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.

No abstract

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.

We are concerned with a class of Nash games in which the players' strategy sets are coupled by a shared constraint. A widely employed solution concept for such games, referred to as generalized Nash games, is the generalized Nash equilibrium (GNE). The variational equilibrium (VE) [6] is a specific kind of GNE given by a solution of the variational inequality formed from the common constraint and the mapping of the gradients of player objectives. Our contribution is a theory that provides sufficient conditions for ensuring that the existence of a GNE implies the existence of a VE; in such an instance, the VE is said to be a refinement of the GNE. For certain games our conditions are shown to be necessary. This theory rests on a result showing that, in both the primal and the primal-dual space, the GNE and the VE are equivalent upto the Brouwer degree of two suitably defined functions, whose zeros are the GNE and VE, respectively. The refinement of the GNE is of relevance to pure, applied and computational game theory. Our results unify some previously known facts pertaining to such equilibria and are utilized in showing that shared-constraint Nash-Cournot games arising in power markets do indeed admit a refinement.