This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.
Since the introduction of mean field games (MFGs) in 2006 by Jean-Michel Lasry and Pierre-Louis Lions [63][64][65] and the simultaneous independent works by Peter E. Caines, Minyi Huang, and Roland P. Malhamé [55][56][57], many authors have proposed MFG models for (or inspired by) the analysis of crowd motion [8,10,20,38,62,68,78]. While the mathematical description of the movement of crowds of people has attracted the interest of several researchers along the years (see, e.g., [7,53,54,58,66,71]), the MFG approach to crowd motion has the particularity that, by considering agents of the crowd to be rational, one may consider that each agent chooses their trajectory based not only on the present or past states of other agents, but also on a rational anticipation of their future behavior. Our aim in this paper is to propose and study an MFG model inspired by crowd motion in which agents interact through a Cucker-Smale-type interaction, in the spirit of [78].
To the authors’ knowledge, the first work to be fully dedicated to a mean field game model for crowd motion is [62], which proposes an MFG model for a two-population crowd with trajectories perturbed by additive Brownian motion and considers both their stationary distributions and their evolution on a prescribed time interval. Other works have later proposed MFG models for crowd motion taking into account different characteristics. To cite a few examples, let us mention [20], which considers the fast exit of a crowd and proposes a mean field game model which is studied numerically; [30], which is not originally motivated by the modeling of crowd motion but considers an MFG model with a density constraint, which is a natural assumption in some crowd motion models (see also [69] for the case of second-order mean field games with density constraints); [1,33], which study a two-population MFG model motivated by urban settlements; [10], which presents numerical simulations for some variational mean field games related to crowd motion; [8], which, inspired by cities crowded by tourists such as Venice, Italy, proposes a mean field game model for the movement of pedestrian tourists; [38], which provides a generalized MFG model for pedestrians with limited predictive abilities; or also [43,44,67,68,75,76], which consider MFG models in which the goal of each agent is to reach a given target set in minimal time, interactions between agents being modeled through a congestion-dependent maximal velocity. Other works studying mean field games motivated by or related to crowd motion include [2, 6,9,32,41].
A common feature of several MFG models for crowd motion is that, at a given time t, an agent at a space position x will interact with other agents only through the distribution m t of their positions in the state space at time t. While this already provides models with interesting features, a position-only interaction fails to capture important aspects of crowd motion. Indeedqsdf, given two agents at a certain distance from one another, it is natural to expect that their interaction will be qualitatively different depending on whether they are moving towards or away from one another, since in the first case they are expected to deviate in order to avoid a collision, while, in the second case, their movements are not expected to have a mutual interference. This motivates the study of MFG models for crowd motion in which the interaction through agents is based not only on their positions, but also on their velocities, or, equivalently for many models, on their controls.
Mean field game models in which the dependence on other agents takes into account not only their positions but also their controls are called MFGs of controls. Their analysis, started in [48,49], has been much developed in recent years (see, e.g., [16,22,29,42,51,52,60,61,78], and, in particular, the PhD thesis [59]). In the context of crowd motion, [78] has proposed a model 1 which assumes that agents evolve in a given set Ω ⊂ R d , that the trajectory γ of an agent choosing a control u is described by the control system γ(t) = u(t), and that an agent chooses their control u in order to minimize a cost of the form
where δ and λ are positive constants, η : R d → R + is an interaction kernel, T > 0 is a fixed time horizon, and Q is a probability measure on the set of all continuous trajectories C([0, T ], Ω) describing the distribution of trajectories of all agents. Given an initial distribution of agents m 0 , described mathematically as a probability measure on Ω, an equilibrium of this MFG model with initial condition m 0 is then a probability measure Q on C([0, T ], Ω) whose evaluation at time 0 is2 m 0 and which is concentrated on trajectories γ that are optimal for (1) with a fixed initial condition. It is proved in [78,Proposition 3.1] that this MFG model is variational (also called potential), i.e., that there exists a function J, defined over the space of probability measures on C([0, T ], Ω)
This content is AI-processed based on open access ArXiv data.