Dynamic games on arbitrary networks with two types of players

We focus on asymmetric games on networks with two types of players characterized by their different intrinsic preferences. In general, it is NP-hard to calculate a pure strategy Nash equilibrium (PNE) for an asymmetric network game, which makes us difficult to predict the evolutionary outcome. In this paper, we study the asymptotic behavior of the tempered best response dynamics (tBRD). We develop an approximate approach that can transform the tBRD into a system of deterministic ordinary differential equations (ODE) for 2 × 2 asymmetric games with arbitrary payoff values and network structures. This then allows us to evaluate the effect of network structure on the evolutionary outcome without calculating the PNE. We highlight the importance of the network homophily index in the strategy evolution. For games with strategic complements, a lower homophily index promotes coordination. In contrast, games with strategic substitutes have a unique stable equilibrium for a higher homophily index and have two stable equilibria for a low homophily index. Finally, for games with a mixture of strategic complements and strategic substitutes, a lower homophily index in general promote cyclic behaviors and inhibits the existence of stable equilibrium. We then apply our approach to analyze the influence of social networks on the election and discuss the electoral strategies for the majority and minority parties.