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Perturbations of Set-Valued Dynamical Systems, with Applications to Game Theory

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Abstract

We present upper-semicontinuity results for attractors and the chain-recurrent set of differential inclusions, in particular w.r.t. discretizations, and applications to game dynamics.

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Notes

  1. Consider a differential inclusion \(\dot{x} \in F(x)\) on a compact domain A∈ℝn, where F is a correspondence with convex values in ℝn and compact graph. Let C be a ‘regular’ compact subset of A for F namely: (1) there exists a basis of neighborhoods which deformation retracts on C, i.e. C is a deformation retract, (2) its Euler characteristic χ(C) exists, (3) there exists a neighborhood V(C) of C such that 0∉F(a) for all a∈V(C)∖C, which is moreover asymptotically stable for the dynamics induced by F, then its Euler characteristic coincides with the index Ind(F,C). This is in particular the case when F is semi-algebraic and C is a (uniformly) attracting component of its set of zeroes. This extends a previous result by Demichelis and Ritzberger [13] dealing with set of zeroes of differential equations.

References

  1. Aubin J-P, Cellina A (1984) Differential inclusions. Springer, Berlin

    Book  MATH  Google Scholar 

  2. Basu K, Weibull J (1991) Strategy subsets closed under rational behavior. Econ Lett 36:141–146

    Article  MathSciNet  MATH  Google Scholar 

  3. Becker RA, Chakrabarti SK, Geller W, Kitchens B, Misiurewicz M (2007) Dynamics of the Nash map in the game of matching pennies. J Differ Equ Appl 13:223–235

    Article  MathSciNet  MATH  Google Scholar 

  4. Bednarik P (2011) Discretized best-response dynamics for cyclic games. Diplomarbeit, University Vienna, Vienna

  5. Benaïm M, Hirsch MW (1999) Mixed equilibria and dynamical systems arising from fictitious play in perturbed games. Games Econ Behav 29:36–72

    Article  MATH  Google Scholar 

  6. Benaïm M, Hofbauer J, Sorin S (2005) Stochastic approximations and differential inclusions. SIAM J Control Optim 44:328–348

    Article  MathSciNet  MATH  Google Scholar 

  7. Benaïm M, Hofbauer J, Sorin S (2006) Stochastic approximations and differential inclusions. Part II: applications. Math Oper Res 31:673–695

    Article  MathSciNet  MATH  Google Scholar 

  8. Blume L (1993) The statistical mechanics of strategic interaction. Games Econ Behav 5:387–424

    Article  MathSciNet  MATH  Google Scholar 

  9. Bronstein IU, Kopanskii AY (1988) Chain recurrence in dynamical systems without uniqueness. Nonlinear Anal Theory Appl 12:147–154

    Article  MathSciNet  MATH  Google Scholar 

  10. Brown GW, von Neumann J (1950) Solutions of games by differential equations. Ann Math Stud 24:73–79

    Google Scholar 

  11. Chen H-C, Friedman JW, Thisse J-F (1997) Boundedly rational Nash equilibrium: a probabilistic choice approach. Games Econ Behav 18:32–54

    Article  MathSciNet  MATH  Google Scholar 

  12. Conley C (1978) Isolated invariant sets and the Morse index. CBMS regional conference series in mathematics, vol 38. Am Math Soc, Providence

    MATH  Google Scholar 

  13. Demichelis S, Ritzberger K (2003) From evolutionary to strategic stability. J Econ Theory 113:51–75

    Article  MathSciNet  MATH  Google Scholar 

  14. Demichelis S, Sorin S (2003) Asymptotic stability for convex-valued differential inclusions. Preprint

  15. Fudenberg D, Kreps D (1993) Learning mixed equilibria. Games Econ Behav 5:320–367

    Article  MathSciNet  MATH  Google Scholar 

  16. Fudenberg D, Levine DK (1995) Consistency and cautious fictitious play. J Econ Dyn Control 19:1065–1089

    Article  MathSciNet  MATH  Google Scholar 

  17. Fudenberg D, Levine DK (1998) The theory of learning in games. MIT Press, Cambridge

    MATH  Google Scholar 

  18. Garay BM, Hofbauer J (1997) Chain recurrence and discretisation. Bull Aust Math Soc 55:63–71

    Article  MathSciNet  MATH  Google Scholar 

  19. Gaunersdorfer A, Hofbauer J (1995) Fictitious play, Shapley polygons, and the replicator equation. Games Econ Behav 11:279–303

    Article  MathSciNet  MATH  Google Scholar 

  20. Harsanyi JC (1973) Oddness of the number of equilibrium points: a new proof. Int J Game Theory 2:235–250

    Article  MathSciNet  MATH  Google Scholar 

  21. Hirsch MW, Smale S (1974) Differential equations, dynamical systems, and linear algebra. Academic Press, San Diego

    MATH  Google Scholar 

  22. Hofbauer J (1995) Stability for the best response dynamics. Mimeo

  23. Hofbauer J (2000) From Nash and Brown to Maynard Smith: equilibria, dynamics, and ESS. Selection 1:81–88

    Article  Google Scholar 

  24. Hofbauer J (2011) Deterministic evolutionary game dynamics. In: Sigmund K (ed) Evolutionary game dynamics. Proceedings of symposia in applied mathematics, vol 69. Am Math Soc, Providence, pp 61–79

    Google Scholar 

  25. Hofbauer J, Sandholm WH (2002) On the global convergence of stochastic fictitious play. Econometrica 70:2265–2294

    Article  MathSciNet  MATH  Google Scholar 

  26. Hofbauer J, Sandholm WH (2009) Stable games and their dynamics. J Econ Theory 144:1665–1693

    Article  MATH  Google Scholar 

  27. Hofbauer J, Sandholm WH (2011) Survival of dominated strategies under evolutionary dynamics. Theor Econ 6:341–377

    Article  MATH  Google Scholar 

  28. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  29. Hofbauer J, Sorin S (2006) Best response dynamics for continuous zero–sum games. Discrete Contin Dyn Syst B6:215–224

    MathSciNet  Google Scholar 

  30. Hofbauer J, Swinkels J (1996) A universal Shapley example. Preprint

  31. Hofbauer J, Sorin S, Viossat Y (2009) Time average replicator and best reply dynamics. Math Oper Res 34:263–269

    Article  MathSciNet  MATH  Google Scholar 

  32. Hurley M (1982) Attractors: persistence, and density of their basins. Trans Am Math Soc 269:247–271

    Article  MathSciNet  MATH  Google Scholar 

  33. Kohlberg E, Mertens J-F (1986) On the strategic stability of equilibria. Econometrica 54:1003–1038

    Article  MathSciNet  MATH  Google Scholar 

  34. Kojima F, Takahashi S (2007) Anti-coordination games and dynamics stability. Int Game Theory Rev 9:667–688

    Article  MathSciNet  MATH  Google Scholar 

  35. Li D, Zhang X (2002) On dynamical properties of general dynamical systems and differential inclusions. J Math Anal Appl 274:705–724

    Article  MathSciNet  MATH  Google Scholar 

  36. Matsui A (1992) Best response dynamics and socially stable strategies. J Econ Theory 57:343–362

    Article  MathSciNet  MATH  Google Scholar 

  37. McKelvey RD, Palfrey TR (1995) Quantal response equilibria for normal form games. Games Econ Behav 10:6–38

    Article  MathSciNet  MATH  Google Scholar 

  38. Nash, J. (1950) Non-cooperative games. Dissertation, Princeton University, Department Mathematics

  39. Nash J (1951) Non-cooperative games. Ann Math 54:287–295

    Article  MathSciNet  Google Scholar 

  40. Smale S (1967) Differentiable dynamical systems. Bull Am Math Soc 73:747–817

    Article  MathSciNet  MATH  Google Scholar 

  41. Swinkels JM (1993) Adjustment dynamics and rational play in games. Games Econ Behav 5:455–484

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

This research was partially supported by grant SNF 2000-21-138242/1 (Switzerland), grant WWTF-MA09-017 (Austria) and grant ANR-08-BLAN-0294-01 (France).

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Authors and Affiliations

  1. Institut de Mathematiques, Université de Neuchâtel, Rue Emile-Argand 11, Suisse, Neuchâtel, Switzerland

    Michel Benaïm

  2. Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090, Vienna, Austria

    Josef Hofbauer

  3. Combinatoire et Optimisation, IMJ, CNRS UMR 7586, Faculté de Mathématiques, Université P. et M. Curie—Paris 6, Tour 15-16, 1 étage, 4 place Jussieu, 75005, Paris, France

    Sylvain Sorin

  4. Laboratoire d’Econométrie, Ecole Polytechnique, Palaiseau, France

    Sylvain Sorin

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  1. Michel Benaïm
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  2. Josef Hofbauer
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  3. Sylvain Sorin
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Correspondence to Josef Hofbauer.

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Benaïm, M., Hofbauer, J. & Sorin, S. Perturbations of Set-Valued Dynamical Systems, with Applications to Game Theory. Dyn Games Appl 2, 195–205 (2012). https://doi.org/10.1007/s13235-012-0040-0

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