In this paper we consider a strategic game played by a group of agents on a set of opinion dynamics models. The models are all Friedkin-Johnsen (FJ) models, which are independent of each other (we call them “parallel FJ models”). The task of an agent is to maximize her overall social power by allocating a given budget of stubbornness across the parallel FJ models. For this game, the cost function is shown to be convex in the action profile set, but discontinuous at some boundary points when for some FJ model only one agent is stubborn (i.e., assigning non-zero stubbornness in the FJ model). Despite the discontinuity, an Nash equilibrium is shown to exist, but is not necessarily unique. Some sufficient conditions that can guarantee the uniqueness are proposed, relying on the strictly monotone pseudo-gradient mappings associated to the game. The conditions are applied to complete graphs with rank-1 weight matrices, for which the link weights are unequal for different agents and on different FJ models. Moreover, for the complete graph case, given the actions of the other agents, the best response of each agent is analytically characterized.

No-regret learning has been widely used to compute a Nash equilibrium in two-person zero-sum games. However, there is still a lack of regret analysis for network stochastic zero-sum games, where players competing in two subnetworks only have access to some local information, and the cost functions are subject to stochastic uncertainty. Such a game model can be found in network interdiction problems, when a group of inspectors work together to detect a group of evaders. In this paper, the authors propose a distributed stochastic mirror descent (D-SMD) method, and establish the regret bounds O(T) and O(log T) in the expected sense for convex-concave and strongly convex-strongly concave costs, respectively. The proposed bounds match those of the best known first-order online optimization algorithms. The authors then prove the convergence of the time-averaged iterates of D-SMD to the set of Nash equilibria. Finally, the authors show that the actual iterates of D-SMD almost surely converge to the Nash equilibrium in the strictly convex-strictly concave setting.