Hernández, P, J.E. Peris and J.A. Silva-Reus
Journal of Mathematical Economics – Forthcoming
Keywords: cost spanning tree, cost allocation, transfer struc- ture, subgame perfect equilibrium, core.
Abstract: We study minimum cost spanning tree problems for a set of users connected to a source. Prim’s algorithm provides a way of finding the minimum cost tree m. This has led to several definitions in the literature, regarding how to distribute the cost. These rules propose different cost allocations, which can be understood as compensations and/or payments between players, with respect to the status quo point: each user pays for the connection she uses to be linked to the source. In this paper we analyze the rationale behind a distribution of the minimum cost by defining an a priori transfer structure. Our first result states the existence of a transfer structure such that no user is willing to choose a different tree from the minimum cost tree. Moreover, given a transfer structure, we implement the above solution as a subgame perfect equilibrium outcome of a game where players decide sequentially with whom to connect. Finally, we obtain that the existence of a transfer structure supporting an allocation characterizes the core of the monotone cooperative game associated with a minimum cost spanning tree problem. This transfer structure is called social transfer structure. Therefore, the minimum cost spanning tree emerges as both a social and individual solution.