Planning problem for continuous-time finite state mean field game with compact action space

TL;DR

This paper investigates the planning problem in continuous-time finite state mean field games with compact action spaces, revealing potential non-existence of solutions and proposing a generalized solution concept based on regret minimization.

Contribution

It introduces a generalized solution framework for the planning problem in finite state mean field games, ensuring existence even when classical solutions do not.

Findings

Classical planning solutions may not exist even if the final distribution is reachable.

A minimal regret solution always exists for the planning problem.

The set of minimal regret solutions is the closure of classical solutions when they exist.

Abstract

The planning problem for the mean field game implies the one tries to transfer the system of infinitely many identical rational agents from the given distribution to the final one using the choice of the terminal payoff. It can be formulated as the mean field game system with the boundary condition only on the measure variable. In the paper, we consider the continuous-time finite state mean field game assuming that the space of actions for each player is compact. It is shown that the planning problem in this case may not admit a solution even if the final distribution is reachable from the initial one. Further, we introduce the concept of generalized solution of the planning problem for the finite state mean field game based on the minimization of regret of the representative player. This minimal regret solution always exists. Additionally, the set of minimal regret solution is the closure of the set of classical solution of the planning problem provided that the latter is nonempty.