Network games with three types of players

In this paper, we analyze a multi-strategy network game with three types of players, conformists, rebels, and stubborn agents. Conformists adopt the strategy that is most common among their neighbors, rebels adopt the least common, and stubborn agents adhere to a fixed strategy. We study the existence and structure of pure strategy Nash equilibrium (PNE). On arbitrary networks, we establish sufficient conditions for PNE existence, and we prove that in large random networks PNE almost surely fails to exist. For several specific network architectures, such as complete network, lines, rings, trees, and stars, we derive necessary and sufficient conditions for PNE existence and fully characterize the equilibrium strategy frequencies. Collectively, these results offer a unified perspective that PNE is likely to exist when every conformist has more conformist and stubborn neighbors, and fails when the network game has numerous conformist-rebel edges.