Abstract
We consider for the first time a stochastic generalized Nash equilibrium problem, i.e., with expected-value cost functions and joint feasibility constraints, under partial-decision information, meaning that the agents communicate only with some trusted neighbors. We propose several distributed algorithms for network games and aggregative games that we show being special instances of a preconditioned forward–backward splitting method. We prove that the algorithms converge to a generalized Nash equilibrium when the forward operator is restricted cocoercive by using the stochastic approximation scheme with variance reduction to estimate the expected value of the pseudogradient.
Highlights
In a stochastic Nash equilibrium problem (SNEP), some agents interact with the aim of minimizing their expected value cost function which is affected by the decision variables of the other agents
We have a stochastic generalized NEP (SGNEP), i.e., the problem of finding a Nash equilibrium where the cost function and the feasible set depend on the decisions of the other agents [7,12,32]
The FB algorithm converges [12,30,34] when the operator used for the forward step is strongly monotone [12,30] or cocoercive as we preliminarily show in the full decision information setup [13]
Summary
In a stochastic Nash equilibrium problem (SNEP), some agents interact with the aim of minimizing their expected value cost function which is affected by the decision variables of the other agents. Consider the gas market where the companies participate in a bounded capacity market [1] or more generally any network Cournot game with market capacity constraints and uncertainty in the demand [35,32] In this case, we have a stochastic generalized NEP (SGNEP), i.e., the problem of finding a Nash equilibrium where the cost function and the feasible set depend on the decisions of the other agents [7,12,32]. In this paper we propose the first distributed algorithms tailored for SGNEPs in partial-decision information and show their convergence to an equilibrium under restricted cocoercivity of the stochastic forward operator. Numerical simulations (Section 9) and conclusion (Section 10) end the paper
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