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Shortest path games

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  • Rosenthal, Edward C.
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    Abstract

    We study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network. Such networks model, among other things, efficient development of a commuter rail system for a growing metropolitan area. We motivate and define these games and provide reasonable conditions for the corresponding rail application. We show that the core of a shortest path game is nonempty and satisfies the given conditions, but that the Shapley value for these games may lie outside the core. However, we show that the shortest path game is convex for the special case of tree networks, and we provide a simple, polynomial time formula for the Shapley value in this case. In addition, we extend our tree results to the case where users of the network travel to nodes other than the source. Finally, we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games, where nodes are added to the network sequentially over time.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0377221712005346
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    Bibliographic Info

    Article provided by Elsevier in its journal European Journal of Operational Research.

    Volume (Year): 224 (2013)
    Issue (Month): 1 ()
    Pages: 132-140

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    Handle: RePEc:eee:ejores:v:224:y:2013:i:1:p:132-140

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    Web page: http://www.elsevier.com/locate/eor

    Related research

    Keywords: Game theory; Shortest paths; Network design;

    References

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    1. Grahn, S., 2001. "Core and Bargaining Set of Shortest Path Games," Papers 2001:03, Uppsala - Working Paper Series.
    2. Mark Voorneveld & Sofia Grahn, 2002. "Cost allocation in shortest path games," Computational Statistics, Springer, vol. 56(2), pages 323-340, November.
    3. Rosenthal, E C, 1990. "Monotonicity of the Core and Value in Dynamic Cooperative Games," International Journal of Game Theory, Springer, vol. 19(1), pages 45-57.
    4. Laporte, Gilbert & Mesa, Juan A. & Perea, Federico, 2010. "A game theoretic framework for the robust railway transit network design problem," Transportation Research Part B: Methodological, Elsevier, vol. 44(4), pages 447-459, May.
    5. S. C. Littlechild & G. Owen, 1973. "A Simple Expression for the Shapley Value in a Special Case," Management Science, INFORMS, vol. 20(3), pages 370-372, November.
    6. Sprumont, Yves, 1990. "Population monotonic allocation schemes for cooperative games with transferable utility," Games and Economic Behavior, Elsevier, vol. 2(4), pages 378-394, December.
    7. Laporte, G. & Mesa, J.A. & Ortega, F.A. & Perea, F., 2011. "Planning rapid transit networks," Socio-Economic Planning Sciences, Elsevier, vol. 45(3), pages 95-104, September.
    8. BergantiƱos, Gustavo & Vidal-Puga, Juan, 2010. "Realizing fair outcomes in minimum cost spanning tree problems through non-cooperative mechanisms," European Journal of Operational Research, Elsevier, vol. 201(3), pages 811-820, March.
    9. Rosenthal, Edward C., 1990. "Monotonicity of solutions in certain dynamic cooperative games," Economics Letters, Elsevier, vol. 34(3), pages 221-226, November.
    10. Grahn, Sofia, 2001. "Core and Bargaining Set of Shortest Path Games," Working Paper Series 2001:3, Uppsala University, Department of Economics.
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    Cited by:
    1. Richard Startz & Kwok Ping Tsang, 2014. "On the Present Value Model in a Cross Section of Stocks," Working Papers e07-47, Virginia Polytechnic Institute and State University, Department of Economics.
    2. Eric Bahel, 2014. "On the core and bargaining set of a veto game," Working Papers e07-48, Virginia Polytechnic Institute and State University, Department of Economics.
    3. Xiaojin Sun & Kwok Ping Tsang, 2013. "Housing Markets, Regulations and Monetary Policy," Working Papers e07-45, Virginia Polytechnic Institute and State University, Department of Economics.
    4. Balazs Sziklai & Tamas Fleiner & Tamas Solymosi, 2013. "The Nucleolus of Directed Acyclic Graph Games," IEHAS Discussion Papers 1321, Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences.
    5. Christian Trudeau, 2013. "Minimum cost spanning tree problems with indifferent agents," Working Papers 1306, University of Windsor, Department of Economics.

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