The Shapley Value for Airport and Irrigation Games
In this paper cost sharing problems are considered. We focus on problems on a rooted tree, we call these problems cost-tree problems, and on the induced transferable utility cooperative games, we call these games 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 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 on a rooted tree. In our characterization results we relate the TU games terminologies to the cost sharing terminologies, so we bridge between the two fields.
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- 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.
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- William Thomson, 2007. "Cost allocation and airport problems," RCER Working Papers 537, University of Rochester - Center for Economic Research (RCER).
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- Aadland, David & Kolpin, Van, 1998. "Shared irrigation costs: An empirical and axiomatic analysis," Mathematical Social Sciences, Elsevier, vol. 35(2), pages 203-218, March.
- René van den Brink, 2002. "An axiomatization of the Shapley value using a fairness property," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(3), pages 309-319.
- 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.
- Granot, D & Maschler, M & Owen, G & Zhu, W.R., 1996. "The Kernel/Nucleolus of a Standard Tree Game," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(2), pages 219-244.
- Moulin, Herve & Shenker, Scott, 1992. "Serial Cost Sharing," Econometrica, Econometric Society, vol. 60(5), pages 1009-1037, September.
- Chun, Youngsub, 1991. "On the Symmetric and Weighted Shapley Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 183-190.
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