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Computing Superdifferentials of Lovász Extension with Application to Coalitional Games

  • Lukáš Adam
  • Tomáš KroupaEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)

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.

Keywords

Coalitional game Lovász extension Choquet integral Core Weber set Superdifferential 

Notes

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).

References

  1. 1.
    Adam, L., Kroupa, T.: The intermediate set and limiting superdifferential for coalition games: between the core and the Weber set. Submitted to International Journal of Game Theory (2015). http://arxiv.org/abs/1504.08195
  2. 2.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol. 264. Springer, London (2013)zbMATHGoogle Scholar
  3. 3.
    Danilov, V., Koshevoy, G.: Cores of cooperative games, superdifferentials of functions, and the Minkowski difference of sets. J. Math. Anal. Appl. 247(1), 1–14 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Grabisch, M.: Set functions over finite sets: transformations and integrals. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 1381–1401. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  5. 5.
    Ichiishi, T.: Super-modularity: applications to convex games and to the greedy algorithm for LP. J. Econ. Theory 25(2), 283–286 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Korte, B., Grötschel, M. (eds.) Mathematical Programming: The State of the Art, pp. 235–257. Springer, Berlin (1983)CrossRefGoogle Scholar
  7. 7.
    Maschler, M., Solan, E., Zamir, S.: Game Theory. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer, Heidelberg (2006)Google Scholar
  9. 9.
    Owen, G.: Game Theory, 3rd edn. Academic Press Inc., San Diego (1995)zbMATHGoogle Scholar
  10. 10.
    Peleg, B., Sudhölter, P.: Introduction to the Theory of Cooperative Games. Theory and Decision Library. Series C: Game Theory, Mathematical Programming and Operations Research, vol. 34, 2nd edn. Springer, Berlin (2007)zbMATHGoogle Scholar
  11. 11.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  12. 12.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  13. 13.
    Sagara, N.: Cores and Weber sets for fuzzy extensions of cooperative games. Fuzzy Sets Syst. 272, 102–114 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shapley, L.S.: Cores of convex games. Int. J. Game Theory 1, 11–26 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Weber, R.J.: Probabilistic values for games. In Roth, A. E., editor, The Shapley Value. Essays in Honor of Lloyd S. Shapley, 101–120. Cambridge University Press (1988)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Information Theory and Automation, Czech Academy of SciencesPragueCzech Republic
  2. 2.Dipartimento di Matematica “Federigo Enriques”Università degli Studi di MilanoMilanoItaly

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