A Mean Field Game Model of Spatial Evolutionary Games.

Evolutionary Game Theory (EGT) studies evolving populations of agents that interact through normal form games whose outcome determines each individual's evolutionary fitness. In many applications, EGT models are extended to include spatial effects in which the agents are located in a structured population such as a graph. We propose a Mean Field Game (MFG) generalization, denoted Pair-MFG, of the spatial evolutionary game model such that the behavior of a given spatial evolutionary game (or more specifically the behavior of its pair approximation) is a special case trajectory of the corresponding MFG. The proposed Pair-MFG model also allows for the formulation of the spatial evolutionary game as a control problem, opening up additional avenues of research into controlling the outcomes of these games. The state evolution equations of the proposed model are highly nonlinear and none of the equations in the system are necessarily convex. This necessitates different numerical methods as compared to those for traditional Linear Quadratic Gaussian MFGs. We provide a method for solving this new Pair-MFG model using fixed point iteration with time-dependent proximal terms and show empirically that this method is capable of finding a solution to a selection of EGT games.