How Likely A Coalition of Voters Can Influence A Large Election?

For centuries, it has been widely believed that the influence of a small coalition of voters is negligible in a large election. Consequently, there is a large body of literature on characterizing the asymptotic likelihood for an election to be influenced, especially by the manipulation of a single voter, establishing an $O(\frac{1}{\sqrt n})$ upper bound and an $\Omega(\frac{1}{n^{67}})$ lower bound for many commonly studied voting rules under the i.i.d.~uniform distribution, known as Impartial Culture (IC) in social choice, where $n$ is the number is voters. In this paper, we extend previous studies in three aspects: (1) we consider a more general and realistic semi-random model, where a distribution adversary chooses a worst-case distribution and then a data adversary modifies up to $\psi$ portion of the data, (2) we consider many coalitional influence problems, including coalitional manipulation, margin of victory, and various vote controls and bribery, and (3) we consider arbitrary and variable coalition size $B$. Our main theorem provides asymptotically tight bounds on the semi-random likelihood of the existence of a size-$B$ coalition that can successfully influence the election under a wide range of voting rules. Applications of the main theorem and its proof techniques resolve long-standing open questions about the likelihood of coalitional manipulability under IC, by showing that the likelihood is $\Theta\left(\min\left\{\frac{B}{\sqrt n}, 1\right\}\right)$ for many commonly studied voting rules. The main technical contribution is a characterization of the semi-random likelihood for a Poisson multinomial variable (PMV) to be unstable, which we believe to be a general and useful technique with independent interest

Resolving the Complexity of Some Fundamental Problems in Computational Social Choice

This thesis is in the area called computational social choice which is an intersection area of algorithms and social choice theory.Comment: Ph.D. Thesi

10101 Abstracts Collection -- Computational Foundations of Social Choice

From March 7 to March 12, 2010, the Dagstuhl Seminar 10101 ``Computational Foundations of Social Choice \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Computational aspects of voting: a literature survey

Preference aggregation is a topic of study in different fields such as philosophy, mathematics, economics and political science. Recently, computational aspects of preference aggregation have gained especial attention and “computational politics” has emerged as a marked line of research in computer science with a clear concentration on voting protocols. The field of voting systems, rooted in social choice theory, has expanded notably in both depth and breadth in the last few decades. A significant amount of this growth comes from studies concerning the computational aspects of voting systems. This thesis comprehensively reviews the work on voting systems (from a computing perspective) by listing, classifying and comparing the results obtained by different researchers in the field. This survey covers a wide range of new and historical results yet provides a profound commentary on related work as individual studies and in relation to other related work and to the field in general. The deliverables serve as an overview where students and novice researchers in the field can start and also as a depository that can be referred to when searching for specific results. A comprehensive literature survey of the computational aspects of voting is a task that has not been undertaken yet and is initially realized here. Part of this research was dedicated to creating a web-depository that contains material and references related to the topic based on the survey. The purpose was to create a dynamic version of the survey that can be updated with latest findings and as an online practical reference

Solving hard problems in election systems

An interesting problem in the field of computational social choice theory is that of elections, in which a winner or set of winners is to be deduced from preferences among a collection of agents, in a way that attempts to maximize the collective well-being of the agents. Besides their obvious use in political science, elections are also used computationally, such as in multiagent systems, in which different agents may have different beliefs and preferences and must reach an agreeable decision. Because the purpose of voting is to gain an understanding of a collection of actual preferences, dishonesty in an election system is often harmful to the welfare of the voters as a whole. Different forms of dishonesty can be performed by the voters (manipulation), by an outside agent affecting the voters (bribery), or by the chair, or administrator, of an election (control). The Gibbard-Satterthwaite theorem shows that in all reasonable election systems, manipulation, or strategic voting, is always inevitable in some cases. Bartholdi, Tovey, and Trick counter by arguing that if finding such a manipulation is NP-hard, then manipulation by computationally-limited agents should not pose a significant threat. However, more recent work has exploited the fact that NP-hardness is only a worst-case measure of complexity, and has shown that some election systems that are NP-hard to manipulate may in fact be easy to manipulate under some reasonable assumptions. We evaluate, both theoretically and empirically, the complexity, worst-case and otherwise, of manipulating, bribing, and controlling elections. Our focus is particularly on scoring protocols. In doing so, we gain an understanding of how these election systems work by discovering what makes manipulation, bribery, and control easy or hard. This allows us to discover the strengths and weaknesses of scoring protocols, and gain an understanding of what properties of election systems are desirable or undesirable. One approach we have used to do this is relating the problems of interest in election systems to problems of known complexity, as well as to problems with known algorithms and heuristics, particularly Satisfiability and Partition. This approach can help us gain an understanding of computational social choice problems in which little is known about the complexity or potential algorithms. Among other results, we show how certain parameters and properties of scoring protocols can make elections easy or hard to manipulate. We find that the empirical complexity of manipulation in some cases have unusual behaviors for its complexity class. For example, it is found that in the case of manipulating the Borda election of unweighted voters with an unbounded candidate cardinality, the encoding of this problem to Satisfiability performs especially well near the boundary cases of this problem and for unsatisfiable instances, both results contrary to the normal behavior of NP-complete problems. Although attempts have been made to design fair election systems with certain properties, another dilemma that this has given rise to is the existence of election systems in which it is hard to elect the winners, at least in the worst case. Two notable election systems in which determining the winners are hard are Dodgson and Young. We evaluate the problem of finding the winners empirically, to extend these complexity results away from the worst case, and determine whether the worst-case complexity of these hard winner problems is truly a computational barrier. We find that, like most NP-complete problems such as Satisfiability, many instances of interest in finding winners of hard election systems are still relatively simple. We confirm that indeed, like Satisfiability, the hard worst-case results occur only in rare circumstances. We also find an interesting complexity disparity between the related problems of finding the Dodgson or Young score of a candidate, and that of finding the set of Dodgson or Young winners. Surprisingly, it appears empirically easier for one to find the set of all winners in a Dodgson or Young election than to score a single candidate in either election

Auctions, Bidding and Exchange Design

The different auction types are outlined using a classification framework along six dimensions. The economic properties that are desired in the design of auction mechanisms and the complexities that arise in their implementation are discussed. Some of the most interesting designs from the literature are analyzed in detail to establish known results and to identify the emerging research directions.Engineering and Applied Science