We consider a class of Nash games, termed as aggregative games, being played over a networked system. In an aggregative game, a player's objective is a function of the aggregate of all the players' decisions. Every player maintains an estimate of this aggregate, and the players exchange this information with their local neighbors over a connected network. We study distributed synchronous and asynchronous algorithms for information exchange and equilibrium computation over such a network. Under standard conditions, we establish the almost-sure convergence of the obtained sequences to the equilibrium point. We also consider extensions of our schemes to aggregative games where the players' objectives are coupled through a more general form of aggregate function. Finally, we present numerical results that demonstrate the performance of the proposed schemes.

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Abstract. Traditionally, a multiuser problem is a constrained optimization problem characterized by a set of users, an objective given by a sum of user-specific utility functions, and a collection of linear constraints that couple the user decisions. The users do not share the information about their utilities, but do communicate values of their decision variables. The multiuser problem is to maximize the sum of the users-specific utility functions subject to the coupling constraints, while abiding by the informational requirements of each user. In this paper, we focus on generalizations of convex multiuser optimization problems where the objective and constraints are not separable by user and instead consider instances where user decisions are coupled, both in the objective and through nonlinear coupling constraints. To solve this problem, we consider the application of gradient-based distributed algorithms on an approximation of the multiuser problem. Such an approximation is obtained through a Tikhonov regularization and is equipped with estimates of the difference between the optimal function values of the original problem and its regularized counterpart. In the algorithmic development, we consider constant steplength primal-dual and dual schemes in which the iterate computations are distributed naturally across the users, i.e., each user updates its own decision only. Convergence in the primal-dual space is provided in limited coordination settings, which allows for differing steplengths across users as well as across the primal and dual space. We observe that a generalization of this result is also available when users choose their regularization parameters independently from a prescribed range. An alternative to primal-dual schemes can be found in dual schemes which are analyzed in regimes where approximate primal solutions are obtained through a fixed number of gradient steps. Per-iteration error bounds are provided in such regimes and extensions are provided to regimes where users independently choose their regularization parameters. Our results are supported by a case-study in which the proposed algorithms are applied to a multi-user problem arising in a congested traffic network.

Abstract-We consider a class of games, termed as aggregative games, being played over a distributed multiagent networked system. In an aggregative game, an agent's objective function is coupled through a function of the aggregate of all agents decisions. Every agent maintains an estimate of the aggregate and agents exchange this information over a connected network. We study the gossip-based distributed algorithm for information exchange and computation of equilibrium decisions of agents over the network. Our primary emphasis is on proving the convergence of the algorithm under an assumption of a diminishing (agent-specific) stepsize sequence. Under standard conditions, we establish the almost-sure convergence of the algorithm to an equilibrium point. Finally, we present numerical results to assess the performance of the gossip algorithm for aggregative games.

Abstract-In this paper, we consider the distributed computation of equilibria arising in monotone stochastic Nash games over continuous strategy sets. Such games arise in settings when the gradient map of the player objectives is a monotone mapping over the cartesian product of strategy sets, leading to a monotone stochastic variational inequality. We consider the application of projection-based stochastic approximation schemes. However, such techniques are characterized by a key shortcoming: they can accommodate strongly monotone mappings only. In fact, standard extensions of stochastic approximation schemes for merely monotone mappings require the solution of a sequence of related strongly monotone problems, a natively two-timescale scheme. Accordingly, we consider the development of single timescale techniques for computing equilibria when the associated gradient map does not admit strong monotonicity. We first show that, under suitable assumptions, standard projection schemes can indeed be extended to allow for strict, rather than strong monotonicity. Furthermore, we introduce a class of regularized stochastic approximation schemes, in which the regularization parameter is updated at every step, leading to a single timescale method. The scheme is a stochastic extension of an iterative Tikhonov regularization method and its global convergence is established. To aid in networked implementations, we consider an extension to this result where players are allowed to choose their steplengths independently and show if the deviation across their choices is suitably constrained, then the convergence of the scheme may be claimed.