Structured Network Games: Leveraging Relational Information in Equilibrium Analysis

We study games with nonlinear best response functions on structured networks. These network structures can emerge from agents' communities or multi-relational interactions, where each relation may follow a different interaction graph. For these structured network games, we establish conditions for uniqueness and stability of pure strategy Nash equilibrium that are stronger yet more computationally efficient to verify than their counterparts in prior research on unstructured (mostly single-relational) network games. Specifically, the network structures enable us to determine Nash equilibrium uniqueness and stability conditions using low-dimensional matrices, often on the order of the number of partitions. This is in contrast to conventional analyses that rely on matrices with dimensions determined by the number of agents multiplied by the action space size. Additionally, we introduce a new degree centrality measure to assess partition influence and use it to establish new Nash equilibrium uniqueness and stability conditions. We compare our findings with prior unstructured network research both analytically and through numerical simulations.