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

2019 ◽  
Vol 25 ◽  
pp. 44 ◽  
Author(s):  
Martino Bardi ◽  
Markus Fischer
Keyword(s):  
Multiple Solutions ◽  
Mean Field ◽  
Current Theory ◽  
Finite Horizon ◽  
Mean Field Games ◽  
Small Data ◽  
Uniqueness Of Solutions ◽  
Time Horizons ◽  
Probabilistic Setting ◽  
Two Populations

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.

scholarly journals Leader formation with mean-field birth and death models

2019 ◽  
Vol 29 (04) ◽  
pp. 633-679 ◽  
Author(s):  
Giacomo Albi ◽  
Mattia Bongini ◽  
Francesco Rossi ◽  
Francesco Solombrino
Keyword(s):  
Social Interactions ◽  
Existence And Uniqueness ◽  
Particle System ◽  
Mean Field ◽  
Jump Process ◽  
Transition Rates ◽  
Global State ◽  
Uniqueness Of Solutions ◽  
Ode System ◽  
Two Populations

We provide a mean-field description for a leader–follower dynamics with mass transfer among the two populations. This model allows the transition from followers to leaders and vice versa, with scalar-valued transition rates depending nonlinearly on the global state of the system at each time. We first prove the existence and uniqueness of solutions for the leader–follower dynamics, under suitable assumptions. We then establish, for an appropriate choice of the initial datum, the equivalence of the system with a PDE–ODE system, that consists of a continuity equation over the state space and an ODE for the transition from leader to follower or vice versa. We further introduce a stochastic process approximating the PDE, together with a jump process that models the switch between the two populations. Using a propagation of chaos argument, we show that the particle system generated by these two processes converges in probability to a solution of the PDE–ODE system. Finally, several numerical simulations of social interactions dynamics modeled by our system are discussed.

scholarly journals Mean field games models of segregation

2017 ◽  
Vol 27 (01) ◽  
pp. 75-113 ◽  
Author(s):  
Yves Achdou ◽  
Martino Bardi ◽  
Marco Cirant
Keyword(s):  
Numerical Methods ◽  
Existence Of Solutions ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Games ◽  
Residential Choice ◽  
Urban Settlements ◽  
Thomas Schelling ◽  
Two Populations ◽  
Large Population Limit

This paper introduces and analyzes some models in the framework of mean field games (MFGs) describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of MFG theory, in the stationary and in the evolutive case. Numerical methods are proposed with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

scholarly journals Stability of McKean–Vlasov stochastic differential equations and applications

2019 ◽  
Vol 20 (01) ◽  
pp. 2050007 ◽  
Author(s):  
Khaled Bahlali ◽  
Mohamed Amine Mezerdi ◽  
Brahim Mezerdi
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Statistical Physics ◽  
Value Function ◽  
Mean Field ◽  
Mean Field Games ◽  
Natural Setting ◽  
Relaxed Control ◽  
Uniqueness Of Solutions ◽  
And Diffusion

We consider McKean–Vlasov stochastic differential equations (MVSDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. This type of SDEs was studied in statistical physics and represents the natural setting for stochastic mean-field games. We will first discuss questions of existence and uniqueness of solutions under an Osgood type condition improving the well-known Lipschitz case. Then, we derive various stability properties with respect to initial data, coefficients and driving processes, generalizing known results for classical SDEs. Finally, we establish a result on the approximation of the solution of a MVSDE associated to a relaxed control by the solutions of the same equation associated to strict controls. As a consequence, we show that the relaxed and strict control problems have the same value function. This last property improves known results proved for a special class of MVSDEs, where the dependence on the distribution was made via a linear functional.

scholarly journals Finite horizon mean field games on networks

2020 ◽  
Vol 59 (5) ◽  
Author(s):  
Yves Achdou ◽  
Manh-Khang Dao ◽  
Olivier Ley ◽  
Nicoletta Tchou
Keyword(s):  
Mean Field ◽  
Finite Horizon ◽  
Mean Field Games

scholarly journals Lax connection and conserved quantities of quadratic mean field games

2021 ◽  
Vol 62 (8) ◽  
pp. 083302
Author(s):  
Thibault Bonnemain ◽  
Thierry Gobron ◽  
Denis Ullmo
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Conserved Quantities ◽  
Quadratic Mean
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