The Banks Set and the Bipartisan Set May Be Disjoint

TL;DR

This paper constructs a specific tournament example showing that the Banks set and the bipartisan set can be disjoint, challenging assumptions about their relationship in social choice theory.

Contribution

It provides the first explicit example of a tournament where the Banks set and the bipartisan set are disjoint, highlighting limitations of certain tournament solution refinements.

Findings

Constructed a tournament of order 36 with disjoint Banks and bipartisan sets

Shows that refinements like the minimal extending set can also be disjoint from the bipartisan set

Impacts understanding of solution stability in social choice models

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.