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Keywords:
noncooperative games; Nash equilibria; differential games; globally convex structure
noncooperative games; Nash equilibria; differential games; globally convex structure
Summary:
Noncooperative games with systems governed by nonlinear differential equations remain, in general, nonconvex even if continuously extended (i. e. relaxed) in terms of Young measures. However, if the individual payoff functionals are “enough” uniformly convex and the controlled system is only “slightly” nonlinear, then the relaxed game enjoys a globally convex structure, which guarantees existence of its Nash equilibria as well as existence of approximate Nash equilibria (in a suitable sense) for the original game.
References:
[1] Balder E. J.: On a useful compactification for optimal control problems. J. Math. Anal. Appl. 72 (1979), 391–398 DOI 10.1016/0022-247X(79)90235-X | MR 0559376 | Zbl 0434.49007
[2] Balder E. J.: A general denseness result for relaxed control theory. Bull. Austral. Math. Soc. 30 (1984), 463–475 DOI 10.1017/S0004972700002185 | MR 0766804 | Zbl 0544.49008
[3] Balder E. J.: Generalized equilibrium results for games with incomplete information. Math. Oper Res. 13 (1988), 265–276 DOI 10.1287/moor.13.2.265 | MR 0942618 | Zbl 0658.90104
[4] Bensoussan A.: Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a $n$ personnes. SIAM J. Control 12 (1974), 460–499 DOI 10.1137/0312037 | MR 0384185 | Zbl 0254.90066
[5] Gabasov R., Kirillova F.: Qualitative Theory of Optimal Processes. Nauka, Moscow 1971 MR 0527204 | Zbl 0339.49002
[7] Kindler J.: Equilibrium point theorems for two-person games. SIAM J. Control Optim. 22 (1984), 671–683 DOI 10.1137/0322042 | MR 0755136 | Zbl 0545.90102
[8] Krasovskiĭ N. N., Subbotin A. I.: Game Theoretical Control Problems. Springer, Berlin 1988 MR 0918771 | Zbl 0649.90101
[9] Lenhart S., Protopopescu V., Stojanovic S.: A minimax problem for semilinear nonlocal competitive systems. Appl. Math. Optim. 28 (1993), 113–132 DOI 10.1007/BF01182976 | MR 1218250 | Zbl 0789.49009
[10] Nash J.: Non–cooperative games. Ann. of Math. 54 (1951), 286–295 DOI 10.2307/1969529 | MR 0043432 | Zbl 0045.08202
[11] Nikaidô H., Isoda K.: Note on non–cooperative equilibria. Pacific J. Math. 5 (1955), 807–815 DOI 10.2140/pjm.1955.5.807 | MR 0073910
[12] Nikol’skiĭ M. S.: On a minimax control problem. In: Optimal Control and Differential Games (L. S. Pontryagin, ed.), Proc. Steklov Inst. Math. 1990, pp. 209–214
[13] Nowak A. S.: Correlated relaxed equilibria in nonzero–sum linear differential games. J. Math. Anal. Appl. 163 (1992), 104–112 DOI 10.1016/0022-247X(92)90281-H | MR 1144709 | Zbl 0778.90102
[14] Patrone F.: Well–posedness for Nash equilibria and related topics. In: Recent Developments in Well–Posed Variational Problems (R. Lucchetti and J. Revalski, eds.), Kluwer, 1995, pp. 211–227 MR 1351746 | Zbl 0849.90131
[15] Roubíček T.: Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin 1997 MR 1458067
[16] Roubíček T.: Noncooperative games with elliptic systems. In: Proc. IFIP WG 7.2 Conf. Control of P.D.E., Chemnitz 1998, accepted MR 1723990
[17] Sainte–Beuve M.-F.: Some topological properties of vector measures with bounded variations and its applications. Ann. Mat. Pura Appl. 116 (1978), 317–379 MR 0506985
[18] Schmidt W. H.: Maximum principles for processes governed by integral equations in Banach spaces as sufficient optimality conditions. Beiträge zur Analysis 17 (1981), 85–93 MR 0663274 | Zbl 0478.49023
[19] Tan K. K., Yu J., Yuan X. Z.: Existence theorems of Nash equilibria for non–cooperative $n$–person games. Internat. J. Game Theory 24 (1995), 217–222 DOI 10.1007/BF01243152 | MR 1349753 | Zbl 0837.90126
[20] Tijs S. H.: Nash equilibria for noncooperative $n$-person game in normal form. SIAM Rev. 23 (1981), 225–237 DOI 10.1137/1023038 | MR 0618639
[21] Warga J.: Optimal Control of Differential and Functional Equations. Academic Press, New York 1972 MR 0372708 | Zbl 0253.49001
[22] Young L. C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, Classe III 30 (1937), 212–234 Zbl 0019.21901
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