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Higher order game dynamics

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  • Laraki, Rida
  • Mertikopoulos, Panayotis
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    Abstract

    Continuous-time game dynamics are typically first order systems where payoffs determine the growth rate of the playersʼ strategy shares. In this paper, we investigate what happens beyond first order by viewing payoffs as higher order forces of change, specifying e.g. the acceleration of the playersʼ evolution instead of its velocity (a viewpoint which emerges naturally when it comes to aggregating empirical data of past instances of play). To that end, we derive a wide class of higher order game dynamics, generalizing first order imitative dynamics, and, in particular, the replicator dynamics. We show that strictly dominated strategies become extinct in n-th order payoff-monotonic dynamics n orders as fast as in the corresponding first order dynamics; furthermore, in stark contrast to first order, weakly dominated strategies also become extinct for n⩾2. All in all, higher order payoff-monotonic dynamics lead to the elimination of weakly dominated strategies, followed by the iterated deletion of strictly dominated strategies, thus providing a dynamic justification of the well-known epistemic rationalizability process of Dekel and Fudenberg [7]. Finally, we also establish a higher order analogue of the folk theorem of evolutionary game theory, and we show that convergence to strict equilibria in n-th order dynamics is n orders as fast as in first order.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Economic Theory.

    Volume (Year): 148 (2013)
    Issue (Month): 6 ()
    Pages: 2666-2695

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    Handle: RePEc:eee:jetheo:v:148:y:2013:i:6:p:2666-2695

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    Web page: http://www.elsevier.com/locate/inca/622869

    Related research

    Keywords: Game dynamics; Higher order dynamical systems; (Weakly) dominated strategies; Learning; Replicator dynamics; Stability of equilibria;

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    References

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    1. Hofbauer, Josef & Weibull, Jörgen W., 1995. "Evolutionary Selection against Dominated Strategies," Working Paper Series 433, Research Institute of Industrial Economics.
    2. Drew Fudenberg & Eddie Dekel, 1987. "Rational Behavior with Payoff Uncertainty," Working papers 471, Massachusetts Institute of Technology (MIT), Department of Economics.
    3. Viossat, Yannick & Sorin, Sylvain & Hofbauer, Josef, 2009. "Time average replicator and best reply dynamics," Economics Papers from University Paris Dauphine 123456789/1014, Paris Dauphine University.
    4. Samuelson, Larry & Zhang, Jianbo, 1992. "Evolutionary stability in asymmetric games," Journal of Economic Theory, Elsevier, vol. 57(2), pages 363-391, August.
    5. Gilboa, Itzhak & Matsui, Akihiko, 1991. "Social Stability and Equilibrium," Econometrica, Econometric Society, vol. 59(3), pages 859-67, May.
    6. Fudenberg, D. & Harris, C., 1992. "Evolutionary dynamics with aggregate shocks," Journal of Economic Theory, Elsevier, vol. 57(2), pages 420-441, August.
    7. Sandholm, William H. & Hofbauer, Josef, 2011. "Survival of dominated strategies under evolutionary dynamics," Theoretical Economics, Econometric Society, vol. 6(3), September.
    8. Ritzberger, Klaus & Weibull, Jörgen W., 1993. "Evolutionary Selection in Normal Form Games," Working Paper Series 383, Research Institute of Industrial Economics.
    9. Sergiu Hart & Andreu Mas-Colell, 2003. "Uncoupled Dynamics Do Not Lead to Nash Equilibrium," American Economic Review, American Economic Association, vol. 93(5), pages 1830-1836, December.
    10. Hofbauer, Josef & Sandholm, William H., 2009. "Stable games and their dynamics," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1665-1693.e, July.
    11. Sandholm, William H. & DokumacI, Emin & Lahkar, Ratul, 2008. "The projection dynamic and the replicator dynamic," Games and Economic Behavior, Elsevier, vol. 64(2), pages 666-683, November.
    12. Rustichini, Aldo, 1999. "Optimal Properties of Stimulus--Response Learning Models," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 244-273, October.
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