Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper, we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow in the Wasserstein metric for an appropriate free-energy functional. Then we use the JKO approach to identify the asymptotics of the metric and the free-energy functional beyond the lowest order for single particle densities in the limit of small particle volumes by matched asymptotic expansions. While we use a propagation of chaos assumption at far distances, we consider correlations at small distance in the expansion. In this way, we obtain a clear picture of the emergence of a macroscopic gradient structure incorporating corrections in the free-energy functional due to the volume exclusion.

Uniformization transforms a pseudo-Poisson process with unequal intensities (leaving rates) into one with uniform intensity. Self-transitions is the price to pay. Two intensities arise when one considers an absorbing barrier of a Markov process as a body in its own right: a pair of Markov processes intertwined by an extended Chapman–Kolmogorov equation naturally arises. We show that Sauer's two-state space empathy theory handles such intertwined processes. The price of self-transitions is also avoided.

A new control strategy for a class of piecewise deterministic processes (PDP) is presented. In this class, PDP stochastic processes consist of ordinary differential equations that are subject to random switches corresponding to a discrete Markov process. The proposed strategy aims at controlling the probability density function (PDF) of the PDP. The optimal control formulation is based on the hyperbolic Fokker–Planck system that governs the time evolution of the PDF of the PDP and on tracking objectives of terminal configuration with a target PDF. The corresponding optimization problems are formulated as a sequence of open-loop hyperbolic optimality systems following a model predictive control framework. These systems are discretized by first-order schemes that guarantee positivity and conservativeness of the numerical PDF solution. The effectiveness of the proposed computational control framework is validated considering PDP with dichotomic noise.

We construct a Galerkin finite element method for the numerical approximation of weaksolutions to a general class of coupled FENE-type finitely extensible nonlinear elasticdumbbell models that arise from the kinetic theory of dilute solutions of polymericliquids with noninteracting polymer chains. The class of models involves the unsteadyincompressible Navier–Stokes equations in a bounded domainΩ ⊂ ℝd, d = 2 or 3, forthe velocity and the pressure of the fluid, with an elastic extra-stress tensor appearingon the right-hand side in the momentum equation. The extra-stress tensor stems from therandom movement of the polymer chains and is defined through the associated probabilitydensity function that satisfies a Fokker–Planck type parabolic equation, a crucial featureof which is the presence of a centre-of-mass diffusion term. We require no structuralassumptions on the drag term in the Fokker–Planck equation; in particular, the drag termneed not be corotational. We perform a rigorous passage to the limit as first the spatialdiscretization parameter, and then the temporal discretization parameter tend to zero, andshow that a (sub)sequence of these finite element approximations converges to a weaksolution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit isperformed under minimal regularity assumptions on the data: a square-integrable anddivergence-free initial velocity datum \hbox{$\absundertilde$}${\mathit{u}}_{\mathrm{0}}{\mathrm{~}}_{}$ for the Navier–Stokes equation and a nonnegative initial probabilitydensity function ψ0 for the Fokker–Planck equation, which hasfinite relative entropy with respect to the Maxwellian M.

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutionsof polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ $\mathbb{R}^d$ , d = 2 or 3, for the velocity andthe pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function $\beta^L(\cdot) :=\min(\cdot,L)$ in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker–Planck–Navier–Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions.We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms,and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm.

We recast Diamond's search equilibrium model into that with a finite number of agents. The state of the model is described by a jump-Markov process, the transition rates of which are functions of the reservation cost, which are endogenously determined by value maximization by rational agents. The existence of stochastic fluctuations causes the fraction of the employed to move from one basin of attraction to the other with positive probabilities when the dynamics have multiple equilibria. Stochastic asymmetric cycles that arise are quite different from the cycles of the set of Diamond–Fudenberg nonlinear deterministic differential equations. By taking the number of agents to infinity, we get a limiting probability distribution over the stationary state equilibria. This provides a natural basis for equilibrium selection in models with multiple equilibria, which is new in the economic literature.