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Population Games, Stable Games, and Passivity

Author

Listed:
  • Michael J. Fox

    (School of Electrical and Computer Engineering, Georgia Institute of Technology, 777 Atlantic Drive NW, Atlanta, GA 30332, USA)

  • Jeff S. Shamma

    (School of Electrical and Computer Engineering, Georgia Institute of Technology, 777 Atlantic Drive NW, Atlanta, GA 30332, USA)

Abstract

The class of “stable games”, introduced by Hofbauer and Sandholm in 2009, has the attractive property of admitting global convergence to equilibria under many evolutionary dynamics. We show that stable games can be identified as a special case of the feedback-system-theoretic notion of a “passive” dynamical system. Motivated by this observation, we develop a notion of passivity for evolutionary dynamics that complements the definition of the class of stable games. Since interconnections of passive dynamical systems exhibit stable behavior, we can make conclusions about passive evolutionary dynamics coupled with stable games. We show how established evolutionary dynamics qualify as passive dynamical systems. Moreover, we exploit the flexibility of the definition of passive dynamical systems to analyze generalizations of stable games and evolutionary dynamics that include forecasting heuristics as well as certain games with memory.

Suggested Citation

  • Michael J. Fox & Jeff S. Shamma, 2013. "Population Games, Stable Games, and Passivity," Games, MDPI, vol. 4(4), pages 1-23, October.
    • Handle: RePEc:gam:jgames:v:4:y:2013:i:4:p:561-583:d:29301
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    References listed on IDEAS

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    1. Sergiu Hart, 2013. "Adaptive Heuristics," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 11, pages 253-287, World Scientific Publishing Co. Pte. Ltd..
    2. Yuzuru Sato & Eizo Akiyama & J. Doyne Farmer, 2001. "Chaos in Learning a Simple Two Person Game," Working Papers 01-09-049, Santa Fe Institute.
    3. Drew Fudenberg & David K. Levine, 2009. "Learning and Equilibrium," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 385-420, May.
    4. Sergiu Hart & Andreu Mas-Colell, 2013. "Uncoupled Dynamics Do Not Lead To Nash Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 7, pages 153-163, World Scientific Publishing Co. Pte. Ltd..
    5. Hofbauer, Josef & Sandholm, William H., 2009. "Stable games and their dynamics," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1665-1693.4, July.
    6. Georgios Chasparis & Jeff Shamma, 2012. "Distributed Dynamic Reinforcement of Efficient Outcomes in Multiagent Coordination and Network Formation," Dynamic Games and Applications, Springer, vol. 2(1), pages 18-50, March.
    7. Hofbauer, Josef & Sandholm, William H., 2007. "Evolution in games with randomly disturbed payoffs," Journal of Economic Theory, Elsevier, vol. 132(1), pages 47-69, January.
    8. Sandholm, William H., 2005. "Excess payoff dynamics and other well-behaved evolutionary dynamics," Journal of Economic Theory, Elsevier, vol. 124(2), pages 149-170, October.
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    Cited by:

    1. Dai Zusai, 2018. "Evolutionary dynamics in heterogeneous populations: a general framework for an arbitrary type distribution," Papers 1805.04897, arXiv.org, revised May 2019.
    2. Dai Zusai, 2018. "Net gains in evolutionary dynamics: A unifying and intuitive approach to dynamic stability," Papers 1805.04898, arXiv.org, revised Oct 2023.
    3. Leslie, David S. & Perkins, Steven & Xu, Zibo, 2020. "Best-response dynamics in zero-sum stochastic games," Journal of Economic Theory, Elsevier, vol. 189(C).
    4. Brendon G. Anderson, 2025. "Choice Paralysis in Evolutionary Games," Papers 2509.10567, arXiv.org, revised Nov 2025.

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