Existence and computation of stationary solutions for congestion-type mean field games via bifurcation theory and forward-forward problems

Time-dependent mean field games are a coupled system of a forward parabolic and backward parabolic partial differential equation. Stationary solutions are of interest, and then naturally the forward-backward structure in time becomes irrelevant. Forward-forward mean field games have been introduced with the rationale that they may be used to straightforwardly compute such stationary solutions. We perform some numerical simulations to find that typically stationary solutions of mean field games are unstable to the forward-forward evolution, i.e. frequently only trivial solutions can be found in this way. We then ask whether there are situations in which one would have reason to believe that the stationary solutions would be stable, and we use the exchange-of-stability phenomenon in bifurcation theory to give a class of examples for which the forward-forward solutions do converge to nontrivial stationary solutions as time increases.