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74
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 39 (13 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 29 (15 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 17 (11 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 17 (13 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.
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 14 (10 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
Feasibility and Approximability of Dodgson's rule
 Auckland University
, 2006
"... It is known that Dodgson's rule is computationally very demanding. Tideman [15] suggested an approximation to it but did not investigate how often his approximation selects the Dodgson winner. We show that under the Impartial Culture assumption the probability that the Tideman winner is the Do ..."
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Cited by 13 (1 self)
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It is known that Dodgson's rule is computationally very demanding. Tideman [15] suggested an approximation to it but did not investigate how often his approximation selects the Dodgson winner. We show that under the Impartial Culture assumption the probability that the Tideman winner is the Dodgson winner tends to 1. However we show that the convergence of this probability to 1 is slow. We suggest another approximation  we call it Dodgson Quick  for which this convergence is exponentially fast. 1
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 11 (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.
The Matrix of Maximum Out Forests of a Digraph and Its Applications
, 2006
"... We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph Γ is a spanning subgraph of Γ that consists of disjoint diverging trees and has the maximum possible number of arcs. If a digraph contains out arborescences, then maximu ..."
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Cited by 10 (2 self)
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We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph Γ is a spanning subgraph of Γ that consists of disjoint diverging trees and has the maximum possible number of arcs. If a digraph contains out arborescences, then maximum out forests coincide with them. We consider Markov chains related to a weighted digraph and prove that the matrix of Cesàro limiting probabilities of such a chain coincides with the normalized matrix of maximum out forests. This provides an interpretation for the matrix of Cesàro limiting probabilities of an arbitrary stationary finite Markov chain in terms of the weight of maximum out forests. We discuss the applications of the matrix of maximum out forests and its transposition, the matrix of limiting accessibilities of a digraph, to the problems of preference aggregation, measuring the vertex proximity, and uncovering the structure of a digraph.
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.