Facets of the cone of exact games

0556097 - ÚTIA 2023 RIV CH eng J - Journal Article
Studený, Milan - Kratochvíl, Václav
Facets of the cone of exact games.
Mathematical Methods of Operations Research. Roč. 95, č. 1 (2022), s. 35-80. ISSN 1432-2994. E-ISSN 1432-5217
R&D Projects: GA ČR(CZ) GA19-04579S
Institutional support: RVO:67985556
Keywords : coalitional game * exact game * totally balanced game * anti-dual of a game * semi-balanced set system * indecomposable min-semi-balanced system
OECD category: Pure mathematics
Impact factor: 1.2, year: 2022
Method of publishing: Limited access
http://library.utia.cas.cz/separaty/2022/MTR/studeny-0556097.pdf https://link.springer.com/article/10.1007/s00186-022-00770-4

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.
Permanent Link: http://hdl.handle.net/11104/0330491