An alternative is said to be a Condorcet winner of an election if it is preferred to any other alternative by a majority of voters. While this is a very attractive solution concept, many elections do not have a Condorcet winner. In this paper, we propose a set-valued relaxation of this concept, which we call a Condorcet winning set: such sets consist of alternatives that collectively dominate any other alternative. We also consider a more general version of this concept, where instead of domination by a majority of voters we require domination by a given fraction \(\theta \) of voters; we refer to such sets as \(\theta \)-winning sets. We explore social choice-theoretic and algorithmic aspects of these solution concepts, both theoretically and empirically.
Some of the results of this paper were previously presented at the 22nd International Joint Conference on Artificial Intelligence (IJCAI’11) under the title “Choosing Collectively Optimal Sets of Alternatives Based on the Condorcet Criterion”, and we would like to thank the anonymous referees of IJCAI’11 and Social Choice and Welfare for their helpful comments. We also thank Bruno Escoffier, Ron Holzman, Christian Laforest, Jean-François Laslier, Hervé Moulin, Remzi Sanver and Bill Zwicker for useful discussions. Abdallah Saffidine thanks the Australian Research Council’s (ARC) Discovery Projects funding scheme (project DP 120102023). Part of this work was done when Edith Elkind was affiliated with Nanyang Technological University (Singapore) and supported by National Research Foundation (Singapore) under grant RF2009-08 and by NTU start-up grant.
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