[Paper Review] Optimal Control of the Mean Field Equilibrium for a Pedestrian Tourists' Flow Model

Fabio Bagagiolo, Silvia Faggian+2 more|arXiv (Cornell University)|Jan 1, 2022

Transportation Planning and Optimization34 references15 citations

TL;DR

This paper formulates a mean field game model for tourist flow in heritage cities, using switching dynamics and continuous controls to represent pedestrian movement across attractions on a network. It proves existence of a mean field equilibrium and solves an optimal control problem where a city authority tunes congestion costs to steer flow toward a desired distribution, ensuring convergence to a target state via parameter optimization.

ABSTRACT

Art heritage cities are popular tourist destinations but for many of them overcrowding is becoming an issue. In this paper, we address the problem of modeling and analytically studying the flow of tourists along the narrow alleys of the historic center of a heritage city. We initially present a mean field game model, where both continuous and switching decisional variables are introduced to respectively describe the position of a tourist and the point of interest that he/she may visit. We prove the existence of a mean field equilibrium. A mean field equilibrium is Nash-type equilibrium in the case of infinitely many players. Then, we study an optimization problem for an external controller who aims to induce a suitable mean field equilibrium.

Motivation & Objective

• To model tourist flow in narrow historic city centers using a mean field game with switching and continuous controls.
• To prove existence of a mean field equilibrium under relaxed assumptions compared to prior work.
• To formulate and solve an optimal control problem where a city authority adjusts congestion cost parameters to guide tourist distribution toward a desired target.

Proposed method

• Models tourist movement on a circular network with three nodes: station, and two attractions.
• Uses continuous control (velocity) and switching variables (1=unvisited, 0=visited) to represent agent state.
• Defines a cost function combining travel effort, congestion, unvisited attractions, and late return penalties.
• Establishes a mean field equilibrium as a fixed point of a map linking distribution to optimal control and back.
• Optimizes controller parameters (α, β) in congestion cost functions to minimize deviation from a target distribution.
• Proves existence of optimal control parameters via compactness and weak-star convergence arguments.

Experimental results

Research questions

• RQ1 Does a mean field equilibrium exist for a pedestrian tourist flow model with switching dynamics and time-varying congestion costs?
• RQ2 Can a city authority optimally control tourist distribution by tuning congestion cost parameters in a mean field game framework?
• RQ3 Is the optimal control problem for congestion cost parameters well-posed, with a solution existing in a compact parameter space?

Key findings

• A mean field equilibrium exists for the proposed tourist flow model under suitable assumptions, generalizing prior results with relaxed conditions.
• The optimal control problem for congestion cost parameters admits a solution, with existence of an optimal pair (α, β) ∈ K proven via compactness and convergence arguments.
• The controller can steer the system toward a target distribution by adjusting (α, β), minimizing uniform deviation from the reference flow.
• Time-dependent congestion costs, piecewise constant over intervals, are compatible with the model and do not disrupt equilibrium existence.
• The approach remains valid when parameters are updated at branch entrances, reflecting real-time control via gates or signals.
• A variant of the problem with worst-case objective (supremum norm) lacks guaranteed existence due to potential non-uniqueness of equilibria.