Abstract

The class of totally balanced games is a class of transferable-utility coalitional games providing important models of cooperative behavior used in mathematical economics. They coincide with market games of Shapley and Shubik and every totally balanced game is also representable as the minimum of a finite set of additive games. In this paper we characterize the polyhedral cone of totally balanced games by describing its facets. Our main result is that there is a correspondence between facet-defining inequalities for the cone and the class of special balanced systems of coalitions, the so-called irreducible min-balanced systems. Our method is based on refining the notion of balancedness introduced by Shapley. We also formulate a conjecture about what are the facets of the cone of exact games, which addresses an open problem appearing in the literature.

AMS classification 91A12 52B12 68R05 90C27

Keywords

coalitional game

totally balanced game

balanced system

polyhedral cone

A pdf version of a preprint (338kB) is available.

The contribution builds on the following publications:

• M. Studeny, T. Kroupa: Core-based criterion for extreme supermodular functions. Discrete Applied Mathematics 206 (2016), pp. 122-151.
• L.S. Shapley: On balanced sets and cores. Naval Research Logistics Quarterly 14 (1967), pp. 453-460.
• E. Quaeghebeur: Learning from samples using coherent lower previsions. PhD thesis, Ghent University, 2009.
• E. Kalai, E. Zemel: Totally balanced games and games of flow. Mathematics of Operations Research 7 (1982), n. 3, pp. 476-478.
• E. Lohmann, P. Borm, P. J.-J. Herings: Minimal exact balancedness. Mathematical Social Sciences 64 (2012), n. 2, pp. 127-135.