Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T12:13:42.595Z Has data issue: false hasContentIssue false
  • English
  • Français

Algorithme de fictitious play et cycles

Published online by Cambridge University Press:  17 August 2016

Richard Baron ,
Jacques Durieu  and
Philippe Solal
Richard Baron
Affiliation:
Université de Saint-Étienne
Jacques Durieu
Affiliation:
Université de Saint-Étienne
Philippe Solal
Affiliation:
Université de Saint-Étienne
Get access

Résumé

Fudenberg et Kreps (1993), Young (1993), et Sela et Herreiner (1999) ont souligné l'insuffisance du critère de convergence en croyances du processus de Fictitious Play dans un cadre d'apprentissage des équilibres de Nash. En conséquence, nous choisissons d'étudier la convergence en straté gies du processus de Fictitious Play dans des jeux de coordination 2x2. Notre propos est de montrer que la convergence en stratégies de ce processus dépend de manière cruciale de la forme des croyances initiales des joueurs. Premièrement, lorsque les croyances initiales forment un profil de stratégies pures, nous établissons que la convergence en stratégies est certaine pour n'importe quelle catégorie de jeux de coordination. Deuxièmement, si les croyances initiales forment un profil de stratégies mixtes, le processus de Fictitious Play converge pour certaines catégories de jeux de coordination. Ainsi, nous caractérisons complètement les conditions assurant la convergence en stratégies.

Summary

Summary

Fudenberg and Kreps (1993), Young (1993), and Sela and Herreiner (1999) suggest that the convergence in beliefs of the Fictitious Play process is not an appopriate notion of what it means to learn an equilibrium. Consequently, we consider the convergence in strategies in two Fictitious Player coordination games. We distinguish two cases according to the initial beliefs of the players. First, if the initial beliefs of the players consist of a pair of pure strategies, we show that the Fictitious Play process converges in strategies in every coordination game. Second, if the initial beliefs of the players consist of a pair of completely mixed strategies, we show that the Fictitious Play process converges in strategies in some categories of coordination games. In this way, we give a complete characterization of the conditions which ensure the convergence in strategies of the Fictitious Play process.

Keywords

ApprentissageCroyancesEquilibre de NashFictitious PlayC72D83BeliefsFictitious PlayLearningNash equilibriumC72D83
Type
Research Article
Information
Recherches Économiques de Louvain/ Louvain Economic Review , Volume 69 , Issue 2 , 2003 , pp. 167 - 180
Copyright
Copyright © Université catholique de Louvain, Institut de recherches économiques et sociales 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Les auteurs remercient Fabien Feschet pour sa suggestion concernant la preuve de la proposition 1 ainsi qu'un rapporteur anonyme pour ses commentaires.

Universitéde Saint-Étienne. CREUSET. 6, rue basse des rives. 42100 Saint-Étienne, France, e-mail: baron@univ-st-etienne.fr. e-mail: durieu@univ-st-etienne.fr. e-mail: solal@univ-st-etienne.fr.

References

Brown, G. (1951), “Iterative Solutions of Fictitious Play”, in Koopmans, T.C. (ed.), Activity Analysis of Production and Allocation, Wiley.Google Scholar
Foster, D. et Young, P. (1998), “On the Nonconvergence of Fictitious Play in Coordination Games”, Games and Economic Behavior, 25, pp.79–91.Google Scholar
Fudenberg, D. et Kreps, D. (1993), “Learning Mixed Equilibria”, Games and Economic Behavior, 5, pp.305–367.Google Scholar
Jordan, J. (1993), “Three Problems in Learning Mixed Strategy Equilibria”, Games and Economic Behavior, 5, pp.368–386.Google Scholar
Milgrom, P. et Roberts, J. (1991), “Adaptive and Sophisticated Learning in Normal Form Games”, Games and Economic Behavior, 3, pp.82–100.Google Scholar
Miyasawa, K. (1961), “On the convergence of the learning process in a 2 ᙮ 2 non-zero-sum two person game”, Research memorandum, 33, Princeton University.Google Scholar
Monderer, D. et Sela, A. (1997), “Fictitious Play and No-cycling Conditions”,Mimeo, University of Mannheim.Google Scholar
Monderer, D. et Sela, A. (1996), “A 2 ᙮2 Game without the Fictitious Play Property”, Games and Economic Behavior, 14, pp.144–148.Google Scholar
Monderer, D. et Shapley, L. (1996), “Fictitious Play Property for Games with Identical Interests” Journal of Economic Theory, 68, pp.258–265.Google Scholar
Robinson, J. (1951), “An Iterative Method of Solving a Game”, Annals of Mathematics, 54, pp.296–301.Google Scholar
Sela, A. et Herreiner, D. (1999), “Fictitious Play in Coordination Games”, International Journal of Game Theory, 28, pp.189–198.Google Scholar
Shapley, L. (1964), “Some Topics in Two-person Games”, in Dresher, M., Shapley, L., Tucker, A. (eds.), Advances in Game Theory, Princeton University Press, Princeton.Google Scholar
Sobel, J. (2000), “Economists’ Models of Learning”, Journal of Economic Theory, 94, pp.241–262.Google Scholar
Young, P. (1993), “The Evolution of Conventions”, Econometrica, 61, pp.57–84.Google Scholar
Walliser, B. (1998), “A Spectrum of Equilibrium Processes in Games” Journal of Evolutionary Economics, 8, pp.67–87.Google Scholar