On the existence of oscillating solutions in non-monotone Mean-Field Games

Cirant, Marco

doi:10.1016/j.jde.2018.12.025

Cited by 24 publications

(18 citation statements)

References 26 publications

(75 reference statements)

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“…Some explicit counterexamples, very different from ours, were shown recently by Briani and Cardaliaguet [9] and Cirant and Tonon [17]. An interesting analysis of multiple oscillating solutions via bifurcations was done very recently by Cirant [16]. For stationary MFG with ergodic cost functional, examples of non-uniqueness are known since the pioneering paper of Lasry and Lions [39], and others more explicit appear in [29,4,7,25].…”

Section: Introductionmentioning

confidence: 79%

On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games

Bardi

Fischer

2019

ESAIM: COCV

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This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons T and convex but non-smooth Hamiltonian H, as well as for smooth H and T large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small T , is proved. Some results are also extended to MFGs with two populations.in the unknowns (v, m), where m(t, ·) is a probability density for each t, and the given running and terminal costs F, G map a subset of R d × P(R d ) into R. In the probabilistic formulation of a Mean

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Section: Introductionmentioning

confidence: 79%

On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games

Bardi

Fischer

2019

ESAIM: COCV

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“…As stated in the Introduction, our result is for short-time horizons, but no assumptions on the behaviour at infinity of H are required. Note finally that C 2 regularity of H is crucial for uniqueness in short-time, while for large T uniqueness may fail in general even when H is smooth (see [3,4,6,7]).…”

Section: Some Parabolic Systems Arising In the Theory Of Mean-field Gmentioning

confidence: 99%

Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games

2019

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We study the local in time existence of a regular solution of a nonlinear parabolic backward-forward system arising from the theory of Mean-Field Games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.

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“…Let us mention that other nonmonotone mean field games have been considered in several works previously: for example, second-order problems have been analyzed in [9,12,13], while [7,18] deal with first-order systems. The works by Ambrose (see [2] and references therein) show the existence of solutions to second-order systems under smallness conditions on the coupling, though these conditions seem to be depending on the time horizon T .…”

Section: Introductionmentioning

confidence: 99%

Long time behavior and turnpike solutions in mildly non-monotone mean field games

Cirant

Porretta

2021

ESAIM: COCV

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We consider mean field game systems in time-horizon (0,T), where the individual cost depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (the aggregation rate of the cost function) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either globally Lipschitz Hamiltonians or quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we give a complete description of the ergodic and long time properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0,T), (ii) the convergence of the system from (0,T) towards (0,\infty), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. We extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

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On the existence of oscillating solutions in non-monotone Mean-Field Games

Cited by 24 publications

References 26 publications

On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games

On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games

Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games

Long time behavior and turnpike solutions in mildly non-monotone mean field games

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