Theoretical and experimental studies of noncooperative games increasingly recognize Nash equilibrium as a limiting outcome of players' repeated interaction. This note, while sharing that view, illustrates and advocates combined use of convex optimization and differential equations, the purpose being to render equilibrium both plausible and stable.

Contact details are reproduced from the original publication and may be historical.

Sjur Didrik Flåm

Dept. of Economics, University of Bergen, 5008 Bergen, Norway
and: Norwegian School of Economics and Business Administration

sjur.flaam@econ.uib.no
  1. Z. Artstein: Irregular cobweb dynamics, Economic Letters 11 (1983) 15–17.https://doi.org/10.1016/0165-1765(83)90155-6Google Scholar
  2. H. Attouch, X. Goudou, P. Redont: The heavy ball with friction method I, The continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system, Communications in Contemporary Mathematics 2 (2000) 1–34.https://doi.org/10.1142/S0219199700000025Google Scholar
  3. J. P. Aubin, A. Cellina: Differential Inclusions, Springer-Verlag, Berlin (1984).https://doi.org/10.1007/978-3-642-69512-4Google Scholar
  4. M. Benaim: A dynamical approach to stochastic approximations, SIAM Journal of Control and Optimization 34 (1996) 437–472.https://doi.org/10.1137/S0363012993253534Google Scholar
  5. M. Benaim, M. W. Hirsch: Mixed equilibria and dynamical systems arising from fictitious play in perturbed games, Games and Economic Behavior 29 (1999) 36–72.https://doi.org/10.1006/game.1999.0717Google Scholar
  6. D. Borwein, J. Borwein: Fixed point iterations for real functions, Journal of Mathematical Analysis and Applications 157 (1991) 112–126.https://doi.org/10.1016/0022-247X(91)90139-QGoogle Scholar
  7. G. W. Brown: Iterative solutions of games by fictitious play, in: Activity Analysis of Production and Allocation, T. J. Koopmans (ed.), J. Wiley, New York (1951).Google Scholar
  8. A. L. Dontschev, R. T. Rockafellar: Ample parametrization of variational inclusions, manuscript, april (2000).Google Scholar
  9. A. Cournot: Recherches sur les principes mathématiques de la théorie des richesses (1838).Google Scholar
  10. Yu. M. Ermoliev, S. P. Uryasiev: Nash equilibrium in n-person games, (in Russian) Kibernetika 3 (1982) 85–88.Google Scholar
  11. S. D. Flåm: Approaches to economic equilibrium, J. Economic Dynamics and Control 20 (1996) 1505–1522.https://doi.org/10.1016/0165-1889(96)00912-8Google Scholar
  12. S. D. Flåm, C. Horvath: Stochastic mean-values, rational expectations, and price movements, Economic Letters 61 (1998) 293–299.https://doi.org/10.1016/S0165-1765(98)00196-7Google Scholar
  13. S. D. Flåm: Averaged predictions and the learning of equilibrium play, Journal of Economic Dynamics and Control 22 (1998) 833–848.https://doi.org/10.1016/S0165-1889(97)00103-6Google Scholar
  14. S. D. Flåm: 2x2 Games, Fictitious Play and Green’s Theorem, Proceedings of the IV Catalan Days of Applied Mathematics, Tarragona (1998) 89–101.Google Scholar
  15. S. D. Flåm: Restricted attention, myopic play, and the learning of equilibrium, Annals of Operations Research 82 (1998) 473–482.https://doi.org/10.1023/A:1018987409039Google Scholar
  16. S. D. Flåm: Learning equilibrium play: a myopic approach, Computational Optimization and Appl. 14 (1999) 87–102.https://doi.org/10.1023/A:1008709129421Google Scholar
  17. S. D. Flåm, M. Sandsmark: Learning to face stochastic demand, Int. Game Theory Review 2 (2000) 259–271.https://doi.org/10.1142/S0219198900000238Google Scholar
  18. S. D. Flåm: Repeated play and Newton’s method, Int. Game Theory Review 2 (2001) 141–154.https://doi.org/10.1142/S0219198900000111Google Scholar
  19. S. D. Flåm: Approaching equilibrium in parallel, in: Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, D. Butnariu, Y. Censor, S Reich (eds.), North-Holland (2001) 267–278.https://doi.org/10.1016/S1570-579X(01)80015-XGoogle Scholar
  20. S. D. Flåm, J. Morgan: Newtonian mechanics and Nash play, manuscript (2001).Google Scholar
  21. R. L. Franks, R. P. Marzec: A theorem on mean-value iterations, Proc. Americ. Math. Soc. 30 (1971) 324–326.https://doi.org/10.1090/S0002-9939-1971-0280656-9Google Scholar
  22. D. Fudenberg, D. K. Levine: The Theory of Learning in Games, MIT Press, Cambridge Mass. (1998).Google Scholar
  23. J. Hofbauer, K. Sigmund: Evolutionary Games and Population Dynamics, Cambridge University Press (1998).https://doi.org/10.1017/CBO9781139173179Google Scholar
  24. M. A. Krasnoselski: Two observations about the method of successive approximations, Usp. Math. Nauk 10 (1955) 123–127.Google Scholar
  25. W. R. Mann: Mean value methods in iteration, Proc. Americ. Math. Soc. 4 (1953) 506–510.https://doi.org/10.1090/S0002-9939-1953-0054846-3Google Scholar
  26. D. Monderer, L. S. Shapley: Potential games, Games and Economic Behavior 14 (1996) 124–143.https://doi.org/10.1006/game.1996.0044Google Scholar
  27. D. Monderer, L. Shapley: Fictitious play property for games with identical interest, Journal of Economic Theory 68 (1996) 258–265.https://doi.org/10.1006/jeth.1996.0014Google Scholar
  28. F. H. Murphy, H. D. Sherali, A. L. Soyster: A mathematical programming approach for determining oligopolistic market equilibrium, Mathematical Programming 24 (1982) 92–106.https://doi.org/10.1007/BF01585096Google Scholar
  29. V. V. Nemytskii, V. V. Stepanov: Qualitative Theory of Differential Equations, Princeton University Press, Princeton (1960).https://doi.org/10.1515/9781400875955Google Scholar
  30. J. von Neumann: Zur Theorie der Gesellschaftsspiele, Mathematische Annalen 100 (1928) 295–320.https://doi.org/10.1007/BF01448847Google Scholar
  31. R. Pemantle: Nonconvergence to unstable points in urn models and stochastic approximations, Ann. Probab. 18 (1990) 698–712.https://doi.org/10.1214/aop/1176990853Google Scholar
  32. J. Robinson: An iterative method for solving a game, Ann. Math. 54 (1951) 296–301.https://doi.org/10.2307/1969530Google Scholar
  33. L. Samuelson: Evolutionary Games and Equilibrium Selection, MIT Press (1997).Google Scholar
  34. J. B. Rosen: Existence and uniqueness of equilibrium points for concave n-person games, Econometrica 33 (1965) 520–534.https://doi.org/10.2307/1911749Google Scholar
  35. F. Vega-Redondo: Extrapolative expectations and market stability, International Economic Review 30 (1989) 513–517.https://doi.org/10.2307/2526770Google Scholar
  36. F. Vega-Redondo: Evolution, Games and Economic Behavior, Oxford University Press (1996).https://doi.org/10.1093/0198774729.001.0001Google Scholar
  37. J. W. Weibull: Evolutionary Game Theory, The MIT Press, Cambridge, Massachusetts (1996).Google Scholar

S. D. Flåm. “Convexity, Differential Equations, and Games.” Journal of Convex Analysis 9 (2002), No. 2, 429–438. https://doi.org/10.68381/jca09025