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Clique games: a family of games with coincidence between the nucleolus and the Shapley value

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  • Trudeau, Christian
  • Vidal-Puga, Juan

Abstract

We introduce a new family of cooperative games for which there is coincidence between the nucleolus and the Shapley value. These so-called clique games are such that agents are divided into cliques, with the value created by a coalition linearly increasing with the number of agents belonging to the same clique. Agents can belong to multiple cliques, but for a pair of cliques, at most a single agent belong to their intersection. Finally, if two agents do not belong to the same clique, there is at most one way to link the two agents through a chain of agents, with any two non-adjacent agents in the chain belonging to disjoint sets of cliques. We provide multiple examples for clique games. Graph-induced games, either when the graph indicates cooperation possibilities or impossibilities, provide us with opportunities to confirm existing results or discover new ones. A particular focus are the minimum cost spanning tree problems. Our result allows us to obtain new coincidence results between the nucleolus and the Shapley value, as well as other cost sharing methods for the minimum cost spanning tree problem.

Suggested Citation

  • Trudeau, Christian & Vidal-Puga, Juan, 2018. "Clique games: a family of games with coincidence between the nucleolus and the Shapley value," MPRA Paper 96710, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:96710
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    1. Kar, Anirban & Mitra, Manipushpak & Mutuswami, Suresh, 2009. "On the coincidence of the prenucleolus and the Shapley value," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 16-25, January.
    2. Sylvain Béal & Eric Rémila & Philippe Solal, 2015. "Axioms of invariance for TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(4), pages 891-902, November.
    3. Subiza, Begoña & Giménez, José Manuel & Peris, Josep E., 2015. "Folk solution for simple minimum cost spanning tree problems," QM&ET Working Papers 15-7, University of Alicante, D. Quantitative Methods and Economic Theory.
    4. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    5. A. van den Nouweland & P. Borm & W. van Golstein Brouwers & R. Groot Bruinderink & S. Tijs, 1996. "A Game Theoretic Approach to Problems in Telecommunication," Management Science, INFORMS, vol. 42(2), pages 294-303, February.
    6. Dutta, Bhaskar & Kar, Anirban, 2004. "Cost monotonicity, consistency and minimum cost spanning tree games," Games and Economic Behavior, Elsevier, vol. 48(2), pages 223-248, August.
    7. Graham, Daniel A & Marshall, Robert C & Richard, Jean-Francois, 1990. "Differential Payments within a Bidder Coalition and the Shapley Value," American Economic Review, American Economic Association, vol. 80(3), pages 493-510, June.
    8. Bergantinos, Gustavo & Vidal-Puga, Juan J., 2007. "A fair rule in minimum cost spanning tree problems," Journal of Economic Theory, Elsevier, vol. 137(1), pages 326-352, November.
    9. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. Tijs, S.H., 2005. "The First Steps with Alexia, the Average Lexicographic Value," Discussion Paper 2005-123, Tilburg University, Center for Economic Research.
    11. Bogomolnaia, Anna & Moulin, Hervé, 2010. "Sharing a minimal cost spanning tree: Beyond the Folk solution," Games and Economic Behavior, Elsevier, vol. 69(2), pages 238-248, July.
    12. Trudeau, Christian, 2012. "A new stable and more responsive cost sharing solution for minimum cost spanning tree problems," Games and Economic Behavior, Elsevier, vol. 75(1), pages 402-412.
    13. Juan J. Vidal-Puga, 2004. "Bargaining with commitments," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(1), pages 129-144, January.
    14. González–Arangüena, E. & Manuel, C. & Owen, G. & del Pozo, M., 2017. "The within groups and the between groups Myerson values," European Journal of Operational Research, Elsevier, vol. 257(2), pages 586-600.
    15. Gomez, Daniel & Gonzalez-Aranguena, Enrique & Manuel, Conrado & Owen, Guillermo & del Pozo, Monica & Tejada, Juan, 2003. "Centrality and power in social networks: a game theoretic approach," Mathematical Social Sciences, Elsevier, vol. 46(1), pages 27-54, August.
    16. Yokote, Koji & Funaki, Yukihiko & Kamijo, Yoshio, 2017. "Coincidence of the Shapley value with other solutions satisfying covariance," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 1-9.
    17. Xiaotie Deng & Toshihide Ibaraki & Hiroshi Nagamochi, 1999. "Algorithmic Aspects of the Core of Combinatorial Optimization Games," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 751-766, August.
    18. 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.
    19. Trudeau, Christian & Vidal-Puga, Juan, 2017. "On the set of extreme core allocations for minimal cost spanning tree problems," Journal of Economic Theory, Elsevier, vol. 169(C), pages 425-452.
    20. Xiaotie Deng & Christos H. Papadimitriou, 1994. "On the Complexity of Cooperative Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 257-266, May.
    21. Youngsub Chun & Nari Park & Duygu Yengin, 2015. "Coincidence of Cooperative Game Theoretic Solutions in the Appointment Problem," School of Economics and Public Policy Working Papers 2015-09, University of Adelaide, School of Economics and Public Policy.
    22. Tijs, S.H., 2005. "The First Steps with Alexia, the Average Lexicographic Value," Other publications TiSEM 0b4f9565-59f7-477f-b8f8-9, Tilburg University, School of Economics and Management.
    23. Kuipers, Jeroen, 1993. "On the Core of Information Graph Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(4), pages 339-350.
    24. Gustavo Bergantiños & Juan Vidal-Puga, 2007. "The optimistic TU game in minimum cost spanning tree problems," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(2), pages 223-239, October.
    25. 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.
    26. Ni, Debing & Wang, Yuntong, 2007. "Sharing a polluted river," Games and Economic Behavior, Elsevier, vol. 60(1), pages 176-186, July.
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    Cited by:

    1. Christian Trudeau, 2023. "Minimum cost spanning tree problems as value sharing problems," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(1), pages 253-272, March.
    2. José-Manuel Giménez-Gómez & Josep E Peris & Begoña Subiza, 2020. "An egalitarian approach for sharing the cost of a spanning tree," PLOS ONE, Public Library of Science, vol. 15(7), pages 1-14, July.
    3. Doudou Gong & Bas Dietzenbacher & Hans Peters, 2024. "One-bound core games," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(3), pages 859-878, September.
    4. Bergantiños, Gustavo & Vidal-Puga, Juan, 2020. "Cooperative games for minimum cost spanning tree problems," MPRA Paper 104911, University Library of Munich, Germany.
    5. Elena Iñarra & Roberto Serrano & Ken-Ichi Shimomura, 2020. "The Nucleolus, the Kernel, and the Bargaining Set: An Update," Revue économique, Presses de Sciences-Po, vol. 71(2), pages 225-266.
    6. Hernández, Penélope & Peris, Josep E. & Vidal-Puga, Juan, 2023. "A non-cooperative approach to the folk rule in minimum cost spanning tree problems," European Journal of Operational Research, Elsevier, vol. 307(2), pages 922-928.
    7. Masrur, Hasan & Khaloie, Hooman & Al-Awami, Ali T. & Ferik, Sami El & Senjyu, Tomonobu, 2024. "Cost-aware modeling and operation of interconnected multi-energy microgrids considering environmental and resilience impact," Applied Energy, Elsevier, vol. 356(C).
    8. Gustavo Bergantiños & Juan Vidal-Puga, 2021. "A review of cooperative rules and their associated algorithms for minimum-cost spanning tree problems," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 12(1), pages 73-100, March.
    9. repec:ehu:ikerla:34464 is not listed on IDEAS

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    More about this item

    Keywords

    nucleolus; Shapley value; clique; minimum cost spanning tree;
    All these keywords.

    JEL classification:

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

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