arXiv subjects

Alexander Aurell

Publications and source records attributed to Alexander Aurell.

7 recordsLinked to original sources

arXiv2021

Finite State Graphon Games with Applications to Epidemics

We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented by a graphon, which can be viewed as the limit of a dense random graph. The player's transition rates between the states depend on their own control and the interaction strengths with the other players. We develop a rigorous mathematical framework for this game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential equations characterizing equilibria. Moreover, we propose a numerical approach based on machine learning tools and show experimental results on different applications to compartmental models in epidemiology.

math.OC↗

arXiv2021

Stochastic Stability of a Recency Weighted Sampling Dynamic

We introduce and study a model of long-run convention formation for rare interactions. Players in this model form beliefs by observing a recency-weighted sample of past interactions, to which they noisily best respond. We propose a continuous state Markov model, well-suited for our setting, and develop a methodology that is relevant for a larger class of similar learning models. We show that the model admits a unique asymptotic distribution which concentrates its mass on some minimal CURB block configuration. In contrast to existing literature of long-run convention formation, we focus on behavior inside minimal CURB blocks and provide conditions for convergence to (approximate) mixed equilibria conventions inside minimal CURB blocks.

econ.TH↗

arXiv2021

Stochastic Graphon Games: II. The Linear-Quadratic Case

In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.

math.PR↗

arXiv2021

Optimal incentives to mitigate epidemics: a Stackelberg mean field game approach

Motivated by models of epidemic control in large populations, we consider a Stackelberg mean field game model between a principal and a mean field of agents evolving on a finite state space. The agents play a non-cooperative game in which they can control their transition rates between states to minimize an individual cost. The principal can influence the resulting Nash equilibrium through incentives so as to optimize its own objective. We analyze this game using a probabilistic approach. We then propose an application to an epidemic model of SIR type in which the agents control their interaction rate and the principal is a regulator acting with non pharmaceutical interventions. To compute the solutions, we propose an innovative numerical approach based on Monte Carlo simulations and machine learning tools for stochastic optimization. We conclude with numerical experiments by illustrating the impact of the agents' and the regulator's optimal decisions in two models: a basic SIR model with semi-explicit solutions and a more complex model with a larger state space.

math.OC↗

arXiv2020

Behavior near walls in the mean-field approach to crowd dynamics

This paper introduces a system of stochastic differential equations (SDE) of mean-field type that models pedestrian motion. The system lets the pedestrians spend time at, and move along, walls, by means of sticky boundaries and boundary diffusion. As an alternative to Neumann-type boundary conditions, sticky boundaries and boundary diffusion have a 'smoothing' effect on pedestrian motion. When these effects are active, the pedestrian paths are semimartingales with first-variation part absolutely continuous with respect to the Lebesgue measure $dt$, rather than an increasing processes (which in general induces a measure singular with respect to $dt$) as is the case under Neumann boundary conditions. We show that the proposed mean-field model for pedestrian motion admits a unique weak solution and that it is possible to control the system in the weak sense, using a Pontryagin-type maximum principle. We also relate the mean-field type control problem to the social cost minimization in an interacting particle system. We study the novel model features numerically and we confirm empirical findings on pedestrian crowd motion in congested corridors.

math.OC↗

arXiv2019

Modeling tagged pedestrian motion: a mean-field type game approach

This paper suggests a model for the motion of tagged pedestrians: pedestrians moving towards a specified targeted destination, which they are forced to reach. It aims to be a decision-making tool for the positioning of fire fighters, security personnel, and other services in a pedestrian environment. Taking interaction with the surrounding crowd into account leads to a differential nonzero-sum game model where the tagged pedestrians compete with the surrounding crowd of ordinary pedestrians. When deciding how to act, pedestrians consider crowd distribution-dependent effects, like congestion and crowd aversion. Including such effects in the parameters of the game, makes it a mean-field type game. The equilibrium control is characterized, and special cases are discussed. Behavior in the model is studied by numerical simulations.

math.OC↗

arXiv2018

Mean-field type modeling of nonlocal crowd aversion in pedestrian crowd dynamics

We extend the class of pedestrian crowd models introduced by Lachapelle and Wolfram (2011) to allow for nonlocal crowd aversion and arbitrarily but finitely many interacting crowds. The new crowd aversion feature grants pedestrians a "personal space" where crowding is undesirable. We derive the model from a particle picture and treat it as a mean-field type game. Solutions to the mean-field type game are characterized via a Pontryagin-type Maximum Principle. The behavior of pedestrians acting under nonlocal crowd aversion is illustrated by a numerical simulation.

math.OC↗