Location of Nash equilibria: A Riemannian geometrical approach

Kristály, Alexandru

doi:10.1090/S0002-9939-09-10145-4

Location of Nash equilibria: A Riemannian geometrical approach
HTML articles powered by AMS MathViewer

by Alexandru Kristály

Proc. Amer. Math. Soc. 138 (2010), 1803-1810

DOI: https://doi.org/10.1090/S0002-9939-09-10145-4

Published electronically: December 21, 2009

PDF | Request permission

Abstract:

Existence and location of Nash equilibrium points are studied for a large class of a finite family of payoff functions whose domains are not necessarily convex in the usual sense. The geometric idea is to embed these non-convex domains into suitable Riemannian manifolds regaining certain geodesic convexity properties of them. By using recent non-smooth analysis on Riemannian manifolds and a variational inequality for acyclic sets, an efficient location result of Nash equilibrium points is given. Some examples show the applicability of our results.

References

Daniel Azagra and Juan Ferrera, Proximal calculus on Riemannian manifolds, Mediterr. J. Math. 2 (2005), no. 4, 437–450. MR 2192524, DOI 10.1007/s00009-005-0056-4
Daniel Azagra, Juan Ferrera, and Fernando López-Mesas, Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220 (2005), no. 2, 304–361. MR 2119282, DOI 10.1016/j.jfa.2004.10.008
Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Shiow-Yu Chang, Maximal elements in noncompact spaces with application to equilibria, Proc. Amer. Math. Soc. 132 (2004), no. 2, 535–541. MR 2022379, DOI 10.1090/S0002-9939-03-07054-0
Pando Gr. Georgiev, Parametric Borwein-Preiss variational principle and applications, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3211–3225. MR 2161143, DOI 10.1090/S0002-9939-05-07853-6
Gábor Kassay, József Kolumbán, and Zsolt Páles, On Nash stationary points, Publ. Math. Debrecen 54 (1999), no. 3-4, 267–279. MR 1694524, DOI 10.5486/pmd.1999.1902
Yu. S. Ledyaev and Qiji J. Zhu, Nonsmooth analysis on smooth manifolds, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3687–3732. MR 2302512, DOI 10.1090/S0002-9947-07-04075-5
Władysław Kulpa and Andrzej Szymanski, Infimum principle, Proc. Amer. Math. Soc. 132 (2004), no. 1, 203–210. MR 2021263, DOI 10.1090/S0002-9939-03-06994-6
J. F. McClendon, Minimax and variational inequalities for compact spaces, Proc. Amer. Math. Soc. 89 (1983), no. 4, 717–721. MR 719003, DOI 10.1090/S0002-9939-1983-0719003-2
Jacqueline Morgan and Vincenzo Scalzo, Pseudocontinuous functions and existence of Nash equilibria, J. Math. Econom. 43 (2007), no. 2, 174–183. MR 2297122, DOI 10.1016/j.jmateco.2006.10.004
John Nash, Non-cooperative games, Ann. of Math. (2) 54 (1951), 286–295. MR 43432, DOI 10.2307/1969529
John F. Nash Jr., Equilibrium points in $n$-person games, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 48–49. MR 31701, DOI 10.1073/pnas.36.1.48
R. Nessah, K. Kerstens, Characterization of the existence of Nash equilibria with non-convex strategy sets. Document du travail LEM, 2008-19, preprint; see http://lem.cnrs.fr/Portals/2/actus/DP_200819.pdf
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
J. E. Tala and E. Marchi, Games with non-convex strategy sets, Optimization 37 (1996), no. 2, 177–181. MR 1416049, DOI 10.1080/02331939608844209
Constantin Udrişte and Constantin Udrişte, Convex functions and optimization methods on Riemannian manifolds, Mathematics and its Applications, vol. 297, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1326607, DOI 10.1007/978-94-015-8390-9
Haomiao Yu and Zhixiang Zhang, Pure strategy equilibria in games with countable actions, J. Math. Econom. 43 (2007), no. 2, 192–200. MR 2298564, DOI 10.1016/j.jmateco.2006.09.008
Abderrahmane Ziad, Pure strategy Nash equilibria of non-zero-sum two-person games: non-convex case, Econom. Lett. 62 (1999), no. 3, 307–310. MR 1684858, DOI 10.1016/S0165-1765(98)00246-8