On the efficiency of equilibria in mean-field oscillator games

A key question in the design of engineered competitive systems has been that of the efficiency of the associated equilibria. Yet, there is little known in this regard in the context of stochastic dynamic games in a large population regime. In this paper, we revisit a class of noncooperative games, arising from the synchronization of a large collection of homogeneous oscillators. In [1], 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 variational problems both for the noncooperative game and its centralized counterpart and employ these problems as a vehicle for conducting this analysis. Using a bifurcation analysis, we analyze the variational solutions and the associated efficiency loss. An expression for the efficiency loss is obtained. Finally, our results are validated through detailed numerics.