Fictitious play in 3 × 3 games: The transition between periodic and chaotic behaviour

Games and Economic Behavior

Volumn 63, Issue 1, 2008, Pages 259-291

  Sparrow, Colin ^a   van Strien, Sebastian ^a   Harris, Christopher ^b  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b Department of Economics ^*  (United Kingdom)

Author keywords

Bifurcation; Chaos; Fictitious pay; Learning process

Indexed keywords

EID: 41649111709     PISSN: 08998256     EISSN: 10902473     Source Type: Journal    
DOI: 10.1016/j.geb.2007.08.005     Document Type: Article

References (27)

1. Aubin J.-P., and Cellina A. Differential inclusions. Set-Valued Maps and Viability Theory. Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences] vol. 264 (1984), Springer-Verlag, Berlin
2. Berger, U., 1995. Replikator- und Best Response Dynamic von 3 × 3-Bimatrixspiele. University of Vienna diploma thesis (unpublished)
3. Berger U. Fictitious play in 2 × n games. J. Econ. Theory 120 2 (2005) 139-154
4. Berger U. Two more classes of games with the continuous-time fictitious play property. Games Econ. Behav. 60 2 (2007) 247-261
5. Brown, G.W., 1949. Some notes on computation of games solutions. The Rand Corporation, P-78, April
6. Brown G.W. Iterative solution of games by fictitious play. Activity Analysis of Production and Allocation. Cowles Commission Monogr. vol. 13 (1951), Wiley, New York, NY 374-376
7. Brown G.W. Notes on the solution of linear systems involving inequalities. Proceedings of a Second Symposium on Large-Scale Digital Calculating Machinery, 1949 (1951), Harvard Univ. Press, Cambridge, MA 137-140
8. Cowan, S., 1992. Dynamical systems arising from game theory. PhD thesis, University of California at Berkeley
9. Ellison G., and Fudenberg D. Learning purified mixed equilibria. J. Econ. Theory 90 1 (2000) 84-115
10. Fudenberg D., and Levine D.K. Consistency and cautious fictitious play. J. Econ. Dynam. Control 19 5-7 (1995) 1065-1089
11. Fudenberg D., and Levine D.K. The Theory of Learning in Games. MIT Press Ser. Econ. Learn. Soc. Evol. vol. 2 (1998), MIT Press, Cambridge, MA
12. Gaunersdorfer A., and Hofbauer J. Fictitious play, Shapley polygons, and the replicator equation. Evolutionary Game Theory in Biology and Economics. Games Econ. Behav. 11 2 (1995) 279-303
13. Hahn S. The convergence of fictitious play in 3 × 3 games with strategic complementarities. Econ. Lett. 64 1 (1999) 57-60
14. Harris C. On the rate of convergence of continuous-time fictitious play. Games Econ. Behav. 22 2 (1998) 238-259
15. Hofbauer, J., 1995. Stability for the best response dynamics. Mimeo, University of Vienna
16. Jordan J.S. Three problems in learning mixed-strategy Nash equilibria. Games Econ. Behav. 5 3 (1993) 368-386
17. Krishna, V., 1992. Learning in games with strategic complementarities. Mimeo, Harvard University
18. Krishna V., and Sjöström T. On the convergence of fictitious play. Math. Operations Res. 23 2 (1998) 479-511
19. Metrick A., and Polak B. Fictitious play in 2 × 2 games: A geometric proof of convergence. Econ. Theory 4 6 (1994) 923-933 Bounded rationality and learning
20. Milgrom P., and Roberts J. Adaptive and sophisticated learning in normal form games. Games Econ. Behav. 3 1 (1991) 82-100
21. Miyasawa, K., 1961. On the convergence of the learning process in a 2 × 2 non-zero-sum two-person game. Research memorandum No. 33. Economic Research Program, Princeton University
22. Monderer D., and Shapley L.S. Fictitious play property for games with identical interests. J. Econ. Theory 68 1 (1996) 258-265
23. Robinson J. An iterative method of solving a game. Ann. Math. (2) 54 (1951) 296-301
24. Rosenmüller J. Über Periodizitätseigenschften spieltheoretischer Lernprozesse. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971) 259-308
25. Sela A. Fictitious play in 2 × 3 games. Games Econ. Behav. 31 1 (2000) 152-162
26. Shapley L.S. Some topics in two-person games. Advances in Game Theory (1964), Princeton Univ. Press, Princeton, NJ 1-28
27. Sparrow, C., van Strien, S., 2007. In preparation