Solving N-player dynamic routing games with congestion: a mean field approach

Both the paper and the model establish MFNE existence via a Kakutani–Fan–Glicksberg fixed-point argument on a compact, convex simplex of mixed path policies. The paper carefully restricts to distributions over finitely many path choices given departure times, proves the best-response map is a KFG map, obtains a fixed point, and then embeds the resulting policy back into the original policy space Π (with an explicit note on finiteness of feasible paths using finite time horizon and positive minimum link travel times). The model’s solution follows the same structure but compresses steps: it identifies the policy space with a finite simplex and invokes Berge’s Maximum Theorem for upper hemicontinuity, without explicitly stating the finiteness assumptions or the path-to-policy embedding. Substantively, the proofs coincide; the model omits some technical details but is logically consistent with the paper’s assumptions.