Tree enterprises and bankruptcy ventures: a game theoretic similarity due to a graph theoretic proof

In a tree enterprise, users reside at the nodes of the tree and their aim is to connect themselves, directly or indirectly, to the root of the tree. The construction costs of arcs of the tree are given by means of the arc-cost-function associated with the tree. Further the bankruptcy venture is described in terms of the estate of the bankrupt firm and the claims of the various creditors. The first objective of the paper is to provide conditions (on the claims and the surplus of the claims in the bankruptcy venture) which are sufficient and necessary for the bankruptcy venture to agree with some tree enterprise. It is established that the bankruptcy venture agrees with some tree enterprise if and only if the surplus of claims in the bankruptcy venture is at most the size of the second smallest claim (in the weak sense). For that purpose, both the tree enterprise as well as the bankruptcy venture are modelled as a cooperative game with transferable utility. Within the framework of cooperative game theory, the proof of the equivalence theorem concerning the tree enterprise game and the bankruptcy game, under the given circumstances, is based on graph-theoretic tools in a tree structure. As an adjunct to the proof of the equivalence theorem, the solution concept of the nucleolus for specific tree enterprises is determine.