding case of the Banzhaf index and check how much the Banzhaf index of a player increases or decreases if it splits up into sub-players. A pseudo-polynomial algorithm to find the optimal split is also provided. Bachrach and Elkind also mentioned manipulation via merging as an open problem. In the paper, we examine the cases where a player annexes other players or merges with them to increase their Banzhaf index or Shapley-Shubik index payoff. We characterize the computational complexity of such manipulations and provide limits to the manipulation. The annexation non-monotonicity paradox is also discovered in the case of the Banzhaf index. The results give insight into coalition formation and manipulation.