A further extension of the Knaster–Kuratowski–Mazurkiwicz–Shapley theorem

Reny and Wooders showed that there is some point in the intersection of sets in Shapley's generalization of the Knaster–Kuratowski–Mazurkiwicz Theorem with the property that the collection of all sets containing that point is partnered as well as balanced. We provide a further extension by showing that the collection of all such sets can be chosen to be strictly balanced, implying the Reny–Wooders result. Our proof is topological, based on the Eilenberg–Montgomery Fixed Point Theorem. Reny and Wooders also show that if the collection of partnered points in the intersection is countable, then at least one of them is minimally partnered. Applying degree theory for correspondences, we show that if this collection is assumed to be zero-dimensional, then there is at least one strictly balanced and minimally partnered point in the intersection. Our approach sheds a new geometric–topological light on the Reny–Wooders results.