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The minimum cost shortest-path tree game

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  • F. Fernández
  • J. Puerto

Abstract

A minimum cost shortest-path tree is a tree that connects the source with every node of the network by a shortest path such that the sum of the cost (as a proxy for length) of all arcs is minimum. In this paper, we adapt the algorithm of Hansen and Zheng (Discrete Appl. Math. 65:275–284, 1996 ) to the case of acyclic directed graphs to find a minimum cost shortest-path tree in order to be applied to the cost allocation problem associated with a cooperative minimum cost shortest-path tree game. In addition, we analyze a non-cooperative game based on the connection problem that arises in the above situation. We prove that the cost allocation given by an ‘à la’ Bird rule provides a core solution in the former game and that the strategies that induce those payoffs in the latter game are Nash equilibrium. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • F. Fernández & J. Puerto, 2012. "The minimum cost shortest-path tree game," Annals of Operations Research, Springer, vol. 199(1), pages 23-32, October.
    • Handle: RePEc:spr:annopr:v:199:y:2012:i:1:p:23-32:10.1007/s10479-011-1043-8
    DOI: 10.1007/s10479-011-1043-8
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    References listed on IDEAS

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    1. Gustavo Bergantiños & Leticia Lorenzo, 2004. "A non-cooperative approach to the cost spanning tree problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(3), pages 393-403, July.
    2. Vito Fragnelli & Ignacio García-Jurado & Luciano Méndez-Naya, 2000. "On shortest path games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 251-264, November.
    3. Gustavo Bergantiños & Leticia Lorenzo, 2005. "Optimal Equilibria in the Non-Cooperative Game Associated with Cost Spanning Tree Problems," Annals of Operations Research, Springer, vol. 137(1), pages 101-115, July.
    4. Fernandez, Francisco R. & Hinojosa, Miguel A. & Puerto, Justo, 2004. "Multi-criteria minimum cost spanning tree games," European Journal of Operational Research, Elsevier, vol. 158(2), pages 399-408, October.
    5. F. Fernández & M. Hinojosa & A. Mármol & J. Puerto, 2009. "Opportune moment strategies for a cost spanning tree game," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(3), pages 451-463, December.
    6. Mark Voorneveld & Sofia Grahn, 2002. "Cost allocation in shortest path games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(2), pages 323-340, November.
    7. María Gómez-Rúa & Juan Vidal-Puga, 2011. "Balanced per capita contributions and level structure of cooperation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 167-176, July.
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    2. Lei Gao & Dong Han, 2020. "Extreme Value Distributions for Two Kinds of Path Sums of Markov Chain," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 279-294, March.

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