Abstract
We develop two efficient procedures for generating cost allocation vectors in the core of a minimum cost spanning tree (m.c.s.t.) game. The first procedure requires O(n 2) elementary operations to obtain each additional point in the core, wheren is the number of users. The efficiency of the second procedure, which is a natural strengthening of the first procedure, stems from the special structure of minimum excess coalitions in the core of an m.c.s.t. game. This special structure is later used (i) to ease the computational difficulty in computing the nucleolus of an m.c.s.t. game, and (ii) to provide a geometric characterization for the nucleolus of an m.c.s.t. game. This geometric characterization implies that in an m.c.s.t. game the nucleolus is the unique point in the intersection of the core and the kernel. We further develop an efficient procedure for generating fair cost allocations which, in some instances, coincide with the nucleolus. Finally, we show that by employing Sterns' transfer scheme we can generate a sequence of cost vectors which converges to the nucleolus.
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References
L.J. Billera, D.C. Heath and J. Raanan, “Internal telephone billing rates—A novel application of non atomic game theory”,Operations Research 26 (1978) 956–965.
C.G. Bird, “On cost allocation for a spanning tree: A game theory approach”,Networks 6 (1976) 335–350.
J. Callen, “Financial cost allocations. A game theoretic approach”,Accounting Review 53 (1978) 303–308.
A. Charnes and K. Kortanek, “On classes of convex and preemptive nuclei forN-person games”, in: H.W. Kuhn, ed.,Proceedings of the 1967 Princeton Mathematical Programming Symposium, Princeton University Press, (Princeton, 1970).
A. Claus and D.J. Kleitman, “Cost allocation for a spanning tree”,Networks 3 (1973) 289–304.
M. Davis and M. Maschler, “The kernel of a cooperative game”,Naval Research Logistics Quarterly 12 (1965) 223–259.
D. Granot, “On the role of cost allocation in locational models”, Faculty of Commerce and Business Administration, University of British Columbia (January 1981).
D. Granot and G. Huberman, “On minimum cost spanning tree games”,Mathematical Programming 21 (1981) 1–18.
D. Granot and G. Huberman, “The relationship between convex games and minimum cost spanning tree games: A case for permutationally convex games”,SIAM Journal of Algebraic and Discrete Methods 3 (1982) 288–292.
D. Granot and G. Huberman, “More cost allocations in the core of a minimum cost spanning tree game”, Working paper No. 695. Faculty of Commerce, University of British Columbia (February 1980, revised December 1981).
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 (Jerusalem, September 1967).
S.C. Littlechild, “A simple expression for the nucleolus in a special case”,International Journal of Game Theory 3 (1974) 21–29.
S.C. Littlechild and G. Owen, “A simple expression for the Shapley value in a special case”,Management Science 20 (1973) 370–372.
M. Maschler, B. Peleg and L.S. Shapley, “Geometric properties of the kernel, nucleolus, and related solution concepts”,Mathematics of Operations Research 4 (4) (1979) 303–338.
N. Megiddo, “Cost allocation for Steiner trees”,Networks 1 (1979) 1–9.
N. Megiddo, “Computational complexity and the game theory approach to cost allocation for a tree”,Mathematics of Operations Research 3 (1978) 189–196.
G. Owen, “A note on the nucleolus”,International Journal of Game Theory 3 (1974) 101–103.
A. Roth and R. Verrecchia, “The Shapley value as applied to cost allocation: A re-interpretation”,Journal of Accounting Research 17 (1979) 295–303.
D. Samet, Y. Tauman and I. Zang. “An application of theA-S prices for cost allocation in transportation problems”,Mathematics of Operations Research, forthcoming.
D. Schmeidler, “The nucleolus of a characteristic function game”,SIAM Journal of Applied Mathematics 17 (1969) 1163–1170.
M. Shubik, “Incentives, decentralized control, the assignment of joint costs and internal pricing”,Management Science 8 (1962) 325–343.
R.E. Sterns, “Convergent transfer schemes forN-person games”, Technical Information Series, No. 67-C-311, R&D Center, General Electric (September 1967).
A. Tamir, “On the core of cost allocation games defined on location problems”, Department of Statistics, Tel-Aviv University (July 1980).
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Part of this research was done while the author was visiting the Department of Operations Research at Stanford University. This research was partially supported by Natural Sciences and Engineering Research Council Canada Grant A-4181.
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Granot, D., Huberman, G. On the core and nucleolus of minimum cost spanning tree games. Mathematical Programming 29, 323–347 (1984). https://doi.org/10.1007/BF02592000
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DOI: https://doi.org/10.1007/BF02592000