Critical Nash Value Nodes for Control Affine Embedded Graphon Mean Field Games*

Abstract

Graphon Mean Field Games (GMFGs) (Caines and Huang (2021)) constitute generalizations of Mean Field Games to the case where the agents form subpopulations associated with the nodes of large graphs. The work in (Foguen-Tchuendom et al. (2021), Foguen-Tchuendom et al. (2022a)) analyzed the stationarity of equilibrium Nash values with respect to node location for large populations of non-cooperative agents with linear dynamics on large graphs embedded in Euclidean space together with their limits (termed embedded graphons). That analysis is extended in this investigation to agent systems lying in the class of control affine non-linear systems (see Isidori (1985)). Specifically, control affine GMFG systems are treated where (i) at each node αeV the drift of each generic agent system is affine in the control function, and (ii) the running costs at each node α ϵ V ⊂ Rm are exponentiated negative inverse quadratic (ENIQ) functions of the difference between a generic state and the local graphon weighted mean Zα,µG where µG:= {µβ,β ϵ V ⊂ Rm} is the globally distributed family of mean fields. The infinite cardinality node and edge limits are considered, where it is assumed that the limit embedded graphon g(α, β), (α, β) ϵ V x V, is continuously differentiable. It is shown that the equilibrium Nash value Vα is stationary with respect to the nodal location α ϵ V if and only if the corresponding mean Zα,µG is stationary with respect to nodal location.