Condorcet winning sets

n 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 θ of voters; we refer to such sets as θ - winning sets . We explore social choice-theoretic and algorithmic aspects of these solution concepts, both theoretically and empirically.