Facets of the cone of exact games

TL;DR

This paper characterizes the cone of exact cooperative games using minimal inequalities linked to semi-balanced coalition systems, introducing indecomposable systems and disproving a previous conjecture about game exactness.

Contribution

It introduces semi-balanced systems, classifies minimal such systems, and establishes a one-to-one correspondence with the cone's facets, advancing understanding of exact game structure.

Findings

Facet-defining inequalities correspond to indecomposable semi-balanced systems.

Minimal semi-balanced systems characterize the cone of exact games.

Counterexample disproves the conjecture linking exactness to total balancedness.

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 secondary 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 exact cone and the cone of totally balanced games.