Solving hard problems in election systems

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Title: Solving hard problems in election systems
Author: Lin, Andrew
Abstract: 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.
Record URI: http://hdl.handle.net/1850/15268
Date: 2012

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