On the Set of Balanced Games

TL;DR

This paper explores the geometric structure of the set of balanced cooperative games with nonempty core, characterizing its shape, extremal elements, and properties, including special cases with fixed values and nonnegativity.

Contribution

It provides a detailed geometric and combinatorial analysis of the set of balanced games, including extremal rays, facets, vertices, and an algorithm for vertex generation.

Findings

The set of balanced games forms a nonpointed polyhedral cone.

The vertices and facets of the convex polytope for nonnegative balanced games are characterized.

An algorithm for uniform random vertex generation and the adjacency graph's Hamiltonian property are developed.

Abstract

We study the geometric structure of the set of cooperative transferable utility games having a nonempty core, characterized by Bondareva and Shapley as balanced games. We show that this set is a nonpointed polyhedral cone, and we find the set of its extremal rays and facets. This study is also done for the set of balanced games whose value for the grand coalition is fixed, which yields an affine nonpointed polyhedral cone. Finally, the case of nonnegative balanced games with fixed value for the grand coalition is tackled. This set is a convex polytope, with remarkable properties. We characterize its vertices and facets, study the adjacency structure of vertices, develop an algorithm for generating vertices in a random uniform way, and show that this polytope is combinatorial and its adjacency graph is Hamiltonian. Last, we give a characterization of the set of games having a core reduced to a singleton. Funding: This work was supported by the Spanish Government [Grant PID2021-124933NB-I00].