Advertisement

International Journal of Game Theory

, Volume 46, Issue 4, pp 891–918 | Cite as

The intermediate set and limiting superdifferential for coalitional games: between the core and the Weber set

  • Lukáš Adam
  • Tomáš Kroupa
Original Paper

Abstract

We introduce the intermediate set as an interpolating solution concept between the core and the Weber set of a coalitional game. The new solution is defined as the limiting superdifferential of the Lovász extension and thus it completes the hierarchy of variational objects used to represent the core (Fréchet superdifferential) and the Weber set (Clarke superdifferential). It is shown that the intermediate set is a non-convex solution containing the Pareto optimal payoff vectors that depend on some chain of coalitions and marginal coalitional contributions with respect to the chain. A detailed comparison between the intermediate set and other set-valued solutions is provided. We compute the exact form of intermediate set for all games and provide its simplified characterization for the simple games and the glove game.

Keywords

Coalitional game Limiting superdifferential Intermediate set Core Weber set 

Mathematics Subject Classification

91A12 49J52 

References

  1. Adam L, Červinka M, Pištěk M (2016) Normally admissible partitions and calculation of normal cones to a finite union of polyhedral sets. Set-Val Var Anal 24(2):207–229CrossRefGoogle Scholar
  2. Aubin J-P (1974) Coeur et valeur des jeux flous à paiements latéraux. Comptes Rendus de l’Académie des Sciences. Série A 279:891–894Google Scholar
  3. Aumann R, Dreze J (1974) Cooperative games with coalition structures. Int J Game Theory 3(4):217–237CrossRefGoogle Scholar
  4. Branzei R, Dimitrov D, Tijs S (2005) Models in cooperative game theory, lecture notes in economics and mathematical systems, vol 556. Springer, BerlinGoogle Scholar
  5. Danilov VI, Koshevoy GA (2000) Cores of cooperative games, superdifferentials of functions, and the Minkowski difference of sets. J Math Anal Appl 247:1–14CrossRefGoogle Scholar
  6. Derks J, Haller H, Peters H (2000) The selectope for cooperative games. Int J Game Theory 29:23–38CrossRefGoogle Scholar
  7. Gerard-Varet L, Zamir S (1987) Remarks on the reasonable set of outcomes in a general coalition function form game. Int J Game Theory 16(2):123–143CrossRefGoogle Scholar
  8. Gilles RP (2010) The cooperative game theory of networks and hierarchies, vol 44. Springer Science & Business Media, BerlinGoogle Scholar
  9. Gilles RP, Owen G, van den Brink R (1992) Games with permission structures: the conjunctive approach. Int J Game Theory 20(3):277–293CrossRefGoogle Scholar
  10. Henrion R, Outrata J (2008) On calculating the normal cone to a finite union of convex polyhedra. Optimization 57(1):57–78CrossRefGoogle Scholar
  11. Ichiishi T (1981) Super-modularity: applications to convex games and to the greedy algorithm for LP. J Econ Theory 25(2):283–286CrossRefGoogle Scholar
  12. Lovász L (1983) Submodular functions and convexity. In: Bachem A, Korte B, Grötschel M (eds) Mathematical programming: the state of the art. Springer, Berlin, pp 235–257CrossRefGoogle Scholar
  13. Mordukhovich BS (2006) Variational analysis and generalized differentiation I. Springer, BerlinGoogle Scholar
  14. Owen G (1995) Game theory, 3rd edn. Academic Press Inc., San DiegoGoogle Scholar
  15. Peleg B, Sudhölter P (2007) 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, BerlinGoogle Scholar
  16. Rockafellar RT (1970) Convex analysis. Princeton University Press, PrincetonCrossRefGoogle Scholar
  17. Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, BerlinCrossRefGoogle Scholar
  18. Sagara N (2015) Cores and Weber sets for fuzzy extensions of cooperative games. Fuzzy Sets Syst 272:102–114CrossRefGoogle Scholar
  19. Selten R (1972) Equal share analysis of characteristic function experiments. In: Sauermann H (ed) Contributions to experimentation in economics. J.C.B. Mohr (Paul Siebeck), Tubingen, pp 130–165Google Scholar
  20. Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26CrossRefGoogle Scholar
  21. Shapley L, Shubik M (1971) The assignment game I: the core. Int J Game Theory 1(1):111–130CrossRefGoogle Scholar
  22. Studený M, Kroupa T (2016) Core-based criterion for extreme supermodular games. Discrete Appl Math 206:122–151CrossRefGoogle Scholar
  23. Tijs S, Lipperts FAS (1982) The hypercube and the core cover of \(n\)-person cooperative games. Cahiers du Centre d‘Études de Researche Opérationelle 24:27–37Google Scholar
  24. Weber RJ (1988) Probabilistic values for games. In: Roth AE (ed) The Shapley value. Essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 101–120CrossRefGoogle Scholar
  25. Ziegler G (1995) Lectures on polytopes, graduate texts in mathematics, vol 152. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Information Theory and AutomationCzech Academy of SciencesPragueCzech Republic
  2. 2.Dipartimento di Matematica “Federigo Enriques”Università degli Studi di MilanoMilanItaly

Personalised recommendations