  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.
