Kybernetika 49 no. 3, 404-419, 2013

Simple games in Łukasiewicz calculus and their cores

Petr Cintula and Tomáš Kroupa

Abstract:

We propose a generalization of simple coalition games in the context of games with fuzzy coalitions. Mimicking the correspondence of simple games with non-constant monotone formulas of classical logic, we introduce simple Łukasiewicz games using monotone formulas of Łukasiewicz logic, one of the most prominent fuzzy logics. We study the core solution on the class of simple Łukasiewicz games and show that cores of such games are determined by finitely-many linear constraints only. The non-emptiness of core is completely characterized in terms of balanced systems and by the presence of strong veto players.

Keywords:

core, simple game, game with fuzzy coalitions, McNaughton function, Łukasiewicz logic

Classification:

91A12, 06D35

References:

  1. S. Aguzzoli: Geometric and Proof-theoretic Issues in {Ł}ukasiewicz Propositional Logics. PhD. Thesis, University of Siena 1998.   CrossRef
  2. J.-P. Aubin: Coeur et valeur des jeux flous à paiements latéraux. Comptes rendus de l'Académie des Sciences, Série A 279 (1974), 891-894.   CrossRef
  3. R. J. Aumann and L. S. Shapley: Values of Non-atomic Games. Princeton University Press, Princeton 1974.   CrossRef
  4. Y. Azrieli and E. Lehrer: On some families of cooperative fuzzy games. Internat. J. Game Theory 36 (2007), 1, 1-15.   CrossRef
  5. R. Branzei, D. Dimitrov and S. Tijs: Models in Cooperative Game Theory. Lecture Notes in Econom. and Math. Systems 556, Springer-Verlag, Berlin 2005.   CrossRef
  6. D. Butnariu and E. P. Klement: Triangular Norm Based Measures and Games with Fuzzy Coalitions. Kluwer, Dordrecht 1993.   CrossRef
  7. D. Butnariu and T. Kroupa: Shapley mappings and the cumulative value for $n$-person games with fuzzy coalitions. European J. Oper. Res. 186 (2008), 1, 288-299.   CrossRef
  8. R. L. O. Cignoli, I. M. L. D'Ottaviano and D. Mundici: Algebraic Foundations of Many-valued Reasoning. Trends in Logic - Studia Logica Library 7, Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  9. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Trends in Logic - Studia Logica Library 8, Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  10. R. McNaughton: A theorem about infinite-valued sentential logic. J. Symbolic Logic 16 (1051), 1-13.   CrossRef
  11. D. Mundici: Averaging the truth-value in Łukasiewicz logic. Studia Logica 55 (1995), 1, 113-127.   CrossRef
  12. B. Peleg and P. Sudh{ö}lter: Introduction to the Theory of Cooperative Games. Second edition. Theory and Decision Library. Series C: Game Theory, Mathematical Programming and Operations Research 34, Springer, Berlin 2007.   CrossRef
  13. B. Rie{č}an and D. Mundici: Probability on {MV}-algebras. In: Handbook of Measure Theory, Vol. I, II, North-Holland, Amsterdam 2002, pp. 869-909.   CrossRef
  14. A. Rose and J. B. Rosser: Fragments of many-valued statement calculi. Trans. AMS 87 (1958), 1-53.   CrossRef
  15. R. Webster: Convexity. Oxford Science Publications. The Clarendon Press Oxford University Press, New York 1994.   CrossRef
  16. I. Wegener: The Complexity of Boolean Functions. Wiley-Teubner Series in Computer Science. John Wiley and Sons, Chichester 1987.   CrossRef