Mathematical Programming

, Volume 21, Issue 1, pp 1–18 | Cite as

Minimum cost spanning tree games

  • Daniel Granot
  • Gur Huberman


We consider the problem of cost allocation among users of a minimum cost spanning tree network. It is formulated as a cooperative game in characteristic function form, referred to as a minimum cost spanning tree (m.c.s.t.) game. We show that the core of a m.c.s.t. game is never empty. In fact, a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem. For m.c.s.t. games with efficient coalition structures we define and construct m.c.s.t. games on the components of the structure. We show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli, respectively, of the induced games on the components of the efficient coalition structure.

Key words

Game Theory Cost Allocation Spanning Tree 


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  1. [1]
    C.G. Bird, “On cost allocation for a spanning tree: A game theory approach”,Networks 6 (1976) 335–350.Google Scholar
  2. [2]
    A. Claus and D.J. Kleitman, “Cost-allocation for a spanning tree”,Networks 3 (1973) 289–304.Google Scholar
  3. [3]
    A. Claus and D. Granot, “Game theory application to cost allocation for a spanning tree”, Working Paper No. 402, Faculty of Commerce and Business Administration, University of British Columbia (June 1976).Google Scholar
  4. [4]
    D. Granot and G. Huberman, “Minimum cost spanning tree games”, Working Paper No. 403, Faculty of Commerce, U.B.C. (June 1976; revised Sept. 1976/August 1977).Google Scholar
  5. [5]
    D. Granot and G. Huberman, “Permutationally convex games and minimum spanning tree games”, Discussion Paper 77-10-3, Simon Fraser University (June 1977).Google Scholar
  6. [6]
    A. Kopelowitz, “Computation of the kernels of simple games and the nucleolus ofn-person games”, Research Memorandum No. 31, Department of Mathematics, The Hebrew University of Jerusalem (September 1967).Google Scholar
  7. [7]
    S.C. Littlechild, “A simple expression for the nucleolus in a special case”,International Journal of Game Theory 3 (1974) 21–29.Google Scholar
  8. [8]
    N. Megiddo, “Computational complexity and the game theory approach to cost allocation for a tree”,Mathematics of Operations Research 3 (1978) 189–196.Google Scholar
  9. [9]
    N. Megiddo, “Cost allocation for Steiner trees”,Networks 8 (1978) 1–6.Google Scholar
  10. [10]
    L.S. Shapley, “A value forn-person games”,Annals of Mathematics Study 28 (1953) 307–317.Google Scholar

Copyright information

© The Mathematical Programming Society 1981

Authors and Affiliations

  • Daniel Granot
    • 1
  • Gur Huberman
    • 2
  1. 1.University of British ColumbiaVancouverCanada
  2. 2.University of ChicagoChicagoUSA

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