An ideal winner in such an election is a Condorcet winner, a candidate that "beats" any other candidate in a pairwise voting contest. Formally, i C is a Condorcet winner if, for every candidate j C\{i}, more than half of the voters prefer i over j. To avoid ties, we assume the number n of voters is odd. Unfortunately, it is easy to construct elections where a Condorcet winner does not exist; indeed typically there is no Condorcet winner. Given this, Elkind, Lang and Saffidine [12] proposed a relaxation where, rather than a single winning candidate, we desire a set of winning candidates that collectively beats any other candidate. Specifically, a set of candidates S C is a Condorcet winning set if, for every candidate j C \S, more than half of the voters prefer some (voter-dependent) candidate in S to j. Elkind et al. [12] showed that a Condorcet winning set always exists provided we allow the winning set to be large enough.

Condorcet winning sets are a set-valued generalization of the well-known concept of a Condorcet winner. As supersets of Condorcet winning sets are always Condorcet winning sets themselves, an interesting property of preference profiles is the size of the smallest Condorcet winning set they admit. This smallest size is called the Condorcet dimension of a preference profile. Since little is known about profiles that have a certain Condorcet dimension, we show in this paper how the problem of finding a preference profile that has a given Condorcet dimension can be encoded as a satisfiability problem and solved by a SAT solver. Initial results include a minimal example of a preference profile of Condorcet dimension 3, improving previously known examples both in terms of the number of agents as well as alternatives. Due to the high complexity of such problems it remains open whether a preference profile of Condorcet dimension 4 exists.

In elections, an alternative is said to be a Condorcet winner 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 setvalued 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 this concept as theta-winning set. We explore social choice-theoretic and algorithmic aspects of these solution concepts, both theoretically and empirically.