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
Allouch, N. and Florenzano, M. (2004) Edgeworth and Walras equilibria of an arbitrage-free economy, Econ. Theory, vol. 23, no. 2, pp. 353–370.
Aubin, J.-P. (1979) Mathematical Methods of Game and Economic Theory, North-Holland (Studies in Mathematics and its Applications, vol. 7, 619 pages.).
Aubin, J.-P. (1981) Cooperative fuzzy games, Math. Op. Res., vol. 6, 1–13.
Aubin, J.-P. (1981) Locally Lipschitz cooperative games, J. Math. Economics, vol. 8, pp. 241–262.
Aubin, J.-P. (1981) A dynamical, pure exchange economy with feedback pricing, J. Economic Behavior and Organizations, 2, pp. 95–127.
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.
Aubin, J.-P. (1983) L’Analyse non linéaire et ses motivations économiques, Masson (English version: Optima and Equilibria, (1993, 1998), Springer-Verlag).
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.
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.
Aubin, J.-P. (1991) Viability Theory, Birkhäuser, Boston, Basel, Berlin.
Aubin, J.-P. (1997) Dynamic Economic Theory: A Viability Approach, Springer-Verlag.
Aubin, J.-P. (2001) Viability Kernels and Capture Basins of Sets under Differential inclusions, SIAM J. Control Optimization, 40, pp. 853–881.
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.
Aubin, J.-P. (2001) Regulation of the evolution of the architecture of a network by connectionist tensors operating on coalitions of actors, preprint.
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.
Aubin, J.-P. & Cellina, A. (1984) Differential Inclusions, Springer-Verlag.
Aubin, J.-P. & da Prato, G. (1995) Stochastic Nagumo’s viability theorem, Stochastic Analysis and Applications, vol. 13, pp. 1–11.
Aubin, J.-P. & da Prato, G. (1998) The viability theorem for stochastic differential inclusions, Stochastic Analysis and Applications, 16, pp. 1–15.
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.
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.
Aubin, J.-P. & Frankowska H. (1990) Set-Valued Analysis, Birkhäuser, Boston, Basel, Berlin.
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.
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.
Aubin, J.-P., Pujal, D. & Saint-Pierre, P. (2001) Dynamic management of portfolios with transaction costs under tychastic uncertainty, preprint.
Başar, T. & Bernhard, P. (1991) H ∞-optimal control and related minimax design problems. A dynamic game approach, Birkhäuser, Boston.
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.
Basile, A. (1993) Finitely additive nonatomic coalition production economies: Core-Walras equivalence, Int. Econ. Rev., 34, pp. 993–995.
Basile, A. (1994) Finitely additive correpondences, Proc. AMS 121, pp. 883–891.
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.
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.
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.
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.
Buckdahn, R., Cardaliaguet, P. & Quincampoix, M. (2000) A representation formula for the mean curvature motion, UBO 08-2000.
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.
Cardaliaguet P. (1994) Domaines dicriminants en jeux différentiels, Thèse de l’Université de Paris-Dauphine.
Cardaliaguet P. (1996) A differential game with two players and one target, SIAM J. on Control and Optimization, 34,4, pp. 1441–1460.
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.
Cardaliaguet P. (2000) Introduction à la théorie des jeux différentiels, Lecture Notes, Université Paris-Dauphine.
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.
da Prato, G. and Frankowska, H. (1994) A stochastic Filippov Theorem, Stochastic Calculus, 12, pp. 409–426.
Doss, H. (1977) Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. Henri Poincaré, Calcul des Probabilités et Statistique, 23, pp. 99–125.
Filar, J.A. & Petrosjan, L.A. (2000) Dynamic cooperative games, International Game Theory Review, 2, pp. 47–65.
Florenzano, M. (1990) Edgeworth equilibria, fuzzy core and equilibria of a production economy without ordered preferences, J. Math. Anal. Appl., 153, pp. 18–36.
Frankowska, H. (1987) L’équation d’Hamilton-Jacobi contingente, Comptes-Rendus de l’Académie des Sciences, PARIS, Série 1, 304, pp. 295–298.
Frankowska, H. (1987) Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations, IEEE, 26th, CDC Conference, Los Angeles, December 9–11.
Frankowska, H. (1989) Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations, Applied Mathematics and Optimization, 19, pp. 291–311.
Frankowska, H. (1989) Hamilton-Jacobi equation: viscosity solutions and generalized gradients, J. of Math. Analysis and Appl. 141, pp. 21–26.
Frankowska, H. (1993) Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation, SIAM J. on Control Optimization, vol. 31, no. 1, pp. 257–272.
Gautier, S. & Thibault, L. (1993) Viability for constrained stochastic differential equations, Differential Integral Equations, 6, pp. 1395–1414.
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.
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.
Isaacs, R. (1965) Differential Games, Wiley, New York.
Leitmann, G. (1980) Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, Vol.32, pp. 569–576.
Mares, M. (2001) Fuzzy cooperative games. Cooperation with vague expectations, Physica Verlag.
Mishizaki, I. & Sokawa, M. (2001) Fuzzy and multiobjective games for conflict resolution, Physica Verlag.
Peirce, C. (1893) Evolutionary love, The Monist.
Petrosjan, L.A. (2004) Dynamic cooperative games, Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control, Birkhäuser, Boston, vol.7.
Petrosjan, L.A. & Zenkevitch, N.A. (1996) Game Theory, World Scientific.
Pujal, D. (2000) Valuation et gestion dynamiques de portefeuilles, Thèse de l’Université de Paris-Dauphine.
Quincampoix, M. (1992) Differential inclusions and target problems, SIAM J. Control Optimization, 30, pp. 324–335.
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.
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.
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.
Rockafellar, R.T. and Wets, R. (1997) Variational Analysis, Springer-Verlag.
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.
Saint-Pierre, P. (1994) Approximation of the viability kernel, Applied Mathematics & Optimisation, vol. 29, pp. 187–209.
Zabczyk, J. (1996) Chance and decision: stochastic control in discrete time, Quaderni, Scuola Normale di Pisa.
Zabczyk, J. (1999) Stochastic invariance and consistency of financial models, preprint, Scuola Normale di Pisa.
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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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DOI: https://doi.org/10.1007/0-8176-4429-6_7
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