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Efficiency and collusion neutrality in cooperative games and networks

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  • van den Brink, René

Abstract

In the literature various axiomatizations of the Shapley and Banzhaf value can be found where the only difference is the Shapley value satisfying efficiency and the Banzhaf value satisfying collusion neutrality. Both properties seem to be desirable. In this paper, we show that there is no solution satisfying efficiency, collusion neutrality and the null player property. After that we show that this impossibility is resolved when cooperation among players is restricted by a cycle-free communication network such that only connected coalitions are feasible. In particular, all hierarchical outcomes introduced by Demange (2004) and their convex combinations satisfy these three properties.

Suggested Citation

  • van den Brink, René, 2012. "Efficiency and collusion neutrality in cooperative games and networks," Games and Economic Behavior, Elsevier, vol. 76(1), pages 344-348.
    • Handle: RePEc:eee:gamebe:v:76:y:2012:i:1:p:344-348
    DOI: 10.1016/j.geb.2012.04.001
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    References listed on IDEAS

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    1. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    2. Herings, P. Jean Jacques & van der Laan, Gerard & Talman, Dolf, 2008. "The average tree solution for cycle-free graph games," Games and Economic Behavior, Elsevier, vol. 62(1), pages 77-92, January.
    3. Le Breton, M & Owen, G & Weber, S, 1992. "Strongly Balanced Cooperative Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(4), pages 419-427.
    4. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-281.
    5. Demange, Gabrielle, 1994. "Intermediate preferences and stable coalition structures," Journal of Mathematical Economics, Elsevier, vol. 23(1), pages 45-58, January.
    6. Marcin Malawski, 2002. "Equal treatment, symmetry and Banzhaf value axiomatizations," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 47-67.
    7. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    8. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    9. Gabrielle Demange, 2004. "On Group Stability in Hierarchies and Networks," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 754-778, August.
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    Cited by:

    1. Jeon, Doh-Shin & Menicucci, Domenico, 2014. "Buyer Group and Buyer Power When Sellers Compete," TSE Working Papers 14-543, Toulouse School of Economics (TSE), revised Nov 2017.
    2. Sylvain Béal & Amandine Ghintran & Eric Rémila & Philippe Solal, 2015. "The sequential equal surplus division for rooted forest games and an application to sharing a river with bifurcations," Theory and Decision, Springer, vol. 79(2), pages 251-283, September.
    3. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2015. "Characterization of the Average Tree solution and its kernel," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 159-165.
    4. Anna Khmelnitskaya & Gerard van der Laan & Dolf Talman, 2016. "Centrality Rewarding Shapley and Myerson Values for Undirected Graph Games," Tinbergen Institute Discussion Papers 16-070/II, Tinbergen Institute.
    5. Sylvain Béal & Sylvain Ferrières & Eric Rémila & Philippe Solal, 2017. "Axiomatic and bargaining foundation of an allocation rule for ordered tree TU-games," Post-Print halshs-01644797, HAL.
    6. Liying Kang & Anna Khmelnitskaya & Erfang Shan & Dolf Talman & Guang Zhang, 2021. "The average tree value for hypergraph games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 437-460, December.
    7. Béal, Sylvain & Ferrières, Sylvain & Rémila, Eric & Solal, Philippe, 2018. "Axiomatization of an allocation rule for ordered tree TU-games," Mathematical Social Sciences, Elsevier, vol. 93(C), pages 132-140.
    8. Jeon, Doh-Shin & Menicucci, Domenico, 2019. "On the unprofitability of buyer groups when sellers compete," Games and Economic Behavior, Elsevier, vol. 115(C), pages 265-288.

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    More about this item

    Keywords

    Cooperative TU-game; Efficiency; Collusion neutrality; Null player property; Communication graph; Hierarchical outcome;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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