Fully Distributed Nash Equilibrium Seeking Over Time-Varying Communication Networks With Linear Convergence Rate

Abstract

We design a distributed algorithm for learning Nash equilibria over time-varying communication networks in a partial-decision information scenario, where each agent can access its own cost function and local feasible set, but can only observe the actions of some neighbors. Our algorithm is based on projected pseudo-gradient dynamics, augmented with consensual terms. Under strong monotonicity and Lipschitz continuity of the game mapping, we provide a simple proof of linear convergence, based on a contractivity property of the iterates. Compared to similar solutions proposed in literature, we also allow for time-varying communication and derive tighter bounds on the step sizes that ensure convergence. In fact, in our numerical simulations, our algorithm outperforms the existing gradient-based methods, when the step sizes are set to their theoretical upper bounds. Finally, to relax the assumptions on the network structure, we propose a different pseudo-gradient algorithm, which is guaranteed to converge on time-varying balanced directed graphs. Index Terms-Game theory, optimization algorithms, networked control systems. I. INTRODUCTIONN ASH equilibrium (NE) problems arise in several network systems, where multiple selfish decision-makers, or agents, aim at optimizing their individual, yet inter-dependent, objective functions. Engineering applications include communication networks [1], demand-side management in the smart grid [2], charging of electric vehicles [3] and demand response in competitive markets [4]. From a game-theoretic perspective, the challenge is to assign the agents behavioral rules that eventually ensure the attainment of a NE, a joint action from which no agent has an incentive to unilaterally deviate.Literature review: Typically, NE seeking algorithms are designed under the assumption that each agent can access

Nash equilibrium seeking under partial-decision information over directed communication networks

“…Compared to similar pseudo-gradient dynamics proposed in the literature [10], [17], the novelty of Algorithm 1 is that the cost related components ∇ xi J i are weighted by the reciprocal of the elements q i of the PF eigenvector. This operation enables convergence on row stochastic graphs, and in fact it is not necessary for doubly stochastic graphs, for which q = 1.…”

“…Remark 4: The condition in ( 13) can always be satisfied by choosing α small enough; an explicit upper bound can be obtained as in [17,Lemma 2].…”

“…Fewer works deal with asymmetric networks. Under the assumption of balanced weights, continuous-time dynamics were proposed in [16] for aggregative games; most recently, we also addressed generally-coupled-cost games via a fixedstep forward-backward method [17]. To the best of our knowledge, the only discrete-time NE seeking algorithm that takes into account non-balanced digraphs is the asynchronous gossip-based scheme in [18].…”

Networked Aggregative Games with Linear Convergence

“…Distributed seeking of NE has extensively been studied in recent years. For games where players' cost functions depend on others' actions in a general manner, researchers have proposed fully distributed algorithms based on alternating direction method of multipliers [8], best-response dynamics [9], and projected-gradient method [10]- [12]. In these algorithms, players utilize local communications to estimate others' actions, and the NE is found by solving equivalent variational inequality problems [13].…”

“…Researchers in [14] and [15] adopted a consensus protocol to dynamically track the aggregate action and used projectedgradient methods to update each player's action. Since only aggregate estimates are exchanged, compared with the algorithms in [8]- [12], there is an n-fold reduction in the amount of data that is transmitted in each iteration, where n is the number of players. However, the convergence was shown only for diminishing step-sizes with strictly monotone pseudo-gradient mappings.…”