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On the Efficiency of Equilibria in Mean-Field Oscillator Games

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  • Huibing Yin
  • Prashant Mehta
  • Sean Meyn
  • Uday Shanbhag

Abstract

A key question in the design of engineered competitive systems has been that of the efficiency loss of the associated equilibria. Yet, there is little known in this regard in the context of stochastic dynamic games, particularly in a large population regime. In this paper, we revisit a class of noncooperative games, arising from the synchronization of a large collection of heterogeneous oscillators. In Yin et al. (Proceedings of 2010 American control conference, pp. 1783–1790, 2010 ), we derived a PDE model for analyzing the associated equilibria in large population regimes through a mean field approximation. Here, we examine the efficiency of the associated mean-field equilibria with respect to a related welfare optimization problem. We construct constrained variational problems both for the noncooperative game and its centralized counterpart and derive the associated nonlinear eigenvalue problems. A relationship between the solutions of these eigenvalue problems is observed and allows for deriving an expression for efficiency loss. By applying bifurcation analysis, a local bound on efficiency loss is derived under an assumption that oscillators share the same frequency. Through numerical case studies, the analytical statements are illustrated in the homogeneous frequency regime; analogous numerical results are provided for the heterogeneous frequency regime. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Huibing Yin & Prashant Mehta & Sean Meyn & Uday Shanbhag, 2014. "On the Efficiency of Equilibria in Mean-Field Oscillator Games," Dynamic Games and Applications, Springer, vol. 4(2), pages 177-207, June.
    • Handle: RePEc:spr:dyngam:v:4:y:2014:i:2:p:177-207
    DOI: 10.1007/s13235-013-0100-0
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    References listed on IDEAS

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    1. Correa, Jose R. & Schulz, Andreas S. & Stier Moses, Nicolas E., 2004. "Computational Complexity, Fairness, and the Price of Anarchy of the Maximum Latency Problem," Working papers 4447-03, Massachusetts Institute of Technology (MIT), Sloan School of Management.
    2. José R. Correa & Andreas S. Schulz & Nicolás E. Stier-Moses, 2004. "Selfish Routing in Capacitated Networks," Mathematics of Operations Research, INFORMS, vol. 29(4), pages 961-976, November.
    3. Ramesh Johari & John N. Tsitsiklis, 2004. "Efficiency Loss in a Network Resource Allocation Game," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 407-435, August.
    4. Daron Acemoglu & Asuman Ozdaglar, 2007. "Competition and Efficiency in Congested Markets," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 1-31, February.
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    Cited by:

    1. René Carmona & Christy V. Graves, 2020. "Jet Lag Recovery: Synchronization of Circadian Oscillators as a Mean Field Game," Dynamic Games and Applications, Springer, vol. 10(1), pages 79-99, March.
    2. Paolo Dai Pra & Elena Sartori & Marco Tolotti, 2023. "Polarization and Coherence in Mean Field Games Driven by Private and Social Utility," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 49-85, July.

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