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Weighted Average-convexity and Cooperative Games

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  • Alexandre Skoda

    (UP1 - Université Paris 1 Panthéon-Sorbonne, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Xavier Venel

    (Dipartimento di Economia e Finanza [Roma] - LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma])

Abstract

We generalize the notion of convexity and average-convexity to the notion of weighted average-convexity. We show several results on the relation between weighted averageconvexity and cooperative games. First, we prove that if a game is weighted averageconvex, then the corresponding weighted Shapley value is in the core. Second, we exhibit necessary conditions for a communication TU-game to preserve the weighted averageconvexity. Finally, we provide a complete characterization when the underlying graph is a priority decreasing tree.

Suggested Citation

  • Alexandre Skoda & Xavier Venel, 2022. "Weighted Average-convexity and Cooperative Games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03717539, HAL.
    • Handle: RePEc:hal:cesptp:halshs-03717539
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03717539
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    References listed on IDEAS

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    1. Inarra, Elena & Usategui, Jose M, 1993. "The Shapley Value and Average Convex Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(1), pages 13-29.
    2. Marco Slikker & Anne van den Nouweland, 2000. "Communication situations with asymmetric players," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(1), pages 39-56, September.
    3. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
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    5. van den Nouweland, Anne & Borm, Peter, 1991. "On the Convexity of Communication Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(4), pages 421-430.
    6. Monderer, Dov & Samet, Dov & Shapley, Lloyd S, 1992. "Weighted Values and the Core," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(1), pages 27-39.
    7. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
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