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  December 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131
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Figure 1.  From Micro to Macroscopic Traffic Flow Models
Figure 2.  From an $ N $-car differential game to MFG (adapted from [39])
Figure 3.  Connections between MFG and LWR
Figure 4.  [MFG-LWR]
Figure 5.  Density Evolution of [MFG-NonSeparable] and [MFG-Separable]
Figure 6.  Fundamental diagram of [MFG-NonSeparable]
Figure 7.  Density, speed and optimal cost profiles for [MFG-NonSeparable] and [MFG-Separable] at $ t = 0 $ and $ t = 1.5 $
Figure 8.  Convergence of solution algorithm in $ L^1 $ norm
Figure 9.  $ N = 21 $ cars' trajectories integrated from the MFE solution of [MFG-NonSeparable]
Figure 10.  MFE-constructed control cost v.s. best response strategy cost, $ N = 21 $ cars
Figure 11.  Accuracy v.s. Number of cars
Table 1.  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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