Linearity of the core correspondence

Bloch and de Clippel (J Econ Theory 145:2424–2434, 2010) characterized sets of balanced TU-games on which the core correspondence is linear by means of an equivalence relation. We characterize maximal regions on which the core correspondence is linear in four different ways. First, by finitely many linear equalities and inequalities; thus, the core is piecewise linear. Second, maximal linear regions coincide with closures of equivalence classes (in the sense of Bloch and de Clippel) that are maximal w.r.t. set inclusion. Third, maximal linear regions coincide with closures of equivalence classes of full dimension. Fourth, for every extreme point of the core of a game in the interior of a maximal linear region, the collection of tight core inequalities constitutes a basis.