# Topics in the mean-field type approach to pedestrian crowd modeling and conventions

**Time: **
Mon 2019-12-16 10.00

**Location: ** Kollegiesalen, Brinellvägen 8, Stockholm (English)

**Doctoral student: **
Alexander Aurell
, Matematisk statistik

**Opponent: **
Professor Roland Malhamé, Polytechnique Montréal, Montréal, Canada

**Supervisor: **
Professor Boualem Djehiche, Matematisk statistik; Professor Xiaoming Hu, Optimeringslära och systemteori

## Abstract

This thesis consists of five appended papers, primarily addressingtopics in pedestrian crowd modeling and the formation of conventions.The first paper generalizes a pedestrian crowd model for competingsubcrowds to include nonlocal interactions and an arbitrary (butfinite) number of subcrowds. Each pedestrian is granted a ’personalspace’ and is effected by the presence of other pedestrians within it.The interaction strength may depend on subcrowd affinity. The paperinvestigates the mean-field type game between subcrowds and derivesconditions for the reduction of the game to an optimization problem.The second paper suggest a model for pedestrians with a predeterminedtarget they have to reach. The fixed and non-negotiablefinal target leads us to formulate a model with backward stochasticdifferential equations of mean-field type. Equilibrium in the game betweenthe tagged pedestrians and a surrounding crowd is characterizedwith the stochastic maximum principle. The model is illustrated by anumber of numerical examples.The third paper introduces sticky reflected stochastic differentialequations with boundary diffusion as a means to include walls andobstacles in the mean-field approach to pedestrian crowd modeling.The proposed dynamics allow the pedestrians to move and interactwhile spending time on the boundary. The model only admits a weaksolution, leading to the formulation of a weak optimal control problem.The fourth paper treats two-player finite-horizon mean-field typegames between players whose state trajectories are given by backwardstochastic differential equations of mean-field type. The paper validatesthe stochastic maximum principle for such games. Numericalexperiments illustrate equilibrium behavior and the price of anarchy.The fifth paper treats the formation of conventions in a large populationof agents that repeatedly play a finite two-player game. Theplayers access a history of previously used action profiles and form beliefson how the opposing player will act. A dynamical model wheremore recent interactions are considered to be more important in thebelief-forming process is proposed. Convergence of the history to acollection of minimal CURB blocks and, for a certain class of games,to Nash equilibria is proven.