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Eliminating the Weakest Link: Making Manipulation Intractable
 In: Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2012
"... Successive elimination of candidates is often a route to making manipulation intractable to compute. We prove that eliminating candidates does not necessarily increase the computational complexity of manipulation. However, for many voting rules used in practice, the computational complexity increase ..."
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Successive elimination of candidates is often a route to making manipulation intractable to compute. We prove that eliminating candidates does not necessarily increase the computational complexity of manipulation. However, for many voting rules used in practice, the computational complexity increases. For example, it is already known that it is NPhard to compute how a single voter can manipulate the result of single transferable voting(the elimination version of plurality voting). We show here that it is NPhard to compute how a single voter can manipulate the result of the elimination version of veto voting, of the closely related Coombs ’ rule, and of the elimination versions of a general class of scoring rules.
Coalitional Manipulation for Schulze’s Rule
"... Schulze’s rule is used in the elections of a large number of organizations including Wikimedia and Debian. Part of the reason for its popularity is the large number of axiomatic properties, like monotonicity and Condorcet consistency, which it satisfies. We identify a potential shortcoming of Schulz ..."
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Schulze’s rule is used in the elections of a large number of organizations including Wikimedia and Debian. Part of the reason for its popularity is the large number of axiomatic properties, like monotonicity and Condorcet consistency, which it satisfies. We identify a potential shortcoming of Schulze’s rule: it is computationally vulnerable to manipulation. In particular, we prove that computing an unweighted coalitional manipulation (UCM) is polynomial for any number of manipulators. This result holds for both the unique winner and the cowinner versions of UCM. This resolves an open question in [14]. We also prove that computing a weighted coalitional manipulation (WCM) is polynomial for a bounded number of candidates. Finally, we discuss the relation between the unique winner UCM problem and the cowinner UCM problem and argue that they have substantially different necessary and sufficient conditions for the existence of a successful manipulation.
Is Computational Complexity a Barrier to Manipulation?
"... Abstract. When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an e ..."
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Abstract. When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NPhard to compute how to manipulate a number of different voting rules. However, NPhardness only bounds the worstcase complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational “phase transitions”. Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NPhard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems. 1
Manipulating Two Stage Voting Rules
"... We study the computational complexity of computing a manipulation of a two stage voting rule. An example of a two stage voting rule is Black’s procedure. The first stage of Black’s procedure selects the Condorcet winner if they exist, otherwise the second stage selects the Borda winner. In general, ..."
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We study the computational complexity of computing a manipulation of a two stage voting rule. An example of a two stage voting rule is Black’s procedure. The first stage of Black’s procedure selects the Condorcet winner if they exist, otherwise the second stage selects the Borda winner. In general, we argue that there is no connection between the computational complexity of manipulating the two stages of such a voting rule and that of the whole. However, we also demonstrate that we can increase the complexity of even a very simple base rule by adding a stage to the front of the base rule. In particular, whilst Plurality is polynomial to manipulate, we show that the two stage rule that selects the Condorcet winner if they exist and otherwise computes the Plurality winner is NPhard to manipulate with 3 or more candidates, weighted votes and a coalition of manipulators. In fact, with any scoring rule, computing a coalition manipulation of the two stage rule that selects the Condorcet winner if they exist and otherwise applies the scoring rule is NPhard with 3 or more candidates and weighted votes. It follows that computing a coalition manipulation of Black’s procedure is NPhard with weighted votes. With unweighted votes, we prove that the complexity of manipulating Black’s procedure is inherited from the Borda rule that it includes. More specifically, a single manipulator can compute a manipulation of Black’s procedure in polynomial time, but computing a manipulation is NPhard for two manipulators. 1
Contents lists available at ScienceDirect Discrete Mathematics
"... journal homepage: www.elsevier.com/locate/disc ..."
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"... Schulze’s rule is used in the elections of a large number of organizations including Wikimedia and Debian. Part of the reason for its popularity is the large number of axiomatic properties, like monotonicity and Condorcet consistency, which it satisfies. We identify a potential shortcoming of Schu ..."
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Schulze’s rule is used in the elections of a large number of organizations including Wikimedia and Debian. Part of the reason for its popularity is the large number of axiomatic properties, like monotonicity and Condorcet consistency, which it satisfies. We identify a potential shortcoming of Schulze’s rule: it is computationally vulnerable to manipulation. In particular, we prove that computing an unweighted coalitional manipulation (UCM) is polynomial for any number of manipulators. This result holds for both the unique winner and the cowinner versions of UCM. This resolves an open question in [14]. We also prove that computing a weighted coalitional manipulation (WCM) is polynomial for a bounded number of candidates. Finally, we discuss the relation between the unique winner UCM problem and the cowinner UCM problem and argue that they have substantially different necessary and sufficient conditions for the existence of a successful manipulation.
Computational voting theory: Gametheoretic and . . .
, 2011
"... For at least two thousand years, voting has been used as one of the most effective ways to aggregate people’s ordinal preferences. In the last 50 years, the rapid development of Computer Science has revolutionize every aspect of the world, including voting. This motivates us to study (1) conceptuall ..."
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For at least two thousand years, voting has been used as one of the most effective ways to aggregate people’s ordinal preferences. In the last 50 years, the rapid development of Computer Science has revolutionize every aspect of the world, including voting. This motivates us to study (1) conceptually, how computational thinking changes the traditional theory of voting, and (2) methodologically, how to better use voting for preference/information aggregation with the help of Computer Science. My Ph.D. work seeks to investigate and foster the interplay between Computer Science and Voting Theory. In this thesis, I will discuss two specific research directions pursued in my Ph.D. work, one for each question asked above. The first focuses on investigating how computational thinking affects the gametheoretic aspects of voting. More precisely, I will discuss the rationale and possibility of using computational complexity to protect voting from a type of strategic behavior of the voters, called manipulation. The second studies a voting setting called Combinatorial Voting, where the set of alternatives is exponentially large and has a combinatorial
ABSTRACT
"... Axioms that govern our choice of voting rules are usually defined by imposing constraints on the rule’s behavior under various transformations of the preference profile. In this paper we adopt a different approach, and view a voting rule as a (multi)coloring of the election graph—the graph whose ve ..."
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Axioms that govern our choice of voting rules are usually defined by imposing constraints on the rule’s behavior under various transformations of the preference profile. In this paper we adopt a different approach, and view a voting rule as a (multi)coloring of the election graph—the graph whose vertices are elections over a given set of candidates, and two vertices are adjacent if they can be obtained from each other by swapping adjacent candidates in one of the votes. Given this perspective, a voting rule F is characterized by the shapes of its “monochromatic components”, i.e., sets of elections that have the same winner under F. In particular, it would be natural to expect each monochromatic component to be convex, or, at the very least, connected. We formalize the notions of connectivity and (weak) convexity for monochromatic components, and say that a voting rule is connected/(weakly) convex if each of its monochromatic components is connected/(weakly) convex. We then investigate which of the classic voting rules have these properties. It turns out that while all voting rules that we consider are connected, convexity and even weak convexity are much more demanding properties. Our study of connectivity suggests a new notion of monotonicity, which may be of independent interest.