Regular and Chaotic Dynamics

, Volume 16, Issue 1–2, pp 128–153 | Cite as

Piecewise linear hamiltonian flows associated to zero-sum games: Transition combinatorics and questions on ergodicity

Article

Abstract

In this paper we consider a class of piecewise affine Hamiltonian vector fields whose orbits are piecewise straight lines. We give a first classification result of such systems and show that the orbit-structure of the flow of such a differential equation is surprisingly rich.

Keywords

Hamiltonian systems non-smooth dynamics Filippov systems piecewise affine Arnol’d diffusion fictitious play best-response dynamics learning process 

MSC2010 numbers

37Jxx 37N40 37Gxx 34A36 34A60 91A20 

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References

  1. 1.
    Aubin, J.-P. and Cellina, A., Differential Inclusions. Set-valued Maps and Viability Theory. Berlin: Springer, 1984.MATHGoogle Scholar
  2. 2.
    Berger, U., Fictitious Play in 2 × n Games, J. Econom. Theory, 2005, vol. 120, pp. 139–154.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brown, G.W., Iterative Solution of Games by Fictitious Play, Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13., New York: John Wiley & Sons, Inc., 1951, pp. 374–376.Google Scholar
  4. 4.
    di Bernardo, M., Budd, C.J., Champneys, A.R., and Kowalczyk, P., Piecewise-smooth Dynamical Systems. Theory and applications., London: Springer, 2008.MATHGoogle Scholar
  5. 5.
    Hofbauer, J., Stability for the Best Response Dynamics, Preprint, August 1995.Google Scholar
  6. 6.
    Kunze, M., Non-smooth Dynamical Systems, Berlin: Springer, 2000.MATHCrossRefGoogle Scholar
  7. 7.
    Leine, R.I. and Nijmeijer, H., Dynamics and Bifurcations of Non-smooth Mechanical Systems, Berlin: Springer, 2004.MATHGoogle Scholar
  8. 8.
    Nash, J., Non-Cooperative Games, Ann. of Math. (2), 1951, vol. 54, pp. 286–295.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Robinson, J., An Iterative Method of Solving a Game, Ann. of Math. (2), 1951, vol. 54, pp. 296–301.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Rosenmüller, J., Über Periodizitätseigenschaften Spieltheoretischer Lernprozesse, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1971, vol. 17, pp. 259–308.MATHCrossRefGoogle Scholar
  11. 11.
    Sparrow, C., van Strien, S., and Harris, Ch., Fictitious Play in 3 × 3 Games: the Transition Between Periodic and Chaotic Behavior, Games Econom. Behav., 2008, vol. 63, pp. 259–291.MATHMathSciNetGoogle Scholar
  12. 12.
    van Strien, S., Hamiltonian Flows with Random-walk Behavior Originating from Zero-sum Games and Fictitious Play, Preprint, 2009.Google Scholar
  13. 13.
    van Strien, S. and Sparrow, C., Fictitious Play in 3 × 3 Games: Chaos and Dithering Behavior, Games Econom. Behav., 2009 (to appear).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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