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Computing Superdifferentials of Lovász Extension with Application to Coalitional Games

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Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 610))

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Abstract

Every coalitional game can be extended from the powerset onto the real unit cube. One of possible approaches is the Lovász extension, which is the same as the discrete Choquet integral with respect to the coalitional game. We will study some solution concepts for coalitional games (core, Weber set) using superdifferentials developed in non-smooth analysis. It has been shown that the core coincides with Fréchet superdifferential and the Weber set with Clarke superdifferential for the Lovász extension, respectively. We introduce the intermediate set as the limiting superdifferential and show that it always lies between the core and the Weber set. From the game-theoretic point of view, the intermediate set is a non-convex solution containing the Pareto optimal payoff vectors, which depend on some ordered partition of the players and the marginal coalitional contributions with respect to the order.

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Notes

  1. 1.

    We may occasionally switch to a simplified notation for coalitions, writing ij in place of \(\{i,j\}\) and similarly.

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Acknowledgments

L. Adam gratefully acknowledges the support from the Grant Agency of the Czech Republic (15-00735S). The work of T. Kroupa was supported by Marie Curie Intra-European Fellowship OASIG (PIEF-GA-2013-622645).

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Authors and Affiliations

  1. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08, Prague, Czech Republic

    Lukáš Adam

  2. Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milano, Italy

    Tomáš Kroupa

Authors
  1. Lukáš Adam
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  2. Tomáš Kroupa
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Corresponding author

Correspondence to Tomáš Kroupa .

Editor information

Editors and Affiliations

  1. INESC-ID,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Joao Paulo Carvalho

  2. LIP 6, Université Pierre et Marie Curie, Paris, France

    Marie-Jeanne Lesot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  4. IDMEC,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Susana Vieira

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Machine Intelligence Institute, Iona College, New Rochelle, New York, USA

    Ronald R. Yager

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Adam, L., Kroupa, T. (2016). Computing Superdifferentials of Lovász Extension with Application to Coalitional Games. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-40596-4_4

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  • DOI: https://doi.org/10.1007/978-3-319-40596-4_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40595-7

  • Online ISBN: 978-3-319-40596-4

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