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Advances in Dynamic Games pp 129–162Cite as

Dynamic Core of Fuzzy Dynamical Cooperative Games

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Part of the book series: Annals of the International Society of Dynamic Games ((AISDG,volume 7))

Abstract

We use in this paper the viability/capturability approach for studying the problem of characterizing the dynamic core of a dynamic cooperative game defined in a characteristic function form. In order to allow coalitions to evolve, we embed them in the set of fuzzy coalitions. Hence, we define the dynamic core as a set-valued map associating with each fuzzy coalition and each time the set of allotments is such that their payoffs at that time to the fuzzy coalition are larger than or equal to the one assigned by the characteristic function of the game. We shall characterize this core through the (generalized) derivatives of a valuation function associated with the game. We shall provide its explicit formula, characterize its epigraph as a viable-capture basin of the epigraph of the characteristic function of the fuzzy dynamical cooperative game, use the tangential properties of such basins for proving that the valuation function is a solution to a Hamilton-Jacobi-Isaacs partial differential equation and use this function and its derivatives for characterizing the dynamic core.

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References

  1. Allouch, N. and Florenzano, M. (2004) Edgeworth and Walras equilibria of an arbitrage-free economy, Econ. Theory, vol. 23, no. 2, pp. 353–370.

    Article  MATH  MathSciNet  Google Scholar 

  2. Aubin, J.-P. (1979) Mathematical Methods of Game and Economic Theory, North-Holland (Studies in Mathematics and its Applications, vol. 7, 619 pages.).

    Google Scholar 

  3. Aubin, J.-P. (1981) Cooperative fuzzy games, Math. Op. Res., vol. 6, 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  4. Aubin, J.-P. (1981) Locally Lipschitz cooperative games, J. Math. Economics, vol. 8, pp. 241–262.

    Article  MATH  MathSciNet  Google Scholar 

  5. Aubin, J.-P. (1981) A dynamical, pure exchange economy with feedback pricing, J. Economic Behavior and Organizations, 2, pp. 95–127.

    Article  Google Scholar 

  6. Aubin, J.-P. (1981) Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Advances in Mathematics, Supplementary Studies, Ed. Nachbin L., pp. 160–232.

    Google Scholar 

  7. Aubin, J.-P. (1983) L’Analyse non linéaire et ses motivations économiques, Masson (English version: Optima and Equilibria, (1993, 1998), Springer-Verlag).

    Google Scholar 

  8. Aubin, J.-P. (1986) A viability approach to Lyapunov’s second methods, In: Dynamical Systems, Eds. A. Kurzhanski and K. Sigmund, Lectures Notes in Economics and Math. Systems, Springer-Verlag, 287, 31–38.

    Google Scholar 

  9. Aubin, J.-P. (1987) Smooth and heavy solutions to control problems, In: Nonlinear and Convex Analysis, Eds. B-L. Lin & Simons S., Proceedings in honor of Ky Fan, Lecture Notes in Pure and Applied Mathematics, June 24–26, 1985.

    Google Scholar 

  10. Aubin, J.-P. (1991) Viability Theory, Birkhäuser, Boston, Basel, Berlin.

    MATH  Google Scholar 

  11. Aubin, J.-P. (1997) Dynamic Economic Theory: A Viability Approach, Springer-Verlag.

    Google Scholar 

  12. Aubin, J.-P. (2001) Viability Kernels and Capture Basins of Sets under Differential inclusions, SIAM J. Control Optimization, 40, pp. 853–881.

    Article  MATH  MathSciNet  Google Scholar 

  13. Aubin, J.-P. (2002) Boundary-value problems for systems of Hamilton-Jacobi-Bellman Inclusions with Constraints, SIAM J. Control Optimization, vol. 41, no. 2, pp. 425–456.

    Article  MATH  MathSciNet  Google Scholar 

  14. Aubin, J.-P. (2001) Regulation of the evolution of the architecture of a network by connectionist tensors operating on coalitions of actors, preprint.

    Google Scholar 

  15. Aubin, J.-P. & Catté, F. (2002) Bilateral fixed-points and algebraic properties of viability kernels and capture basins of sets, Set-Valued Analysis, vol. 10, no. 4, pp. 379–416.

    Article  MATH  MathSciNet  Google Scholar 

  16. Aubin, J.-P. & Cellina, A. (1984) Differential Inclusions, Springer-Verlag.

    Google Scholar 

  17. Aubin, J.-P. & da Prato, G. (1995) Stochastic Nagumo’s viability theorem, Stochastic Analysis and Applications, vol. 13, pp. 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  18. Aubin, J.-P. & da Prato, G. (1998) The viability theorem for stochastic differential inclusions, Stochastic Analysis and Applications, 16, pp. 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  19. Aubin, J.-P., da Prato, G. & Frankowska, H. (2000) Stochastic invariance for differential inclusions, Set-Valued Analysis vol. 8, No. 1–2, pp. 181–201.

    Article  MATH  MathSciNet  Google Scholar 

  20. Aubin, J.-P. & Dordan, O. (1996) Fuzzy systems, viability theory and toll sets, In Handbook of Fuzzy Systems, Modeling and Control, Hung Nguyen Ed.. Kluwer, pp. 461–488.

    Google Scholar 

  21. Aubin, J.-P. & Frankowska H. (1990) Set-Valued Analysis, Birkhäuser, Boston, Basel, Berlin.

    MATH  Google Scholar 

  22. Aubin, J.-P. & Frankowska, H. (1996) The viability kernel algorithm for computing value functions of infinite horizon optimal control problems, J.Math. Anal. Appl., 201, pp. 555–576.

    Article  MATH  MathSciNet  Google Scholar 

  23. Aubin, J.-P., Louis-Guerin, C. & Zavalloni, M. (1979) Comptabilité entre conduites sociales réelles dans les groupes et les représentations symboliques de ces groupes: un essai de formalisation mathématique, Math. Sci. Hum., 68, pp. 27–61.

    MATH  MathSciNet  Google Scholar 

  24. Aubin, J.-P., Pujal, D. & Saint-Pierre, P. (2001) Dynamic management of portfolios with transaction costs under tychastic uncertainty, preprint.

    Google Scholar 

  25. Başar, T. & Bernhard, P. (1991) H ∞-optimal control and related minimax design problems. A dynamic game approach, Birkhäuser, Boston.

    Google Scholar 

  26. Basile A., de Simone, A. & Graziano, M.G. (1996) On the Aubin-like characterization of competitive equilibria in infinite-dimensional economies, Rivista di Matematica per le Scienze Economiche e Sociali, 19, pp. 187–213.

    Article  MATH  MathSciNet  Google Scholar 

  27. Basile, A. (1993) Finitely additive nonatomic coalition production economies: Core-Walras equivalence, Int. Econ. Rev., 34, pp. 993–995.

    Article  MathSciNet  Google Scholar 

  28. Basile, A. (1994) Finitely additive correpondences, Proc. AMS 121, pp. 883–891.

    Article  MATH  MathSciNet  Google Scholar 

  29. Basile, A. (1998) On the ranges of additive correspondences, In: Functional analysis and economic theory. Based on the special session of the conference on nonlinear analysis and its applications in engineering and economics, Samos, Greece, July 1996, dedicated to Charalambos Aliprantis on the occasion of his 50th birthday. (ed.) Abramovich, Yuri et al., Springer, Berlin, pp. 47–60.

    Google Scholar 

  30. Buckdahn, R., Quincampoix, M. & Rascanu, A. (1997) Propriétés de viabilité pour des équations différentielles stochastiques rétrogrades et applications à des équations aux dérivées partielles, Comptes-Rendus de l’Académie des Sciences, Paris, 235, pp. 1159–1162.

    Article  MathSciNet  Google Scholar 

  31. Buckdahn, R., Quincampoix, M. & Rascanu, A. (1998) Stochastic viability for backward stochastic differential equations and applications to partial differential equations, Un. Bretagne Occidentale, 01-1998.

    Google Scholar 

  32. Buckdahn, R., Peng, S., Quincampoix, M. & Rainer, C. (1998) Existence of stochastic control under state constraints, Comptes-Rendus de l’Académie des Sciences, Paris, 327, pp. 17–22.

    Article  MATH  MathSciNet  Google Scholar 

  33. Buckdahn, R., Cardaliaguet, P. & Quincampoix, M. (2000) A representation formula for the mean curvature motion, UBO 08-2000.

    Google Scholar 

  34. Cardaliaguet P., Quincampoix M. & Saint-Pierre, P. (1995) Contribution à l’étude des jeux différentiels quantitatifs et qualitatifs avec contrainte sur l’état, Comptes-Rendus de l’Académie des Sciences, Paris, 321, pp. 1543–1548.

    MATH  MathSciNet  Google Scholar 

  35. Cardaliaguet P. (1994) Domaines dicriminants en jeux différentiels, Thèse de l’Université de Paris-Dauphine.

    Google Scholar 

  36. Cardaliaguet P. (1996) A differential game with two players and one target, SIAM J. on Control and Optimization, 34,4, pp. 1441–1460.

    Article  MATH  MathSciNet  Google Scholar 

  37. Cardaliaguet P. (1997) On the regularity of semi-permeable surfaces in control theory with application to the optimal exit-time problem (Part II), SIAM J. on Control Optimization, vol. 35, no. 5, pp. 1638–1652.

    Article  MATH  MathSciNet  Google Scholar 

  38. Cardaliaguet P. (2000) Introduction à la théorie des jeux différentiels, Lecture Notes, Université Paris-Dauphine.

    Google Scholar 

  39. Cardaliaguet, P., Quincampoix, M. & Saint-Pierre, P. (1999) Set-valued numerical methods for optimal control and differential games, In Stochastic and differential games. Theory and numerical methods, Annals of the International Society of Dynamical Games, pp. 177–247, Birkhäuser.

    Google Scholar 

  40. da Prato, G. and Frankowska, H. (1994) A stochastic Filippov Theorem, Stochastic Calculus, 12, pp. 409–426.

    MATH  Google Scholar 

  41. Doss, H. (1977) Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. Henri Poincaré, Calcul des Probabilités et Statistique, 23, pp. 99–125.

    MathSciNet  Google Scholar 

  42. Filar, J.A. & Petrosjan, L.A. (2000) Dynamic cooperative games, International Game Theory Review, 2, pp. 47–65.

    MATH  MathSciNet  Google Scholar 

  43. Florenzano, M. (1990) Edgeworth equilibria, fuzzy core and equilibria of a production economy without ordered preferences, J. Math. Anal. Appl., 153, pp. 18–36.

    Article  MATH  MathSciNet  Google Scholar 

  44. Frankowska, H. (1987) L’équation d’Hamilton-Jacobi contingente, Comptes-Rendus de l’Académie des Sciences, PARIS, Série 1, 304, pp. 295–298.

    MATH  MathSciNet  Google Scholar 

  45. Frankowska, H. (1987) Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations, IEEE, 26th, CDC Conference, Los Angeles, December 9–11.

    Google Scholar 

  46. Frankowska, H. (1989) Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations, Applied Mathematics and Optimization, 19, pp. 291–311.

    Article  MATH  MathSciNet  Google Scholar 

  47. Frankowska, H. (1989) Hamilton-Jacobi equation: viscosity solutions and generalized gradients, J. of Math. Analysis and Appl. 141, pp. 21–26.

    Article  MATH  MathSciNet  Google Scholar 

  48. Frankowska, H. (1993) Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation, SIAM J. on Control Optimization, vol. 31, no. 1, pp. 257–272.

    Article  MATH  MathSciNet  Google Scholar 

  49. Gautier, S. & Thibault, L. (1993) Viability for constrained stochastic differential equations, Differential Integral Equations, 6, pp. 1395–1414.

    MATH  MathSciNet  Google Scholar 

  50. Haurie, A. (1975) On some properties of the characteristic function and core of multistage game of coalitions, IEEE Trans. Automatic Control, vol. 20, pp. 238–241.

    Article  MATH  MathSciNet  Google Scholar 

  51. Gaitsgory, V.V. & Leizarowitz, A. (1999) Limit occoputional measures set for a control system and averaging of singularly perturbed control sysytems, J. Math. Anal, Appl., 233, pp. 461–475.

    Article  MATH  MathSciNet  Google Scholar 

  52. Isaacs, R. (1965) Differential Games, Wiley, New York.

    MATH  Google Scholar 

  53. Leitmann, G. (1980) Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, Vol.32, pp. 569–576.

    Article  MATH  MathSciNet  Google Scholar 

  54. Mares, M. (2001) Fuzzy cooperative games. Cooperation with vague expectations, Physica Verlag.

    Google Scholar 

  55. Mishizaki, I. & Sokawa, M. (2001) Fuzzy and multiobjective games for conflict resolution, Physica Verlag.

    Google Scholar 

  56. Peirce, C. (1893) Evolutionary love, The Monist.

    Google Scholar 

  57. Petrosjan, L.A. (2004) Dynamic cooperative games, Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control, Birkhäuser, Boston, vol.7.

    Google Scholar 

  58. Petrosjan, L.A. & Zenkevitch, N.A. (1996) Game Theory, World Scientific.

    Google Scholar 

  59. Pujal, D. (2000) Valuation et gestion dynamiques de portefeuilles, Thèse de l’Université de Paris-Dauphine.

    Google Scholar 

  60. Quincampoix, M. (1992) Differential inclusions and target problems, SIAM J. Control Optimization, 30, pp. 324–335.

    Article  MATH  MathSciNet  Google Scholar 

  61. Quincampoix, M. (1992) Enveloppes d’invariance pour des inclusions différentielles Lipschitziennes: applications aux problèmes de cibles, Comptes-Rendus de l’Académie des Sciences, Paris, 314, pp. 343–347.

    MATH  MathSciNet  Google Scholar 

  62. Quincampoix, M. (1990) Frontières de domaines d’invariance et de viabilité pour des inclusions différentielles avec contraintes, Comptes-Rendus de l’Académie des Sciences, Paris, 311, pp. 411–416.

    MATH  MathSciNet  Google Scholar 

  63. Quincampoix, M. & Saint-Pierre P. (1995) An algorithm for viability kernels in Hölderian case: Approximation by discrete viability kernels, J. Math. Syst. Estimation and Control, pp. 115–120.

    Google Scholar 

  64. Rockafellar, R.T. and Wets, R. (1997) Variational Analysis, Springer-Verlag.

    Google Scholar 

  65. Runggaldier, W.J. (2000) Adaptive and robust control peocedures for risk minimization under uncertainty, In: Optimal Control and Partial Differential Equations, pp. 511–520, IOS Press.

    Google Scholar 

  66. Saint-Pierre, P. (1994) Approximation of the viability kernel, Applied Mathematics & Optimisation, vol. 29, pp. 187–209.

    Article  MATH  MathSciNet  Google Scholar 

  67. Zabczyk, J. (1996) Chance and decision: stochastic control in discrete time, Quaderni, Scuola Normale di Pisa.

    Google Scholar 

  68. Zabczyk, J. (1999) Stochastic invariance and consistency of financial models, preprint, Scuola Normale di Pisa.

    Google Scholar 

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Authors and Affiliations

  1. Place du Maréchal, De Lattre de Tessigny, Université Paris Dauphine, 75775, Paris Cedex 16, France

    Jean-Pierre Aubin

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  1. Jean-Pierre Aubin
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Editors and Affiliations

  1. Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wypiańskiego 27, 50-370, Wrocław, Poland

    Andrzej S. Nowak  & Krzysztof Szajowski  & 

  2. Faculty of Mathematics, Computer Science, and Econometrics, University of Zielona Góra, Podgórna 50, 65-246, Zielona Góra, Poland

    Andrzej S. Nowak

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Aubin, JP. (2005). Dynamic Core of Fuzzy Dynamical Cooperative Games. In: Nowak, A.S., Szajowski, K. (eds) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol 7. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4429-6_7

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