Convergence of Imitation Dynamics for Public Goods Games on Networks

We perform convergence analysis on networks of agents playing public goods games, choosing between the strategies cooperation and defection, and updating asynchronously according to the (unconditional) imitation update rule. The agents earn payoffs by participating in multiplayer games, which greatly differs from the situation when the interactions are pairwise, and hence, requires a more sophisticated analysis. We show that, regardless of the initial condition and the order of the activated agents, in typical topologies including the star, ring and well-mixed networks, an equilibrium state is reached under the imitation dynamics in finite time. The proof for the ring networks is algorithmic; namely, we design a computational algorithm to find particular quadruple or quintuplet strategy patterns, whose amount in the entire network serves as a potential-like function. We also study the final strategy state in the well-mixed and star cases, and show that cooperation is more likely to survive in less-connected networks such as the star compared to completely-connected networks. Our results shed light on how network structure affects the evolution of cooperation among imitative individuals participating in multiplayer games, and serves as a stepping stone to controlling such decision-making populations.