# Legal Framework for Rear-End Crashes in Mixed-Traffic Platooning: A Matrix Game Approach

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Contributions of This Paper

- We propose a legal framework that incorporates liability rules to the rear-end crash problems in mixed-traffic platoons.
- We leverage a matrix game approach for the rear-end crash problem to model interactions in three vehicle-encounter scenarios: HV-HV, AV-HV, and AV-AV scenarios.
- We perform sensitivity analysis and investigate what factors may impact the equilibrium results of the rear-end crash game.

## 2. Preliminaries

#### 2.1. Rear-End Crash in a Platoon

#### 2.2. Liability Rule

- No-fault rule [30]: The rule was first utilized to determine crash loss between automobiles in New York state. It is applied to any cyclist, pedestrian, passenger, or driver injured by a motor vehicle. There are now 12 states adopting the no-fault rule. The crash loss L is assigned to drivers, regardless of who is at fault in a car crash. Mathematically,$$\begin{array}{c}\hfill {S}_{N-1}={S}_{N}=\frac{1}{2}\end{array}$$
- Contributory rule [22]: The crash loss L is assigned to drivers according to the regime of negligence and non-negligence. We adopt the reaction time r to define the regime. Mathematically,$$\begin{array}{c}\hfill \left\{\begin{array}{c}{S}_{N-1}=0,{S}_{N}=1,if\phantom{\rule{4pt}{0ex}}{r}_{N-1}<\overline{r},{r}_{N}>\overline{r}\hfill \\ {S}_{N-1}=1,{S}_{N}=0,if\phantom{\rule{4pt}{0ex}}{r}_{N-1}>\overline{r},{r}_{N}<\overline{r}\hfill \\ {S}_{N-1}={S}_{N}=\frac{1}{2},if\phantom{\rule{4pt}{0ex}}({r}_{N-1}-\overline{r})({r}_{N}-\overline{r})\u2a7e0\hfill \end{array}\right.\end{array}$$$\overline{r}$ is a baseline to identify negligence and non-negligence conditions according to drivers’ reaction times. In this work, we assume $\overline{r}=1.5\mathrm{s}$, which is the average reaction time in brake-to-stop events obtained from real-world scenarios [12].
- Comparative rule [23,24]: The crash loss L is assigned to drivers according to their contributions to the rear-end crash. In this work, we use reaction time to measure drivers’ contributions to a car accident. Mathematically,$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {S}_{N-1}=\frac{{e}^{{r}_{N-1}}}{{e}^{{r}_{N-1}}+{e}^{{r}_{N}}},{S}_{N}=\frac{{e}^{{r}_{N}}}{{e}^{{r}_{N-1}}+{e}^{{r}_{N}}}\hfill \end{array}$$

## 3. Matrix Game Approach

#### 3.1. Assumptions

- The rear-end crash only happens between two vehicles. Crashes among three or more vehicles are not considered in this work.
- Vehicle $1,\dots ,N-2$ are not involved in the crash. They are non-strategic players whose reaction times are predetermined.
- All vehicles in the car platoon share the same initial velocity, break rate, and headway. We have: ${v}_{i}={v}_{0},{a}_{i}={a}_{0},{h}_{i}=\overline{h},i=1,\dots ,N$.
- Players in different encounter scenarios know whether their opponents are HVs or AVs. In other words, an AV’s reaction time in one scenario does not affect its choice in other scenarios. This is different from the assumption [14] that the decision making of AVs is predetermined by an AV manufacturer.

#### 3.2. Game Formulation

**HH scenario**: The rear-end crash happens between two human drivers. We specify elements in the rear-end crash game as follows:

- Players: Human drivers play a symmetric matrix game.
- Decision variables: Reaction time measures the level of precaution (i.e., care level [12,14]) for drivers when navigating roads. ${r}_{N-1}^{H}\in {R}^{H}$ and ${r}_{N}^{H}\in {R}^{H}$ represent the reaction time of human drivers for car $N-1$ and car N, respectively. ${R}^{H}=\{{\overline{r}}_{a}^{H},{\overline{r}}_{p}^{H}\}$ is a discrete feasible set for players. ${\overline{r}}_{a}^{H}$ indicates risk-averse behavior with a short reaction time, and ${\overline{r}}_{p}^{H}$ is risk-prone behavior with a long reaction time.
- Utility: The utility of drivers in the rear-end crash captures the effects of reaction time, and the crash loss assigned to drivers. ${U}_{N-1}^{\left(HH\right)}$ and ${U}_{N}^{\left(HH\right)}$ represent the utility of drivers in cars $N-1$ and N, respectively. We have$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {U}_{N-1}^{\left(HH\right)}({r}_{N-1}^{H},{r}_{N}^{H})={\beta}_{h}\xb7{f}_{h}\left({r}_{N-1}^{H}\right)+(1-{\beta}_{h})\xb7(-L\xb7{S}_{N-1}),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {U}_{N}^{\left(HH\right)}({r}_{N-1}^{H},{r}_{N}^{H})={\beta}_{h}\xb7{f}_{h}\left({r}_{N}^{H}\right)+(1-{\beta}_{h})\xb7(-L\xb7{S}_{N}),\hfill \end{array}$$
- Payoff Matrix: Given players’ decision variables and utility functions, we can formulate the payoff matrix for cars $N-1$ and N in the HH scenario as follows:
Car N ${\overline{r}}_{a}^{H}$ ${\overline{r}}_{p}^{H}$ Car $N-1$ ${\overline{r}}_{a}^{H}$ ${U}_{N-1}^{\left(HH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{a}^{H}),{U}_{N}^{\left(HH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{a}^{H})$ ${U}_{N-1}^{\left(HH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{p}^{H}),{U}_{N}^{\left(HH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{p}^{H})$ ${\overline{r}}_{p}^{H}$ ${U}_{N-1}^{\left(HH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{a}^{H}),{U}_{N}^{\left(HH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{a}^{H})$ ${U}_{N-1}^{\left(HH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{p}^{H}),{U}_{N}^{\left(HH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{p}^{H})$ - Game Equilibrium: At equilibrium, no human drivers can improve the utility by unilaterally changing the reaction time.

**AH scenario**: We now specify elements in the AH scenario:

- Players: An AV and an HV play an asymmetric matrix game. Note that there are two cases in the AH scenario: (car $N-1$, car N) is (HV, AV) and (car $N-1$, car N) is (AV, HV). For simplicity, we present the case when (car $N-1$, car N) is (HV, AV) in this section. In numerical experiments, we investigate both cases.
- Decision variables: ${r}_{N-1}^{H}\in {R}^{H}$ and ${r}_{N}^{A}\in {R}^{A}$ denote the reaction time. ${R}^{A}=\{{\overline{r}}_{a}^{A},{\overline{r}}_{p}^{A}\}$ and ${\overline{r}}_{a}^{A}$, ${\overline{r}}_{p}^{A}$ indicate risk-averse and risk-prone behaviors, respectively.
- Utility: The utility of cars $N-1$ and N is given by:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {U}_{N-1}^{\left(AH\right)}({r}_{N-1}^{H},{r}_{N}^{A})={\beta}_{h}\xb7{f}_{h}\left({r}_{N-1}^{H}\right)+(1-{\beta}_{h})\xb7(-L\xb7{S}_{N-1}),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {U}_{N}^{\left(AH\right)}({r}_{N-1}^{H},{r}_{N}^{A})={\beta}_{a}\xb7{f}_{a}\left({r}_{N}^{A}\right)+(1-{\beta}_{a})\xb7(-L\xb7{S}_{N}),\hfill \end{array}$$
- Payoff Matrix: Given players’ decision variables and utility functions, we can formulate the payoff matrix for cars $N-1$ and N in the AH scenario as follows:
Car N ${\overline{r}}_{a}^{A}$ ${\overline{r}}_{p}^{A}$ Car $N-1$ ${\overline{r}}_{a}^{H}$ ${U}_{N-1}^{\left(AH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{a}^{A}),{U}_{N}^{\left(AH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{a}^{A})$ ${U}_{N-1}^{\left(AH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{p}^{A}),{U}_{N}^{\left(AH\right)}({\overline{r}}_{a}^{H},{\overline{r}}_{p}^{A})$ ${\overline{r}}_{p}^{H}$ ${U}_{N-1}^{\left(AH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{a}^{A}),{U}_{N}^{\left(AH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{a}^{A})$ ${U}_{N-1}^{\left(AH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{p}^{A}),{U}_{N}^{\left(AH\right)}({\overline{r}}_{p}^{H},{\overline{r}}_{p}^{A})$ - Game Equilibrium: At equilibrium, no HV or AV can improve the utility by unilaterally changing reaction time.

**AA scenario**: We now specify elements in the AA scenario:

- Players: Two AVs play a symmetric game.
- Decision variables: ${r}_{N-1}^{A},{r}_{N}^{A}\in {R}^{A}$ denote the reaction time of AVs.
- Utility: The utility of cars $N-1$ and N is given by:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {U}_{N-1}^{\left(AA\right)}({r}_{N-1}^{A},{r}_{N}^{A})={\beta}_{a}\xb7{f}_{a}\left({r}_{N-1}^{A}\right)+(1-{\beta}_{a})\xb7(-L\xb7{S}_{N-1}),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {U}_{N}^{\left(AA\right)}({r}_{N-1}^{A},{r}_{N}^{A})={\beta}_{a}\xb7{f}_{a}\left({r}_{N}^{A}\right)+(1-{\beta}_{a})\xb7(-L\xb7{S}_{N}).\hfill \end{array}$$
- Payoff Matrix: Given players’ decision variables and utility functions, we can formulate the payoff matrix for cars $N-1$ and N in the AA scenario as follows:
Car N ${\overline{r}}_{a}^{A}$ ${\overline{r}}_{p}^{A}$ Car $N-1$ ${\overline{r}}_{a}^{A}$ ${U}_{N-1}^{\left(AA\right)}({\overline{r}}_{a}^{A},{\overline{r}}_{a}^{A}),{U}_{N}^{\left(AA\right)}({\overline{r}}_{a}^{A},{\overline{r}}_{a}^{A})$ ${U}_{N-1}^{\left(AA\right)}({\overline{r}}_{a}^{A},{\overline{r}}_{p}^{A}),{U}_{N}^{\left(AA\right)}({\overline{r}}_{a}^{A},{\overline{r}}_{p}^{A})$ ${\overline{r}}_{p}^{A}$ ${U}_{N-1}^{AA}({\overline{r}}_{p}^{A},{\overline{r}}_{a}^{A}),{U}_{N}^{\left(AA\right)}({\overline{r}}_{p}^{A},{\overline{r}}_{a}^{A})$ ${U}_{N-1}^{\left(AA\right)}({\overline{r}}_{p}^{A},{\overline{r}}_{p}^{A}),{U}_{N}^{\left(AA\right)}({\overline{r}}_{p}^{A},{\overline{r}}_{p}^{A})$ - Game Equilibrium: At equilibrium, no AVs can improve the utility by unilaterally changing reaction time.

**Remark 1.**

- 2.
- Mixed Nash equilibrium may exist in the rear-end crash game. For example, the mixed Nash equilibrium for an AV is to choose action ${\overline{r}}_{a}^{A}$ with probability p and ${\overline{r}}_{p}^{A}$ with probability $1-p$. We then use the average policy to denote the equilibrium ${r}^{A*}$ for the AV, i.e., ${r}^{A*}=p\xb7{\overline{r}}_{a}^{A}+(1-p)\xb7{\overline{r}}_{p}^{A}$.

#### 3.3. Performance Measure

**Definition 1.**

## 4. Numerical Experiments

- How do liability rules impact the equilibrium results in rear-end crash games?
- Under what circumstances does a moral hazard exist for human drivers in the platoon?
- What factors may influence the reaction time of HVs and AVs at equilibrium?

- There exists moral hazards for human drivers if risk-averse drivers are in the platoon. This is mainly because risk-averse drivers enlarge the available stopping distance, making drivers in the following vehicles less cautious in brake-to-stop events.
- Compared to HVs, AVs execute a smaller reaction time in rear-end crashes, indicating that AVs are more conservative than HVs. Human drivers tend to be less attentive by increasing their reaction time when encountering AVs.
- Compared to the no-fault rule, contributory and comparative rules make road users have more incentives to reduce their reaction time and improve the road safety in platooning.
- The reaction time at equilibrium has a positive relationship with the headway in the car platoon. A longer headway creates a safer driving environment where a longer reaction time can be executed.

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Chen, X.; Di, X.
Legal Framework for Rear-End Crashes in Mixed-Traffic Platooning: A Matrix Game Approach. *Future Transp.* **2023**, *3*, 417-428.
https://doi.org/10.3390/futuretransp3020025

**AMA Style**

Chen X, Di X.
Legal Framework for Rear-End Crashes in Mixed-Traffic Platooning: A Matrix Game Approach. *Future Transportation*. 2023; 3(2):417-428.
https://doi.org/10.3390/futuretransp3020025

**Chicago/Turabian Style**

Chen, Xu, and Xuan Di.
2023. "Legal Framework for Rear-End Crashes in Mixed-Traffic Platooning: A Matrix Game Approach" *Future Transportation* 3, no. 2: 417-428.
https://doi.org/10.3390/futuretransp3020025