This paper considers the case where the opinion of agents in a social network is influenced not only by the other agents, but also by two marketers in competition. The main contributions of this work is to propose a dynamical game formulation of the problem and to conduct the corresponding equilibrium analysis. Due to the competition between the marketers, the opinions never reach consensus but are spread between the desired opinions of the two marketers. Our analysis provides practical insights to know how a marketer should exploit its knowledge about the social network to design the control of opinions using results from optimal control theory. Numerical examples illustrate the analysis.

This work focuses on the design of decentralized feedback control gains that aims at optimizing individual costs in a multi-agent synchronization problem. As reported in the literature, the optimal control design for synchronization of agents using local information is NP-hard. Consequently, we relax the problem and use the notion of satisfaction equilibrium from game theory to ensure that each individual cost is guaranteed to be lower than a given threshold. Our main results provide conditions in the form of linear matrix inequalities (LMIs) to check if a given set of control gains are in satisfaction equilibrium i.e. all individual costs are upper-bounded by the imposed threshold. Moreover, we provide an algorithm in order to synthesize gains that are in satisfaction equilibrium. Finally, we illustrate this algorithm with numerical examples.

We study multi-agent optimization problems described by ordinal potential games. The objective is to reach a Nash equilibrium (NE) in a distributed manner. It is well known that an asynchronous best response dynamics (ABRD) will always converge to a pure NE in this case. However, computing the exact best response at every step of the algorithm may be computationally heavy, if not impossible. Therefore, instead of computing the exact best response, we propose an algorithm that performs a "better response", which decreases the local cost rather than minimizing it. The agents perform a distributed asynchronous gradient descent algorithm, in which only a finite number of iterations of the gradient descent are performed by each player. We prove that this algorithm always converges to a NE and demonstrate via simulations that the computational time to reach the NE can be much shorter than with the classical ABRD. Taking into account the time required for each agent to communicate, the proposed algorithm is shown to also outperform a distributed synchronous gradient descent in simulations. I. INTRODUCTIONGame theory has been widely applied in various fields like economics [1], wireless communication [2] and automatic control [3]. A special class of games called potential games was introduced and studied by Monderer in [4], and several conditions guaranteeing the existence of a pure Nash equilibrium (NE) were provided. Potential games find many applications, e.g., in traffic management, wireless design, model predictive control, see [5].In [6], it was shown that in potential games, the asynchronous best response dynamics (ABRD), always converges to a pure NE. Recall that the ABRD involves players updating their actions to be one that minimizes their local cost asynchronously, while the other player actions are fixed. However, in many situations, the exact best responses which are obtained by minimizing a certain cost are hard or impossible to compute, and instead, algorithms may be used to find approximate best responses. Recent works have studied the convergence of approximate best response dynamics for special applications. In [7], a framework for iterative approximate best response algorithms is provided for distributed constraint optimization problems. In [8], the authors focus on approximate best response dynamics in interference games, which are a special class of games often studied in wireless communication. On the other hand, [9] looks at stochastic algorithms which are used to approximate the NE.