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The Shapley value for airport and irrigation games

  • Márkus, Judit
  • Pintér, Miklós
  • Radványi, Anna

In this paper cost sharing problems are considered. We focus on problems given by rooted trees, we call these problems cost-tree problems, and on the induced transferable utility cooperative games, called irrigation games. A formal notion of irrigation games is introduced, and the characterization of the class of these games is provided. The well-known class of airport games Littlechild and Thompson (1977) is a subclass of irrigation games. The Shapley value Shapley (1953) is probably the most popular solution concept for transferable utility cooperative games. Dubey (1982) and Moulin and Shenker (1992) show respectively, that Shapley's Shapley (1953) and Young (1985)'s axiomatizations of the Shapley value are valid on the class of airport games. In this paper we show that Dubey (1982)'s and Moulin and Shenker (1992)'s results can be proved by applying Shapley (1953)'s and Young (1985)'s proofs, that is those results are direct consequences of Shapley (1953)'s and Young (1985)'s results. Furthermore, we extend Dubey (1982)'s and Moulin and Shenker (1992)'s results to the class of irrigation games, that is we provide two characterizations of the Shapley value for cost sharing problems given by rooted trees. We also note that for irrigation games the Shapley value is always stable, that is it is always in the core Gillies (1959).

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 30031.

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Date of creation: 2011
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Handle: RePEc:pra:mprapa:30031
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  1. S.C. Littlechild & G.F. Thompson, 1977. "Aircraft Landing Fees: A Game Theory Approach," Bell Journal of Economics, The RAND Corporation, vol. 8(1), pages 186-204, Spring.
  2. Moulin, Herve & Shenker, Scott, 1992. "Serial Cost Sharing," Econometrica, Econometric Society, vol. 60(5), pages 1009-37, September.
  3. Granot, D, et al, 1996. "The Kernel/Nucleolus of a Standard Tree Game," International Journal of Game Theory, Springer, vol. 25(2), pages 219-44.
  4. S. H. Tijs & M. Koster & E. Molina & Y. Sprumont, 2002. "Sharing the cost of a network: core and core allocations," International Journal of Game Theory, Springer, vol. 30(4), pages 567-599.
  5. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
  6. Koster, M.A.L. & Molina, E. & Sprumont, Y. & Tijs, S.H., 2001. "Sharing the cost of a network : Core and core allocations," Other publications TiSEM 20f62f3f-75ba-4fcd-abdc-a, Tilburg University, School of Economics and Management.
  7. Chun, Youngsub, 1991. "On the Symmetric and Weighted Shapley Values," International Journal of Game Theory, Springer, vol. 20(2), pages 183-90.
  8. René van den Brink, 2002. "An axiomatization of the Shapley value using a fairness property," International Journal of Game Theory, Springer, vol. 30(3), pages 309-319.
  9. 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.
  10. Aadland, David & Kolpin, Van, 1998. "Shared irrigation costs: An empirical and axiomatic analysis," Mathematical Social Sciences, Elsevier, vol. 35(2), pages 203-218, March.
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