Abstract
This contribution is in the framework of non-cooperative games with focus on the main solution concept: the Nash equilibrium (NE). The properties of Tikhonov well-posedness for Nash equilibria (briefly T-wp for NE) will be analized with a particular attention to its generalization: Tikhonov well-posedness in value (T w-wp) which is an ordinal property. Metric characterizations of T w-wp will be discussed and known results, which give existence and uniqueness of NE in oligopoly model, will be proved to guarantee T-wp too.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M. Bertero, P. Boccacci, Introduction to inverse problems in imaging, Institute of Physics Publishing (1998).
E. Cavazzuti, Cobwebs and something else, in Decision Processes in Economics (G. Ricci ed.), Proc. Conf. Modena/Italy 1989, Springer. Berlin (1990), 34–43.
E. Cavazzuti, J. Morgan, Well-Posed Saddle Point Problems, in Optimization, theory and algorithms (J. B. Hirriart-Urruty, W. Oettli and J. Stoer eds.), Proc. Conf. Confolant/France 1981, (1983), 61–76.
G. Costa. P.A. Mori, Introduzione alla Teoria dei Giochi, Il Mulino, 1994.
A. Cournot, Mathematical principles of the theory of wealth, english traslation by N. O. Bacon, New York: The Macmillan Company, 1927.
A. Dontchev, T. Zolezzi, Well-Posed Optimization Problems, Springer, Berlin, 1993.
M. Furi, A. Vignoli, About well-posed optimization problems for functionals in metric spaces, J. Optimization Theory Appl. 5 (1970), 225–229.
D. Fudenberg, J. Tirole, Game Theory, MIT Press, Cambridge (Massachusetts), 1991.
D. Gabay, H. Moulin, On the uniqueness and stability of Nashequilibria in non-cooperative games, Applied Stochastic Control in Econometrics and Management Science (A. Bensoussan, et al..Eds), New York: North Holland.
R. Gibbons, A Primer in Game Theory, Il Mulino, Bologna, Hemel Hempstead etc. Prentice Hall International, 1994.
J.C. Harsanyi, Games with incomplete information played by Bayesian players, Management Science 14 (1967–1968).
S. Karamardian, The nonlinear complementarity problem with applications, J. Opt. Th. Appl. 4 (1969), 87–98 and 167–181.
R. Lucchetti, F. Patrone. A characterization of Tyhonov wellposedness for minimum problems, with applications to variational inequalities, Numer. Funct. Analysis Optimiz. 3 (1981), 461–476.
R. Lucchetti, F. Patrone, Some properties of “well-posed” variational inequalities governed by linear operators, Numer. Funct. Analysis Optimiz. 5 (1982–1983), 349–361.
R. Lucchetti, J. Revalski (eds.), Recent Developments in Well-Posed Variational Problems, Kluwer, Dordrecht, 1995.
M. Margiocco, F. Patrone, L. Pusillo Chicco, A New Approach to Tikhonov Well-Posedness for Nash Equilibria, Optimization 40 (1997), 385–400.
M. Margiocco, F. Patrone, L. Pusillo Chicco, Metric characterizations of Tikhonov well-posedness in value, J. Opt. Theory Appl. 100,n. 2, (1999), 377–387.
R.B. Myerson, Game Theory: Analysis of Conflict, Harvard University Press, Cambridge (MA), 1991.
J. von Neumann, O. Morgenstern, Theory of Games and economic behavior, Princeton University Press, 1944.
M. Osborne, A. Rubistein, Bargaining and markets, Academic Press, London, 1990.
F. Patrone, Well-Posedness as an Ordinal Property, Rivista di Matematica pura ed applicata 1 (1987), 95–104.
F. Patrone F, Well-posed minimum problems for preorders, Rend. Sem. Mat. Univ. Padova 84 (1990), 109–121.
F. Patrone, L. Pusillo Chicco, Antagonism for two-person games: taxonomy and applications to Tikhonov well-posedness, preprint.
J.P. Revalski, Variational inequalities with unique solution, in Mathematics and Education in Mathematics, Proc. 14th Spring Confer. of the Union of Bulgarian Mathematicians, Sofia, 1985.
J.B. Rosen, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica 33 (1965), 520–534.
R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Int. J. Game Theory 4 (1975), 25–55.
F. Szidarovszky, S. Yakowitz, A new proof of the existence and uniqueness of the Cournot equilibrium, Int. Economic Rev. 18 (1977), 787–789.
A.N. Tikhonov, On the stability of the functional optimization problem, USSR J. Comp. Math. Math. Phys. 6(1966), 631–634.
A. Watts, On the uniqueness of equilibrium in Cournot oligopoly and other games, Games and Economic Behavior 13 (1996), 269–285.
R. Wilson, Computing equilibria of n-person games, SIAM J. Appl. Math. 21 (1971), 80–87.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Chicco, L.P. (2001). Approximate Solutions and Tikhonov Well-Posedness for Nash Equilibria. In: Giannessi, F., Maugeri, A., Pardalos, P.M. (eds) Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Nonconvex Optimization and Its Applications, vol 58. Springer, Boston, MA. https://doi.org/10.1007/0-306-48026-3_15
Download citation
DOI: https://doi.org/10.1007/0-306-48026-3_15
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4020-0161-1
Online ISBN: 978-0-306-48026-3
eBook Packages: Springer Book Archive