We prove that for any constant k and any ϵ<1, there exist bimatrix
win-lose games for which every ϵ-WSNE requires supports of cardinality
greater than k. To do this, we provide a graph-theoretic characterization of
win-lose games that possess ϵ-WSNE with constant cardinality supports.
We then apply a result in additive number theory of Haight to construct
win-lose games that do not satisfy the requirements of the characterization.
These constructions disprove graph theoretic conjectures of Daskalakis, Mehta
and Papadimitriou, and Myers