# American Institute of Mathematical Sciences

December  2020, 25(12): 4869-4903. doi: 10.3934/dcdsb.2020131

## A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games

 1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, United States 2 Department of Civil Engineering and Engineering Mechanics and Data Science Institute, Columbia University, New York, NY 10027, United States 3 Department of Applied Physics and Applied Mathematics and Data Science Institute, Columbia University, New York, NY 10027, United States 4 Department of Computer Science, Columbia University, New York, NY 10027, United States

* Corresponding author: Xuan Di

Received  November 2019 Revised  January 2020 Published  April 2020

This paper proposes an efficient computational framework for longitudinal velocity control of a large number of autonomous vehicles (AVs) and develops a traffic flow theory for AVs. Instead of hypothesizing explicitly how AVs drive, our goal is to design future AVs as rational, utility-optimizing agents that continuously select optimal velocity over a period of planning horizon. With a large number of interacting AVs, this design problem can become computationally intractable. This paper aims to tackle such a challenge by employing mean field approximation and deriving a mean field game (MFG) as the limiting differential game with an infinite number of agents. The proposed micro-macro model allows one to define individuals on a microscopic level as utility-optimizing agents while translating rich microscopic behaviors to macroscopic models. Different from existing studies on the application of MFG to traffic flow models, the present study offers a systematic framework to apply MFG to autonomous vehicle velocity control. The MFG-based AV controller is shown to mitigate traffic jam faster than the LWR-based controller. MFG also embodies classical traffic flow models with behavioral interpretation, thereby providing a new traffic flow theory for AVs.

Citation: Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131
##### References:

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##### References:
From Micro to Macroscopic Traffic Flow Models
From an $N$-car differential game to MFG (adapted from [39])
Connections between MFG and LWR
[MFG-LWR]
Density Evolution of [MFG-NonSeparable] and [MFG-Separable]
Fundamental diagram of [MFG-NonSeparable]
Density, speed and optimal cost profiles for [MFG-NonSeparable] and [MFG-Separable] at $t = 0$ and $t = 1.5$
Convergence of solution algorithm in $L^1$ norm
$N = 21$ cars' trajectories integrated from the MFE solution of [MFG-NonSeparable]
MFE-constructed control cost v.s. best response strategy cost, $N = 21$ cars
Accuracy v.s. Number of cars
Classification of macroscopic traffic flow models
 Speed Acceleration rate Traditional First-order (e.g., LWR) Higher-order (e.g., PW/ARZ) Game-theoretic First-order MFGs Higher-order MFGs
 Speed Acceleration rate Traditional First-order (e.g., LWR) Higher-order (e.g., PW/ARZ) Game-theoretic First-order MFGs Higher-order MFGs
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