Summary
In this paper we consider a two-stage three-players game: in the first stage one of the players chooses an optimal strategy knowing that, at the second stage, the other two players react by playing a noncooperative game which may admit more than one Nash equilibrium. We investigate continuity properties of the set-valued function defined by the Nash equilibria of the (second stage) two players game and of the marginal functions associated to the first stage optimization problem. By using suitable approximations of the mixed extension of the Nash equilibrium problem, we obtain without convexity assumption the lower semicontinuity of the set-valued function defined by the considered approximate Nash equilibria and the continuity of the associate approximate average marginal functions when the second stage corresponds to a particular class of noncooperative games called antipotential games.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Key Words
References
Altman, E., Boulogne, T., El-Azouzi, R. and Jimenez, T. (2004), A survey on networking games in telecommunications, Computers and Operation Research.
Aubin, J. P. and Frankowska, H. (1990) Set-valued Analysis, Birkhauser, Boston.
Basar, T. and Olsder, G.J. (1995) Dynamic Noncooperative Game Theory, Academic Press. New York.
Borel, E. (1953) The theory of play and integral equations with skew symmetric kernels, Econometrica vol. 21, pp. 97–100.
Breton, M., Alj, A. and Haurie, A. (1988), Sequential Stackelberg equilibria in two-person games, Journal of Optimization Theory and Applications vol. 59, pp. 71–97.
Dempe, S. (2002) Foundations of Bilevel Programming, Kluwer Academic Publishers, Dordrecht.
Dempe, S. (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization vol. 52, pp. 333–359.
Facchini, G., Van Megen, F., Borm, P. and Tijs, S. (1997), Congestion models and weighted Bayesian potential games, Theory and Decision vol. 42, pp. 193–206.
Lignola, M.B. and Morgan, J. (1992) Semi-continuities of marginal functions in a sequential setting, Optimization vol. 24, pp. 241–252.
Loridan, P. and Morgan, J. (1989) On strict ɛ-solutions for a two-level optimization problem, Proceedings of the international Conference on Operation Research 90 in Vienna, Ed. By W. Buhler, G. Feichtinger, F. Harti, F.J. Radermacher, P. Stanley, Springer Verlag, Berlin, pp. 165–172.
Loridan, P. and Morgan, J. (1996) Weak via strong Stackelberg problems: new results, Journal of Global Optimization vol. 8, pp. 263–287.
Luo, Z.-Q., Pang, J.-S., Ralph, D. (1996) Mathematical programs with equilibrium constraints, Cambridge University Press, Cambridge.
Mallozzi, L. and Morgan, J. (2001) Mixed strategies for hierarchical zero-sum games. In: Advances in dynamic games and applications (Maastricht, 1998), Annals of the International Society on Dynamic Games, Birkhauser Boston MA vol. 6, pp. 65–77.
Mallozzi, L. and Morgan, J. (2005) On equilibria for oligopolistic markets with leadership and demand curve having possible gaps and discontinuities, Journal of Optimization Theory and Applications vol. 125, n.2, pp. 393–407.
Marcotte, P. and Blain, M. (1991) A Stackelberg-Nash model for the design of deregulated transit system, Dynamic Games in Economic Analysis, Ed. by R.H. Hamalainen and H.K. Ethamo, Lecture Notes in Control and Information Sciences, Springer Verlag, Berlin, vol. 157.
Monderer, D. and Shapley, L.S. (1996) Potential games, Games and Economic Behavior vol. 14, pp. 124–143.
Morgan, J. and Raucci, R. (1999) New convergence results for Nash equilibria, Journal of Convex Analysis vol. 6, n. 2, pp. 377–385.
Morgan, J. and Raucci, R. (2002) Lower semicontinuity for approximate social Nash equilibria, International Journal of Game Theory vol. 31, pp. 499–509.
Nash, J. (1951) Non-cooperative games, Annals of Mathematics vol. 54, pp. 286–295.
Petit, M.L. and Sanna-Randaccio, F. (2000) Endogenous R & D and foreign direct investment in international oligopolies, International Journal of Industrial Organization vol. 18, pp. 339–367.
Sheraly, H.D., Soyster, A.L. and Murphy, F.H. (1983) Stackelberg-Nash-Cournot Equilibria: characterizations and computations, Operation Research vol. 31, pp. 253–276.
Tobin, R.L. (1992) Uniqueness results and algorithm for Stackelberg-Cournot-Nash equilibria, Annals of Operation Research vol. 34, pp. 21–36.
von Neumann, J. and Morgenstern, O. (1944) Theory of Games and Economic Behavior, New York Wiley.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer Science + Business Media, LLC
About this chapter
Cite this chapter
Mallozzi, L., Morgan, J. (2006). On approximate mixed Nash equilibria and average marginal functions for two-stage three-players games. In: Dempe, S., Kalashnikov, V. (eds) Optimization with Multivalued Mappings. Springer Optimization and Its Applications, vol 2. Springer, Boston, MA . https://doi.org/10.1007/0-387-34221-4_5
Download citation
DOI: https://doi.org/10.1007/0-387-34221-4_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-34220-7
Online ISBN: 978-0-387-34221-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)