Optimal Influence Strategies in an Oligopolistic Competition Network Environment

This paper presents a non-linear optimization methodology for determining the Nash Equilibrium (NE) solutions of a non-cooperative two-player game. Each player, in particular, is trying to maximize a rational profit function within a continuous action space. The game arises in the context of a duopolistic network environment where two identical rival firms are competing to maximize their influence over a single consumer. Specifically, we consider a weighted and strongly connected network which mediates the opinion formation processes concerning the perceived qualities of their products. Obtaining the NE solutions for such a game is an extremely difficult task which cannot be analytically addressed, even if additional simplifying assumptions are imposed on the exogenous parameters of the model. Our approach, obtains the required NE solutions by combining the Karush-Kuhn-Tucker (KKT) conditions associated with the original optimization tasks into a single-objective nonlinear maximization problem under nonlinear constrains. The resulting optimization problem is, ultimately, solved through the utilization of the Sequential Quadratic Programming (SQP) algorithm which constitutes a state-of-the-art method for nonlinear optimization problems. The validity of our work is justified through the conduction of a series of experiments in which we simulated the best response-based dynamical behaviour of the two agents in the network that make strategic decisions. Juxtaposing the intersection points of the acquired best response curves against the NE solutions obtained by the proposed nonlinear optimization methodology verifies that the corresponding solution points coincide.