Discrete-Space Social Interaction Models: Stability and Continuous Limit

We study the equilibrium properties, including stability, of discrete-space social interaction models with a single type of agents, and their continuous limit. We show that, even though the equilibrium in discrete space can be non-unique for all finite degree of discretization, any sequence of discrete-space models' equilibria converges to the continuous-space model's unique equilibrium as the discretization of space is refined. Showing the existence of multiple equilibria resorts to the stability analysis of equilibria. A general framework for studying equilibria and their stability is presented by characterizing the discrete-space social interaction model as a potential game.