Mathematical Modeling of Intelligent Traffic Flow Using Mean Field Games

Abstract

Urban traffic congestion represents a multi-billion-dollar economic drain, a major contributor to global carbon emissions, and a profound engineering challenge for modern smart cities. Traditional traffic macroscopic and microscopic models often fail to capture the strategic, anticipatory behavior of modern drivers or autonomous vehicles (AVs). This paper presents a comprehensive framework utilizing Mean Field Games (MFG) to model intelligent traffic flow in an environment featuring mixed autonomy. By scaling individual driver decisions into a continuous distribution, MFG bypasses the computational intractability of classical N-player game theory. We formulate a coupled system consisting of a backward Hamilton-Jacobi-Bellman (HJB) equation, which dictates optimal trajectory and velocity planning for a representative agent, and a forward Fokker-Planck-Kolmogorov (FPK) equation, which governs the time evolution of the macroscopic traffic density. Uniquely, this work incorporates heterogeneous driver behavior categorized into aggressive, defensive, and autonomous agents and introduces a pricing-based routing mechanism designed to mitigate congestion at major bottlenecks. Numerical simulations demonstrate that the proposed MFG framework successfully eliminates phantom traffic jams, reduces average travel times by up to 22%, and smooths the velocity transition across bottleneck zones. The results offer valuable architectural insights for municipal traffic management systems and decentralized cooperative driving protocols in next-generation intelligent transportation networks.