Abstract

The class of exact transferable utility coalitional games, introduced in 1972 by Schmeidler, has been studied both in the context of game theory and in the context of imprecise probabilities. We characterize the cone of exact games by describing the minimal set of linear inequalities defining this cone; these facet-defining inequalities for the exact cone appear to correspond to certain set systems (= systems of coalitions). We noticed that non-empty proper coalitions having non-zero coefficients in these facet-defining inequalities form set systems with particular properties. More specifically, we introduce the concept of a semi-balanced system of coalitions, which generalizes the classic concept of a balanced coalitional system in cooperative game theory. The semi-balanced coalitional systems provide valid inequalities for the exact cone and minimal semi-balanced systems (in the sense of inclusion of set systems) characterize this cone. We also introduce basic classification of minimal semi-balanced systems, their pictorial representatives and a substantial concept of an indecomposable (minimal) semi-balanced system of coalitions. The main result of the paper is that indecomposable semi-balanced systems are in one-to-one correspondence with facet-defining inequalities for the exact cone. The second relevant result is the rebuttal of a former conjecture claiming that a coalitional game is exact iff it is totally balanced and its anti-dual is also totally balanced. We additionally characterize those inequalities which are facet-defining both for the cone of exact games and for the cone of totally balanced games.

AMS classification 91A12 52B12 68R05 90C27

Keywords

coalitional game

exact game

totally balanced game

anti-dual of a game

semi-balanced set system

indecomposable min-semi-balanced set system

A pdf version of a preprint (448kB) is available.

The contribution builds on the following publications:

• E. Lohmann, P. Borm, P. J.-J. Herings: Minimal exact balancedness. Mathematical Social Sciences 64 (2012), n. 2, pp. 127-135.
• L.S. Shapley: On balanced sets and cores. Naval Research Logistics Quarterly 14 (1967), pp. 453-460.
• T. Kroupa, M. Studeny: Facets of the cone of totally balanced games. Mathematical Methods of Operations Research 90 (2019), pp. 271-300.
• M. Studeny, V. Kratochvil: Linear criterion for testing the extremity of an exact game based on its finest min-representation. International Journal of Approximate Reasoning 101 (2018), pp. 49-68.
• M. Studeny, V. Kratochvil, J. Vomlel: On irreducible min-balanced set systems.. In Symbolic and Quantitative Approaches to Reasoning with Uncertainty (G. Kern-Isberner, Z. Ognjanovic eds.), Lecture Notes in Artificial Intelligence 1126, Springer-Verlag, Cham 2019, pp. 444-454.