A game-theoretical formulation of influence networks

An influence network consists of a set of interacting agents, each of whose actions produces effects or influence on his neighbors' actions. In general, the effects can be arbitrary functions of the neighbors' joint action; and they can be inhomogeneous among agents. Such effects are akin to the externalities (positive or negative) engendered on the action level (which further impacts one's utilities), and capture the commonality of a variety of networks in economics and engineering. Therefore, the study of strategic interactions among agents in an influence network and a characterization of an equilibrium can be of great value and applicability. In this paper, we formulate a simple game-theoretical model of influence networks that aims to study strategic interactions among agents in light of such influence. We then establish a connection between the resulting multi-player game with the well-known nonlinear complementarity problem (NCP). This connection not only places our work in the existing literature of a well-studied subject, thus deepening the understanding of both problems, but also, to a certain extent, allows us to leverage the tools in the NCP literature to identify different sufficient conditions for the existence and uniqueness of a Nash equilibrium in the multi-player game. We then characterize two broadly-scoped classes of influence networks with natural and intuitive interpretations and draw tools from fixed point theory to show the existence of a Nash equilibrium, as the existing results in the NCP literature are not directly applicable. We finally comment on the distributed, model-agnostic best response dynamics and show that under certain cases, they converge to a Nash equilibrium.