Strict-control representation of relaxed equilibria

Determine whether a relaxed mean field game equilibrium obtained under nonunique individual best responses can be represented by a strict control, under suitable convexity or purification assumptions.

Background

The paper identifies relaxation of the strict-concavity and unique-Hamiltonian-maximizer assumptions as a natural extension. Without a unique best response, the consistency map becomes set-valued, motivating a relaxed-control formulation in which mixed strategies restore compactness and convexity and permit the use of the Kakutani–Fan–Glicksberg fixed-point theorem.

The unresolved issue is whether an equilibrium in this relaxed, mixed-strategy formulation can be converted into an equilibrium implemented by a strict control. The paper indicates that additional convexity or purification assumptions may be relevant to such a representation.