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Article Contents

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

• * Corresponding author: Xuan Di
• 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.

Mathematics Subject Classification: Primary: 49N90, 90B20; Secondary: 35Q91.

 Citation:

• 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

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