Decentralized Noncooperative Games with Coupled Decision-Dependent Distributions

Distribution variations in machine learning, driven by the dynamic nature of deployment environments, significantly impact the performance of learning models. This paper explores endogenous distribution shifts in learning systems, where deployed models influence environments and subsequently alter data distributions. This phenomenon is formulated by a decision-dependent distribution mapping within the recently proposed framework of performative prediction (PP) Perdomo et al. (2020). We investigate the performative effect in a decentralized noncooperative game, where players aim to minimize private cost functions while simultaneously managing coupled inequality constraints. Under performativity, we examine two equilibrium concepts for the studied game: performative stable equilibrium (PSE) and Nash equilibrium (NE), and establish sufficient conditions for their existence and uniqueness. Notably, we provide the first upper bound on the distance between the PSE and NE in the literature, which is challenging to evaluate due to the absence of strong convexity on the joint cost function. Furthermore, we develop a decentralized stochastic primal-dual algorithm for efficiently computing the PSE point. By carefully bounding the performative effect in theoretical analysis, we prove that the proposed algorithm achieves sublinear convergence rates for both performative regrets and constraint violation and maintains the same order of convergence rate as the case without performativity. Numerical experiments validate the effectiveness of our algorithm and theoretical results.