A potential approach for planning mean-field games in one dimension

<p style='text-indent:20px;'>This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These are comprised of a Hamilton–Jacobi equation coupled with Fokker–Planck equation. Applying Poincaré's Lemma to the equation, we deduce existence potential. Rewriting in terms potential, obtain system Euler–Lagrange equations certain variational problems. Instead problem (MFP), study this problem. By direct method calculus variations, prove uniqueness solutions The approach has advantage eliminating continuity equation.</p><p style='text-indent:20px;'>We also consider first-order MFP congestion. We that congestion weak solution by introducing potential relying on theory inequalities. end paper presenting an application Hughes' model.</p>