On the uniqueness and stability of equilibria of network games

We study a class of games played on networks with general (non-linear) best-response functions. Specifically, we let each agent's payoff depend on a linearly weighted sum of her neighbors' actions through a non-linear interaction function. We identify conditions on the network structure underlying the game given which (i) the Nash equilibrium of the game is unique, and (ii) the Nash equilibria are stable under perturbations in the model's parameters. We find that both the uniqueness and stability of the Nash equilibria are related to the lowest eigenvalue of suitably defined matrices, which are determined by the network's adjacency matrix, as well as the slopes of the interaction functions. We show that our uniqueness result generalizes an existing uniqueness condition for games of linear best-responses to games with general best-response functions. We further identify the classes of agents that are instrumental in the spread of shocks over the network. In particular, for small shocks, we show that agents that are strictly inactive at a given equilibrium can be precluded from the equilibrium's stability analysis, irrespective of their network position or links.