A Two-Point Boundary Value Formulation of a Class of Multi-Population Mean-Field Games

We consider a multi-agent system consisting of several populations. The interaction between large populations of agents seeking to regulate their state on the basis of the distribution of the neighboring populations is studied. Examples of such interactions can typically be found in social networks and opinion dynamics, where heterogeneous agents or clusters are present and decisions are influenced by individual objectives as well as by global factors. In this paper, such a problem is posed as a multi-population mean-field game, for which solutions depend on two partial differential equations, namely the Hamilton-Jacobi-Bellman equation and the Fokker-Planck-Kolmogorov equation. The case in which the distributions of agents are sums of polynomials and the value functions are quadratic polynomials is considered. It is shown that for this class of problems, which can be considered as approximations of more general problems, a set of ordinary differential equations, with two-point boundary value conditions, can be solved in place of the more complicated partial differential equations characterizing the solution of the multi-population mean-field game.