Dynamic Games and Applications

, Volume 4, Issue 2, pp 177–207 | Cite as

On the Efficiency of Equilibria in Mean-Field Oscillator Games

  • Huibing Yin
  • Prashant G. Mehta
  • Sean P. Meyn
  • Uday V. Shanbhag
Article

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.

Keywords

Mean-field Efficiency Oscillators 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Huibing Yin
    • 1
  • Prashant G. Mehta
    • 1
  • Sean P. Meyn
    • 2
  • Uday V. Shanbhag
    • 3
  1. 1.Coordinated Science Laboratory and Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-Champaign (UIUC)ChampaignUSA
  2. 2.Department of Electrical and Computer EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Department of Industrial and Manufacturing EngineeringPennsylvania State UniversityState CollegeUSA

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