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55
Winner determination in sequential majority voting
 In Proceedings of the ECAI2006 Multidisciplinary Workshop on Advances in Preference Handling
, 2007
"... Preferences can be aggregated using a voting rule. Each agent gives their preference orderings over a set of candidates, and a voting rule is used to compute the winner. We consider voting rules which perform a sequence of pairwise comparisons between two candidates, where the result of each is comp ..."
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Cited by 38 (12 self)
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Preferences can be aggregated using a voting rule. Each agent gives their preference orderings over a set of candidates, and a voting rule is used to compute the winner. We consider voting rules which perform a sequence of pairwise comparisons between two candidates, where the result of each is computed by a majority vote. The winner thus depends on the chosen sequence of comparisons, which can be represented by a binary tree. There are candidates that will win in some trees (called possible winners) or in all trees (called Condorcet winners). While it is easy to find the possible and Condorcet winners, we prove that it is difficult if we insist that the tree is balanced. This restriction is therefore enough to make voting difficult for the chair to manipulate. We also consider the situation where we lack complete informations about preferences, and determine the computational complexity of computing possible and Condorcet winners in this extended case. 1
Preferences for multiattributed alternatives: Traces, Dominance, and Numerical Representations
 JOURNAL OF MATHEMATICAL PSYCHOLOGY
, 2002
"... This paper analyzes conjoint measurement models allowing for intransitive and/or incomplete preferences. This analysis is based on the study of marginal traces induced on coordinates by the preference relation and uses conditions guaranteeing that these marginal traces are complete. Within the ..."
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Cited by 25 (14 self)
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This paper analyzes conjoint measurement models allowing for intransitive and/or incomplete preferences. This analysis is based on the study of marginal traces induced on coordinates by the preference relation and uses conditions guaranteeing that these marginal traces are complete. Within the
A Computational Analysis of the Tournament Equilibrium Set
, 2009
"... A recurring theme in the mathematical social sciences is how to select the “most desirable ” elements given a binary dominance relation on a set of alternatives. Schwartz’s tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions prop ..."
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Cited by 15 (9 self)
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A recurring theme in the mathematical social sciences is how to select the “most desirable ” elements given a binary dominance relation on a set of alternatives. Schwartz’s tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions proposed so far. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive solution concept refining both the Banks set and Dutta’s minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NPhard, and thus does not admit a polynomialtime algorithm unless P equals NP. Furthermore, we propose a heuristic that significantly outperforms the naive algorithm for computing TEQ.
Computing the minimal covering set
 In Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge
, 2007
"... We present the first polynomialtime algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists fo ..."
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Cited by 14 (11 self)
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We present the first polynomialtime algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set, the minimal upward covering set and the minimal downward covering set, unless P equals NP. Finally, we observe a strong relationship between von NeumannMorgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other.
Dealing with Incomplete Agents ’ Preferences and an Uncertain Agenda in Group Decision Making via Sequential Majority Voting
"... We consider multiagent systems where agents ’ preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents ’ preferences is common i ..."
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Cited by 12 (7 self)
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We consider multiagent systems where agents ’ preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents ’ preferences is common in many reallife settings due to privacy issues or an ongoing elicitation process. In addition, there may be uncertainty about how the preferences are aggregated. For example, the agenda (a tree whose leaves are labelled with the decisions being compared) may not yet be known or fixed. We therefore study how to determine collectively optimal decisions (also called winners) when preferences may be incomplete, and when the agenda may be uncertain. We show that it is computationally easy to determine if a candidate decision always wins, or may win, whatever the agenda. On the other hand, it is computationally hard to know whether a candidate decision wins in at least one agenda for at least one completion of the agents ’ preferences. These results hold even if the agenda must be balanced so that each candidate decision faces the same number of majority votes. Such results are useful for reasoning about preference elicitation. They help understand the complexity of tasks such as determining if a decision can be taken collectively, as well as knowing if the winner can be manipulated by appropriately ordering the agenda.
Cloning in Elections
"... We consider the problem of manipulating elections via cloning candidates. In our model, a manipulator can replace each candidate c by one or more clones, i.e., new candidates that are so similar to c that each voter simply replaces c in his vote with the block of c’s clones. The outcome of the resul ..."
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Cited by 10 (3 self)
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We consider the problem of manipulating elections via cloning candidates. In our model, a manipulator can replace each candidate c by one or more clones, i.e., new candidates that are so similar to c that each voter simply replaces c in his vote with the block of c’s clones. The outcome of the resulting election may then depend on how each voter orders the clones within the block. We formalize what it means for a cloning manipulation to be successful (which turns out to be a surprisingly delicate issue), and, for a number of prominent voting rules, characterize the preference profiles for which a successful cloning manipulation exists. We also consider the model where there is a cost associated with producing each clone, and study the complexity of finding a minimumcost cloning manipulation. Finally, we compare cloning with the related problem of control via adding candidates.
Minimal Stable Sets in Tournaments
, 2009
"... We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, which encompasses the top cycle, the uncove ..."
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Cited by 9 (8 self)
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We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, which encompasses the top cycle, the uncovered set, the Banks set, the minimal covering set, the tournament equilibrium set, the Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set. 1
A New Perspective on Implementation by Voting Trees
"... Voting trees describe an iterative procedure for selecting a single vertex from a tournament. They provide a very general abstract model of decisionmaking among a group of individuals, and it has therefore been studied which voting rules have a tree that implements them, i.e., chooses according to ..."
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Cited by 9 (1 self)
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Voting trees describe an iterative procedure for selecting a single vertex from a tournament. They provide a very general abstract model of decisionmaking among a group of individuals, and it has therefore been studied which voting rules have a tree that implements them, i.e., chooses according to the rule for every tournament. While partial results concerning implementable rules and necessary conditions for implementability have been obtained over the past forty years, a complete characterization of voting rules implementable by trees has proven surprisingly hard to find. A prominent rule that cannot be implemented by trees is the Copeland rule, which singles out vertices with maximum degree. In this paper, we suggest a new angle of attack and reexamine the implementability of the Copeland solution using paradigms and techniques that are at the core of theoretical computer science. We study the extent to which voting trees can approximate the maximum degree in a tournament, and give upper and lower bounds on the worstcase ratio between the degree of the vertex chosen by a tree and the maximum degree, both for the deterministic model concerned with a single fixed tree, and for randomizations over arbitrary sets of trees. Our main positive result is a randomization over surjective trees of polynomial size that provides an approximation ratio of at least 1/2. The proof is based on a connection between a randomization over caterpillar trees and a rapidly mixing Markov chain.
Following the traces: An introduction to conjoint measurement without transitivity and additivity
, 2003
"... This paper presents a selfcontained introduction to a general conjoint measurement framework for the analysis of nontransitive and/or incomplete binary relations on product sets. It is based on the use of several kinds of marginal traces on coordinates induced by the binary relation. This framework ..."
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Cited by 9 (3 self)
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This paper presents a selfcontained introduction to a general conjoint measurement framework for the analysis of nontransitive and/or incomplete binary relations on product sets. It is based on the use of several kinds of marginal traces on coordinates induced by the binary relation. This framework
Computational Aspects of Covering in Dominance Graphs
, 2008
"... Various problems in AI and multiagent systems can be tackled by finding the “most desirable” elements of a set given some binary relation. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Some particularly attractive solution sets are defined in term ..."
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Cited by 7 (4 self)
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Various problems in AI and multiagent systems can be tackled by finding the “most desirable” elements of a set given some binary relation. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Some particularly attractive solution sets are defined in terms of a covering relation—a transitive subrelation of the original relation. We consider three different types of covering (upward, downward, and bidirectional) and the corresponding solution concepts known as the uncovered set and the minimal covering set. We present the first polynomialtime algorithm for finding the minimal bidirectional covering set (an acknowledged open problem) and prove that deciding whether an alternative is in a minimal upward or downward covering set is NPhard. Furthermore, we obtain various settheoretical inclusions, which reveal a strong connection between von NeumannMorgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other hand. In particular, we show that every stable set is also a minimal upward covering set.