Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash Games
本文提出随机逼近方案来估计随机纳什博弈中的稳定价格，通过开发随机块坐标外梯度方法处理非光滑凸目标与单调变分不等式约束，并给出误差界与复杂度分析，对设计通信网络和电力市场等高效网络系统有参考价值。
The goal in this article is to approximate the Price of Stability (PoS) in stochastic Nash games using stochastic approximation (SA) schemes. PoS is among the most popular metrics in game theory and provides an avenue for estimating the efficiency of Nash games. In particular, evaluating the PoS can help with designing efficient networked systems, including communication networks and power market mechanisms. Motivated by the absence of efficient methods for computing the PoS, first we consider stochastic optimization problems with a nonsmooth and merely convex objective function and a merely monotone stochastic variational inequality (SVI) constraint. This problem appears in the numerator of the PoS ratio. We develop a randomized block-coordinate stochastic extra-(sub)gradient method where we employ a novel iterative penalization scheme to account for the mapping of the SVI in each of the two gradient updates of the algorithm. We obtain an iteration complexity of the order ϵ -4 that appears to be best known result for this class of constrained stochastic optimization problems, where ϵ denotes an arbitrary bound on suitably defined infeasibility and suboptimality metrics. Second, we develop an SA-based scheme for approximating the PoS and derive lower and upper bounds on the approximation error. To validate the theoretical findings, we provide preliminary simulation results on a networked stochastic Nash Cournot competition.