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Nash Stable Outcomes in Fractional Hedonic Games: Existence, Efficiency and Computation

Author

Listed:
  • Vittorio Bilò

    (Dipartimento di Matematica Ennio De Giorgi - Università del Salento = University of Salento [Lecce], Università del Salento = University of Salento [Lecce])

  • Angelo Fanelli

    (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

  • Michele Flammini

    (DISIM - Dipartimento di Ingegneria, Scienza dell'Informazione e Matematica - UNIVAQ - Università degli Studi dell'Aquila = University of L'Aquila, UNIVAQ - University of L'Aquila [Italy])

  • Gianpiero Monaco

    (DI - Dipartimento di Informatica [Italy] - UNIVAQ - Università degli Studi dell'Aquila = University of L'Aquila)

  • Luca Moscardelli

    (Department of Economic Studies - University of Chieti-Pescara, Dipartimento di Scienze - Universita di Chieti-Pescara - UNICH - Universita' degli Studi "G. d'Annunzio" Chieti-Pescara)

Abstract

We consider fractional hedonic games, a subclass of coalition formation games that can be succinctly modeled by means of a graph in which nodes represent agents and edge weights the degree of preference of the corresponding endpoints. The happiness or utility of an agent for being in a coalition is the average value she ascribes to its members. We adopt Nash stable outcomes as the target solution concept; that is we focus on states in which no agent can improve her utility by unilaterally changing her own group. We provide existence, efficiency and complexity results for games played on both general and specific graph topologies. As to the efficiency results, we mainly study the quality of the best Nash stable outcome and refer to the ratio between the social welfare of an optimal coalition structure and the one of such an equilibrium as to the price of stability. In this respect, we remark that a best Nash stable outcome has a natural meaning of stability, since it is the optimal solution among the ones which can be accepted by selfish agents. We provide upper and lower bounds on the price of stability for different topologies, both in case of weighted and unweighted edges. Beside the results for general graphs, we give refined bounds for various specific cases, such as triangle-free, bipartite graphs and tree graphs. For these families, we also show how to efficiently compute Nash stable outcomes with provable good social welfare.

Suggested Citation

  • Vittorio Bilò & Angelo Fanelli & Michele Flammini & Gianpiero Monaco & Luca Moscardelli, 2018. "Nash Stable Outcomes in Fractional Hedonic Games: Existence, Efficiency and Computation," Post-Print hal-02089363, HAL.
    • Handle: RePEc:hal:journl:hal-02089363
    DOI: 10.1613/jair.1.11211
    Note: View the original document on HAL open archive server: https://hal.science/hal-02089363v1
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    References listed on IDEAS

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    1. Krzysztof R. Apt & Andreas Witzel, 2009. "A Generic Approach To Coalition Formation," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 11(03), pages 347-367.
    2. Francis Bloch & Effrosyni Diamantoudi, 2011. "Noncooperative formation of coalitions in hedonic games," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(2), pages 263-280, May.
    3. Ballester, Coralio, 2004. "NP-completeness in hedonic games," Games and Economic Behavior, Elsevier, vol. 49(1), pages 1-30, October.
    4. Bogomolnaia, Anna & Jackson, Matthew O., 2002. "The Stability of Hedonic Coalition Structures," Games and Economic Behavior, Elsevier, vol. 38(2), pages 201-230, February.
    5. Aziz, Haris & Brandt, Felix & Harrenstein, Paul, 2013. "Pareto optimality in coalition formation," Games and Economic Behavior, Elsevier, vol. 82(C), pages 562-581.
    6. Tayfun Sönmez & Suryapratim Banerjee & Hideo Konishi, 2001. "Core in a simple coalition formation game," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(1), pages 135-153.
    7. Dreze, J H & Greenberg, J, 1980. "Hedonic Coalitions: Optimality and Stability," Econometrica, Econometric Society, vol. 48(4), pages 987-1003, May.
    8. Vittorio Bilò & Angelo Fanelli & Michele Flammini & Gianpiero Monaco & Luca Moscardelli, 2014. "Nash stability in fractional hedonic games," Post-Print hal-01103984, HAL.
    9. 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.
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    Cited by:

    1. Edith Elkind & Angelo Fanelli & Michele Flammini, 2020. "Price of Pareto Optimality in hedonic games," Post-Print hal-02932135, HAL.
    2. Martin Gairing & Rahul Savani, 2019. "Computing Stable Outcomes in Symmetric Additively Separable Hedonic Games," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 1101-1121, August.
    3. Gianpiero Monaco & Luca Moscardelli & Yllka Velaj, 2021. "Additively Separable Hedonic Games with Social Context," Games, MDPI, vol. 12(3), pages 1-14, September.

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