International Journal of Game Theory (1995) 24:147-163 Hierarchical Organization Structures and Constraints on Coalition Formation 1 JEAN J.M. DERKS Department of Mathematics, University of Limburg, P.O. Box 616, 6200 MD Maastricht, The Nether- lands ROBERT P. GILLES Department of Economics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061- 0316, USA Abstract: This paper studies the constraints in coalition formation that result from a hierarchical organization structure on the class of players in a cooperative game with transferable utilities. If one assumes that the superiors of a certain individual have to give permission to the actions undertaken by the individual, then one arrives at a limited collection of formable or autonomous coalitions. This resulting collection is a lattice of subsets on the player set. We show that if the collection of formable coalitions is limited to a lattice, the core allows for (infinite) exploitation of subordinates. For discerning lattices we are able to generalize the results of Weber (1988), namely the core is a subset of the convex hull of the collection of all attainable marginal contribution vectors plus a fixed cone. This relation is an equality if and only if the game is convex. This extends the results of Shapley (1971) and Ichiishi (1981). 1 Introduction In social sciences and game theory one regularly addresses the problem of coopera- tive behavior assuming that every decision maker is an autonomous acting individual. Under this hypothesis one traditionally arrives at a model in which, in principle, every group of individuals has to be regarded as a formable or autonomous coalition. This is the viewpoint as reflected in the traditional definition of the core, the Shapley value, and the bargaining set in a cooperative game with transferable utilities. If one discards the assumption that individuals are completely free in forming coalitions, one arrives at refinements which incorporate certain constraints in coalition formation. Aumann and Dreze (1974) describe a situation in which certain exogenous conditions result in a partition of individuals into a finite number of jurisdictions. These coalition structures lead to well specified constraints on coalition formation, namely groups within a jurisdiction can be formed freely, but individuals cannot over- step the boundaries as set by the partition of the individuals into jurisdictions. Another approach is to derive constraints in coalition formation from certain so- cial defects of the individuals such as limited abilities to communicate with other individuals. This is the starting point of Myerson (1977), who introduces an undi- rected graph as a description of the limited communication possibilities open to the individuals. Now a coalition can only form if it does not depend on outside individuals We would like to thank Rene van den Brink, Hans Hailer, Nancy Lutz, Bruno Parigi, and an anonymous referee for their remarks on a previous draft of this paper and Alan Kirman for his questions that raised the issue as discussed in this paper. 0020-7276/95/2/147-163 $ 2.50 9 1995 Physica-Verlag, Heidelberg