M. Studeny, V. Kratochvil:
Facets of the cone of exact games.
Mathematical Methods of Operation Research
95 (2022), n. 1, pp. 35800.
 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 facetdefining inequalities for the exact cone appear to correspond to certain set systems (= systems of coalitions). We noticed that nonempty proper coalitions having nonzero coefficients in these facetdefining inequalities form set systems with particular properties.
More specifically, we introduce the concept of a semibalanced system of coalitions, which generalizes the classic concept of a balanced coalitional system in cooperative game theory. The semibalanced coalitional systems provide valid inequalities for the exact cone and minimal semibalanced systems (in the sense of inclusion of set systems) characterize this cone. We also introduce basic classification of minimal semibalanced systems, their pictorial representatives and a substantial concept of an indecomposable (minimal)
semibalanced system of coalitions. The main result of the paper is that indecomposable semibalanced systems are in onetoone correspondence with facetdefining 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 antidual is also totally balanced. We additionally characterize those inequalities which are facetdefining 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
 antidual of a game
 semibalanced set system
 indecomposable minsemibalanced set system
 A
pdf version of a preprint (448kB) is available.
Moreover, the result of the work is also an
interactive electronic catalogue of indecomposable minsemibalanced system on at most 6 variables.
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. 127135.
 L.S. Shapley:
On balanced sets and cores.
Naval Research Logistics Quarterly
14 (1967), pp. 453460.
 T. Kroupa, M. Studeny:
Facets of the cone of totally balanced games.
Mathematical Methods of Operation Research
90 (2019), pp. 271300.

M. Studeny, V. Kratochvil:
Linear criterion for testing the extremity of an exact game based on its finest
minrepresentation.
International Journal of Approximate Reasoning
101 (2018), pp. 4968.
 M. Studeny, V. Kratochvil, J. Vomlel:
On irreducible minbalanced set systems..
In Symbolic and Quantitative Approaches to Reasoning with Uncertainty
(G. KernIsberner, Z. Ognjanovic eds.),
Lecture Notes in Artificial Intelligence 1126,
SpringerVerlag, Cham 2019, pp. 444454.