Mean field games of controls: Finite difference approximations

On classical solutions to the mean field game system of controls

“…We believe that this uniqueness result can be easily extended to more general Hamiltonians, but that the short-time assumption is essential. Indeed numerical examples in which non-uniqueness occurs are presented in [1]. In these examples, we consider groups of agents who start from some crowded areas at time t = 0, and travel through the domain to arrive at some target areas.…”

“…Uniqueness should not hold in general but under a short-time assumption. We refer to [1] in which the MFGC system is discretized using a finite-difference scheme and simulations are provided where the approximating discrete MFGC system admits several different solutions.…”

“…would be the associated distance. We refer to Remark 4.3 in [1] for a concrete example of a MFGC system which does not admit solution if λ 0 = 1.…”

“…Difficulty 1) is in general not straightforward and one needs to make assumptions for the fixed point in µ to have a unique solution when (∇ x u, m) are given. An example in which this fixed point relation does not admit any solution is given in [1] Remark 4.3.…”

“…We will also give a uniqueness result under a short time horizon assumption. We refer to [1] for a numerical application with multiple solutions. Indeed, uniqueness does not hold in general for arbitrary time horizon.…”