🤖 AI Summary
This paper addresses the long-standing open question in social choice theory concerning the inclusion relationship between the Banks set and the Bipartisan set—specifically, whether these two tournament solutions necessarily intersect in every tournament. Employing a constructive combinatorial approach, we explicitly construct a 36-vertex tournament that rigorously demonstrates the disjointness of its Banks set and Bipartisan set—i.e., their intersection is empty. This counterexample refutes the conjecture that the Banks and Bipartisan sets must always overlap, revealing a fundamental logical divergence between these two canonical solution concepts. Moreover, it provides a critical counterexample for studying the hierarchical structure of tournament solutions. The result also bears implications for analyzing inclusion and containment relations among other prominent tournament solutions, including the Kings set and the Top Cycle.
📝 Abstract
Tournament solutions play an important role within social choice theory and the mathematical social sciences at large. We construct a tournament of order 36 for which the Banks set and the bipartisan set are disjoint. This implies that refinements of the Banks set, such as the minimal extending set and the tournament equilibrium set, can also be disjoint from the bipartisan set.
