Modeling of lane-changing behaviors of human-machine co-driving vehicles based on evolutionary game

Ronghan Yao; Xiaomei Wang; Bing Yang; Yang Chen; Feng Sun; Hu Zhang; Ronghan Yao; Xiaomei Wang; Bing Yang; Yang Chen; Feng Sun; Hu Zhang

doi:10.48130/dts-0026-0001

To alleviate the conflicts resulting from lane-changing in human-machine co-driving, this study investigates four game scenarios: human-driven lane-changing vehicles (HLVs) and human-driven target vehicles (HTVs), HLVs and Autonomous target vehicles (ATVs), autonomous lane-changing vehicles (ALVs) and HTVs, and ALVs and ATVs. An evolutionary game model is formulated by integrating safety, efficiency, comfort, fatigue, and energy-saving utilities. The strategy selection process of decision-makers is simulated by replicator dynamics equations. Based on the Friedman method, the stability of equilibrium points in different situations is analyzed, and the sensitivity of evolutionary results to utility parameters is clarified. Results show that the strategy profile of (lane-changing, not-giving-way) that causes conflicts only exists temporarily under certain conditions and will not become a long-term stable point. The game characteristics under different scenarios are obviously different; that is, the game evolutionary paths for autonomous vehicles are the most stable, while the adjustment of game strategies for human-driven vehicles is more frequent. Additionally, energy-saving promotes the formation of the strategy profile of (lane-changing, giving-way), while fatigue limits the frequent lane-changing of human-driven vehicles. The evolutionary game model proposed can provide theoretical support for improving the performance of the human-machine co-driving system.

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Introduction

In recent years, the National Development and Reform Commission, the Cyberspace Administration of China (CAC), and 11 other departments have issued The Strategy for Innovative Development of Intelligent Vehicles^[1], and the General Office of the Ministry of Transport has issued The Guidelines on Transport Safety Services of Autonomous Vehicles (Trial Implementation)^[2]. This clearly indicates that autonomous driving, driven by artificial intelligence, is a global innovation hotspot and an aim for future development. With the development of science and technology, the development and popularization of autonomous vehicles is expected to reduce traffic accidents^[3], to improve traffic efficiency^[4,5], and to reduce traffic emissions^[6,7]. Artificial intelligence has opened new possibilities for optimizing traffic operations^[8]. Autonomous driving is classified into six levels based on the degree of automation: L0 (no driving automation), L1 (driver assistance), L2 (partial driving automation), L3 (conditional driving automation), L4 (high driving automation), and L5 (full driving automation)^[9]. At present, autonomous driving technology has developed to L4, while L2 and L3 have become mature. However, due to immature technology and imperfect laws and regulations, there must be human-driven vehicles and different grades of autonomous vehicles in road traffic systems during a long period before fully autonomous driving is realized.

Driving decision is the key to ensuring that autonomous vehicles can accurately and smoothly complete various driving tasks^[10], and a very important decision of a vehicle is a lane-changing decision. Most of the existing studies on vehicle lane-changing decision-making focus on theoretical and practical discussions such as logical rules, machine learning, and game theory. Lane-changing models based on logical rules make decisions according to lane-changing intentions and conditions. These models have a simple structure and clear logic. The earliest rule-based lane-changing model was put forward by Gipps^[11]. Later, some scholars improved safety constraints^[12] and lane-changing benefits^[13], but such models fail to reflect the interaction between vehicles during lane-changing, and their decision-making accuracy relies on rule settings, which lack sufficient flexibility and struggle to adapt to complex traffic environments^[14].

Many scholars have begun using machine learning to train models on large-scale datasets, optimizing lane-changing models for vehicles to adapt to more complex traffic environments and enhance the accuracy of both decision-making and recognition^[15]. Mahajan et al.^[16] built an end-to-end machine learning model, which effectively captures the motivation of lane-changing for vehicles. On this basis, some scholars have proposed the MSIF-DRL strategy based on deep reinforcement learning and multi-source information fusion^[17,18], as well as a reward function model for neural network training^[19], thereby improving the efficiency and environmental adaptability of vehicle lane-changing decision-making. However, such models rely on large amounts of high-quality training data, which makes the modeling process complicated, parameters calibration difficult, and generalization to unknown scenarios limited^[20−23].

As an effective modeling tool, game theory can simulate and analyze the interactive behaviors and decision-making processes between human-machine co-driving vehicles^[24]. In recent years, many scholars have thoroughly explored lane-changing decision-making based on game theory. In the existing studies, the classical game theory constructs an interactive decision-making framework between vehicles through the collision time, acceleration utility, conflict risk, and other factors, and realizes the optimization of lane-changing strategy^[25−27]. It also introduces several methods, such as a quantum reaction equilibrium framework and receding horizon control^[28], which provide new perspectives on the application of game theory in autonomous driving. However, these studies assume that any driver is completely rational, which is inconsistent with the reality that any driver has limited rationality, and the decision is easily affected by fatigue, or other factors in a real driving environment. To overcome such problems in the classical game theory, some researchers put forward the lane-changing decision of autonomous vehicles based on the evolutionary game^[29,30]. In the aspect of the consideration of drivers' utilities, most studies consider three utilities, i.e., safety, efficiency, and comfort^[31,32], which more exactly describe the decision metrics.

Nevertheless, there are still some problems with the existing research. First, an adaptive framework is not designed for the human-machine co-driving scenarios, which cannot consider the heterogeneous decision-making needs of human-driven vehicles and autonomous vehicles. Second, although the existing research has made progress in the consideration of drivers' utilities, it ignores the key factors that affect the rationality of decision-making^[33−35], which leads to the incomplete design of utility functions, and the deviation between theoretical and real decision-making results. To solve such problems, the utilities and utility functions of human-driven vehicles and autonomous vehicles are described, and the heterogeneous decision-making and dynamic interaction behaviors of these two types of vehicles are further accurately expressed. At the same time, the evolutionary game model of lane-changing of human-driven or autonomous vehicles is formulated by integrating the safety, efficiency, comfort, fatigue, and energy-saving utilities, so that autonomous vehicles can adjust their lane changing strategies in real-time according to the surrounding environments. Thus, the performance of the human-machine co-driving system can be enhanced.

Evolutionary stable strategies

Equilibrium points

By solving the differential Eq. (15) about p and q, composed of Eqs (13) and (14), five equilibrium points of the game between LVs and TVs can be obtained:

(0,0) (not-lane-changing, not-giving-way) is a steady state where the benefit of lane-changing is lower than its risk cost, with the core purpose of avoiding ineffective lane-changing. (1,1) (lane-changing, giving-way) represents the optimal cooperative steady state for vehicles; by balancing multi-dimensional utilities, it can be translated into improved traffic flow efficiency. (0,1) (not-lane-changing, giving-way) is a transitional equilibrium, reflecting that a decision-maker who choose giving-way fails to enable effective lane-changing. (1,0) (lane-changing, not-giving-way) is a conflict-oriented temporary equilibrium that only exists momently under specific initial conditions. Such a point confirms the unstable nature of conflict strategies and underscores the rigid demand of the safety utility in transportation systems. The mixed-strategy equilibrium (x, y) is a probabilistic steady state in uncertain environments, where x and y correspond to the probabilities of LVs choosing to change lanes and TVs choosing to give way. Where (x,y) = $ (\dfrac{\Delta U_2^{\mathrm{TVs}}}{\Delta U_2^{\mathrm{TVs}}-\Delta U_1^{\mathrm{TVs}}},\dfrac{\Delta U_2^{\mathrm{LVs}}}{\Delta U_2^{\mathrm{LVs}}-\Delta U_1^{\mathrm{LVs}}}) $.

$ \left\{\begin{aligned} & \partial \textit{p}/\partial \textit{t}=0\\ &\partial q/\partial \textit{t}=0 \end{aligned}\right. ,\;\;p,\;q\in [0,1] $

(15)

Furthermore, the Jacobian matrix J of the dynamic game system is Eq. (16).

$ \begin{aligned} & J=\left(\begin{matrix} {{\dfrac{\partial {F}^{\mathrm{LVs}}(p,\textit{q})}{\partial p}}} & {{\dfrac{\partial {F}^{\mathrm{LVs}}(p,\textit{q})}{\partial q}}}\\ {{\dfrac{\partial {F}^{\mathrm{TVs}}(p,\textit{q})}{\partial p}}} &{{ \dfrac{\partial {F}^{\mathrm{TVs}}(p,\textit{q})}{\partial q}}} \end{matrix} \right)\\ &=\left(\begin{matrix} {{\left(1-2p\right)\left[\left(\Delta U_{1}^{\mathrm{LVs}}-\Delta U_{2}^{\mathrm{LVs}}\right)q+\Delta U_{2}^{\mathrm{LVs}}\right]}} & {{p\left(1-p\right)\left(\Delta U_{1}^{\mathrm{LVs}}-\Delta U_{2}^{\mathrm{LVs}}\right)}}\\ {{ q\left(1-q\right)\left(\Delta U_{1}^{\mathrm{TVs}}-\Delta U_{2}^{\mathrm{TVs}}\right)}} & {{\left(1-2q\right)\left[\left(\Delta U_{1}^{\mathrm{TVs}}-\Delta U_{2}^{\mathrm{TVs}}\right)p+\Delta U_{2}^{\mathrm{TVs}}\right]}} \end{matrix} \right) \end{aligned}$

(16)

Stability of equilibrium points

Whether an equilibrium point is stable, saddle, center, or unstable depends on the determinant and trace of the Jacobian matrix. According to the Friedman method for determining the type of an equilibrium point, when the determinant of the Jacobian matrix is greater than 0, and the trace is less than 0, the equilibrium point is a stable point, and the strategy chosen by each decision-maker at this time is an evolutionary stable strategy. From the above analyses, it can be seen that the signs of the determinant and trace of the Jacobian matrix depend on the signs of $ \Delta U_{1}^{\mathrm{LVs}} $, $ \Delta U_{2}^{\mathrm{LVs}} $, $ \Delta U_{1}^{\mathrm{TVs}} $ and $ \Delta U_{2}^{\mathrm{TVs}} $. Table 2 lists the types of the equilibrium points corresponding to 16 situations, and the combination of evolutionary stable strategies can be subdivided into six types, namely {(0,0), (0,1), (1,0), (1,1), (0,0) and (1,1), (0,1) and (1,0)}.

Table 2. Stability analyses of equilibrium points of evolutionary game.

Equilibrium points	Situation 1	Situation 2	Situation 3	Situation 4
	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $<0	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $<0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $<0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $<0
	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $<0	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $<0
(0,0)	Saddle point	Stable point	Saddle point	Stable point
(0,1)	Saddle point	Instability point	Stable point	Saddle point
(1,0)	Instability point	Instability point	Instability point	Instability point
(1,1)	Stable point	Stable point	Saddle point	Saddle point
(x,y)	—	Central point	—	—
Equilibrium points	Situation 5	Situation 6	Situation 7	Situation 8
	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $<0	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $<0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $<0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $<0
	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $<0	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $<0
(0,0)	Saddle point	Stable point	Saddle point	Stable point
(0,1)	Saddle point	Instability point	Stable point	Saddle point
(1,0)	Saddle point	Saddle point	Saddle point	Saddle point
(1,1)	Saddle point	Saddle point	Instability point	Instability point
(x,y)	Saddle point	—	—	—
Equilibrium points	Situation 9	Situation 10	Situation 11	Situation 12
	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $>0	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $>0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $>0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $>0
	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $<0	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $>0, $ \Delta U_{2}^{\mathrm{TVs}} $<0
(0,0)	Instability point	Saddle point	Instability point	Saddle point
(0,1)	Saddle point	Instability point	Stable point	Saddle point
(1,0)	Saddle point	Saddle point	Saddle point	Saddle point
(1,1)	Stable point	Stable point	Saddle point	Saddle point
(x,y)	—	—	—	Saddle point
Equilibrium points	Situation 13	Situation 14	Situation 15	Situation 16
	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $>0	$ \Delta U_{1}^{\mathrm{LVs}} $>0, $ \Delta U_{2}^{\mathrm{LVs}} $>0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $>0	$ \Delta U_{1}^{\mathrm{LVs}} $<0, $ \Delta U_{2}^{\mathrm{LVs}} $>0
	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $<0	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $>0	$ \Delta U_{1}^{\mathrm{TVs}} $<0, $ \Delta U_{2}^{\mathrm{TVs}} $<0
(0,0)	Instability point	Saddle point	Instability point	Saddle point
(0,1)	Saddle point	Instability point	Stable point	Saddle point
(1,0)	Stable point	Stable point	Stable point	Stable point
(1,1)	Saddle point	Saddle point	Instability point	Instability point
(x,y)	—	—	Central point	—

LVs make behavioral decisions according to the strategies corresponding to the evolutionary game results. Firstly, each decision-maker identifies the evolutionary stable strategy. Then, the decision-maker judges whether the currently selected strategy is an evolutionary stable strategy based on the current running state: if the strategy is stable, the decision-maker maintains the currently selected strategy; otherwise, the decision-maker selects the evolutionary stable strategy. If there are multiple evolutionary stable strategies, the decision-maker chooses the evolutionary stable strategy with the highest total utility.

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LVs	TVs
LVs	Giving-way	Not-giving-way
Lane-changing	$ (U_{11}^{\mathrm{LVs}},U_{11}^{\mathrm{TVs}}) $	$ (U_{12}^{\mathrm{LVs}},U_{12}^{\mathrm{TVs}}) $
Not-lane-changing	$ (U_{21}^{\mathrm{LVs}},U_{21}^{\mathrm{TVs}}) $	$ (U_{22}^{\mathrm{LVs}},U_{22}^{\mathrm{TVs}}) $

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Modeling of lane-changing behaviors of human-machine co-driving vehicles based on evolutionary game

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