On the Set of Balanced Games

• Published in: Mathematics of operations research Vol. 50; no. 3; pp. 2047 - 2072
• Main Authors: Garcia-Segador, Pedro, Grabisch, Michel, Miranda, Pedro
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.08.2025; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; Apexes; balanced games; Combinatorial analysis; combinatorial polytope; Combinatorics; convex polyhedra; cooperative TU games; core; Games; Geometry; Graph theory; Mathematics; Polytopes; Primary 91A12; Random variables; secondary 52B05; Mathematics:Sets:Polyhedra; core; cooperative TU-games; balanced games; convex polyhedra; combinatorial polytope. MSC2000 subject classification: Primary 91A12; Secondary 52B05 OR/MS subject classification: Games/group decisions:Cooperative
• ISSN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2023.0379

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].