# A Study on Pedestrian–Vehicle Conflict at Unsignalized Crosswalks Based on Game Theory

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Characteristic Analysis of Traffic Participants on Unsignalized Sections

#### 3.1. Traffic Characteristic Analysis

#### 3.1.1. Pedestrian Crossing Walking Speed

#### 3.1.2. Characteristics of Pedestrians When Crossing the Street

#### 3.1.3. Risk Assessment and Waiting Delay

#### 3.2. Behavioral Characteristic Analysis

#### 3.2.1. Pedestrian Behavior Characteristics

#### 3.2.2. Driver Behavior Characteristics

#### 3.3. Psychological Characteristic Analysis

## 4. Pedestrian–Vehicle Dynamic Game Model

#### 4.1. Notation

#### 4.2. Utility Values of Pedestrian Crossing Decision Behavior

#### 4.3. Pedestrian–Vehicle Dynamic Game Model and Solution

#### 4.3.1. Game Model of Pedestrian First Decision and Equilibrium Solution

_{R}

_{1}, P

_{R}

_{2}and P

_{R}

_{3}(i.e., the initial phase of the pedestrian), and information on the stage’s probability distribution of pedestrian A is shared with vehicle B. Pedestrian A decides first to “cross” or “wait” according to the state of the pedestrian and road conditions, while vehicle B decides after receiving pedestrian A’s decision information to get the utility payment ${u}_{1}({P}_{Ri};({c}_{1},{c}_{2}))\text{}and\text{}{u}_{2}({P}_{Ri};({c}_{1},{c}_{2}))$, as shown in Figure 6a.

#### 4.3.2. Game Model of Vehicle First Decision and Equilibrium Solution

- (1)
- When the relationship of probability between pedestrians and vehicles satisfies $3{P}_{R1}+{P}_{R2}<3{P}_{R3}$, the strategy combination is (P, W), that is, the vehicle chooses to “pass” first and the pedestrian decides to “wait” after observing vehicles passing without slowing down, which ensure safe passage of vehicles with minimal risk to the pedestrian.
- (2)
- When $3{P}_{R1}+{P}_{R2}<3{P}_{R3}$, the strategy combination is (A, T), which means that the vehicle chooses to “avoid” first and then the pedestrian chooses to “cross”, so that the pedestrian can cross the street safely with lower risk.
- (3)
- When $3{P}_{R1}+{P}_{R2}=3{P}_{R3}$, the equilibrium is the game equilibrium of the vehicle’s first decision under the mixed strategy, and the equilibrium solution is the probability of strategy selection. The specific strategy combination is illustrated in Table 8.

## 5. Conclusions and Prospects

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**General form of the unfolding game. (

**a**) Pedestrian first decision. (

**b**) Vehicle first decision.

(a) Number of pauses for street crossings | |||||

Number of stops | 0 | 1 | 2 | 3 | More than 3 |

Proportion | 12.28% | 39.30% | 33.33% | 9.47% | 5.61% |

(b) Waiting time for street crossings | |||||

Waiting time/s | less than 10 | 10–20 | 20–30 | 30–45 | more than 45 |

Proportion | 18.95% | 42.46% | 27.72% | 8.77% | 2.11% |

(c) The relationship between waiting time and rush crossing | |||||

Waiting time/s | less than 10 | 10–20 | 20–30 | 30–45 | more than 45 |

Rush crossing ratio | 1.40% | 8.42% | 32.63% | 27.37% | 30.18% |

(a) Tensions that arise when crossing the street | ||||

Frequency | Often | Sometimes | Seldom | |

Proportion | 27.37% | 41.40% | 31.23% | |

(b) Crossing the street with a tension psychology | ||||

Mode | Average speed | Adjustment at any time | Fast Crossing | |

Proportion | 35.90% | 16.67% | 47.43% | |

(c) Options for tense people at risk of conflict | ||||

Options | Fast Crossing | Back off | Stop in place | |

Proportion | 11.54% | 80.77% | 7.69% | |

(d) Average daily number of street crossings for tense people | ||||

Number | 0 to 2 times | 2 to 6 times | More than 6 times | |

Proportion | 71.93% | 22.46% | 5.61% | |

(e) Waiting time for tense crowds | ||||

Waiting time | 30 to 60 s | 60 to 90 s | 90 to 120 s | More than 120 s |

Proportion | 10.18% | 24.91% | 23.51% | 41.40% |

(a) Proportion of people choosing to cross the street in groups | ||||

Scale | One person | 2 to 4 people | Group | |

Proportion | 55.09% | 32.28% | 12.63% | |

(b) Number of road observations under group crossing | ||||

Number | Frequently | Lower than single person | Rarely | |

Proportion | 43.16% | 51.93% | 4.91% | |

(c) Group waiting time for crossing the street | ||||

Waiting time | 30 to 60 s | 60 to 90 s | 90 to 120 s | More than 120 s |

Proportion | 5.56% | 25.00% | 16.67% | 52.77% |

Symbol | Definition |
---|---|

A and P | Decision making by vehicles in front of unsignalized crosswalks; A stands for “avoid”; P stands for “pass”; |

W and T | Decision making by pedestrian; W stands for “wait”; T stands for “ cross”; |

${v}_{i}(\sigma )$ | Expected utility function of game subject i; |

${u}_{i}(s)$ | Expected utility of game subject i; |

${c}_{i}$ | Decision numbers for pedestrians and vehicles crossing the street; |

${P}_{Ri}$ | Probability of waiting stage i for pedestrians, according to different waiting times; i = 1, 2, 3; |

${\sigma}_{i}$ | Decision-making subject strategy options; |

${a}_{Ri}\text{}\mathrm{and}\text{}{a}_{ci}$ | Represent the utility payment values for pedestrian and vehicle delay payments, respectively; |

${b}_{Ri}\text{}\mathrm{and}\text{}{b}_{ci}$ | Represent the utility payment values for pedestrian and vehicle risk payments, respectively; |

${c}_{Ri}\text{}\mathrm{and}\text{}{c}_{ci}$ | Represents the utility payment value for safe pedestrian and vehicle passage, respectively; |

${s}_{i}$ | Set of selection strategies for decision subjects; |

T’ | Waiting time for pedestrians to cross the street; |

${t}_{i}$ | Critical time range values for different waiting phases; i = 1, 2, 3; |

d | Payment utility when choosing a particular strategy; |

X and Y | Under the game model of pedestrian first decision, the probability that the decision maker chooses a certain strategy; X and Y represent pedestrian and vehicle, respectively; |

${q}_{1}\text{}\mathrm{and}\text{}{q}_{2}$ | Under the game model of vehicle first decision, the probability that the decision maker chooses a certain strategy; q_{1} and q_{2} represent pedestrian and vehicle, respectively; |

Vehicles | A(avoid) | P(pass) | |
---|---|---|---|

Pedestrian | |||

W(wait) | −1, −1 | −1, 1 | |

T(cross) | 1, −1 | −4, −4 |

Stages | Spacing Range | Critical Point Range | Percentage |
---|---|---|---|

The first stage | 0 < t ≤ t_{1} | 15 s <t_{1}≤ 25 s | P_{R1} |

The second stage | t_{1} < t ≤ t_{2} | 25 s <t_{2} ≤ 40 s | P_{R2} |

The third stage | t_{2} < t ≤ t_{3} | 40 s <t_{3} ≤ 60 s | P_{R3} |

Stages | The First Stage | The Second Stage | The Third Stage | |
---|---|---|---|---|

Delay payments ${a}_{Ri}$ | −1 | −2 | −4 | |

Pedestrian | Risk payment ${b}_{Ri}$ | −3 | −5 | −6 |

Safe passage ${c}_{Ri}$ | 1 | 1 | 1 | |

Delay payments ${a}_{ci}$ | −4 | −3 | −1 | |

Vehicles | Risk payment ${b}_{ci}$ | −7 | −9 | −10 |

Safe passage ${c}_{ci}$ | 1 | 1 | 1 |

Strategy Portfolio | Probability of Different Strategy Combinations | |
---|---|---|

Pedestrian First Decision | Vehicle First Decision | |

${c}_{1}(T),{c}_{2}(P)$ | $(\frac{2.42+0.43\xb7d}{-1.71-0.57\xb7d},0.67)$ | $(\frac{{P}_{R1}+2{P}_{R2}+4{P}_{R3}}{{P}_{R2}+3{P}_{R3}},1-\frac{4{P}_{R1}+3{P}_{R2}+{P}_{R3}}{3{P}_{R1}+2{P}_{R2}})$ |

${c}_{1}(T),{c}_{2}(A)$ | $(\frac{2.42+0.43\xb7d}{-1.71-0.57\xb7d},0.33)$ | $(1-\frac{{P}_{R1}+2{P}_{R2}+4{P}_{R3}}{{P}_{R2}+3{P}_{R3}},1-\frac{4{P}_{R1}+3{P}_{R2}+{P}_{R3}}{3{P}_{R1}+2{P}_{R2}})$ |

${c}_{1}(W),{c}_{2}(P)$ | $(1+\frac{2.42+0.43\xb7d}{+1.71+0.57\xb7d},0.33)$ | $(\frac{{P}_{R1}+2{P}_{R2}+4{P}_{R3}}{{P}_{R2}+3{P}_{R3}},\frac{4{P}_{R1}+3{P}_{R2}+{P}_{R3}}{3{P}_{R1}+2{P}_{R2}})$ |

${c}_{1}(W),{c}_{2}(A)$ | $(1+\frac{2.42+0.43\xb7d}{1.71+0.57\xb7d},0.67)$ | $(1-\frac{{P}_{R1}+2{P}_{R2}+4{P}_{R3}}{{P}_{R2}+3{P}_{R3}},\frac{4{P}_{R1}+3{P}_{R2}+{P}_{R3}}{3{P}_{R1}+2{P}_{R2}})$ |

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**MDPI and ACS Style**

Sun, X.; Lin, K.; Wang, Y.; Ma, S.; Lu, H.
A Study on Pedestrian–Vehicle Conflict at Unsignalized Crosswalks Based on Game Theory. *Sustainability* **2022**, *14*, 7652.
https://doi.org/10.3390/su14137652

**AMA Style**

Sun X, Lin K, Wang Y, Ma S, Lu H.
A Study on Pedestrian–Vehicle Conflict at Unsignalized Crosswalks Based on Game Theory. *Sustainability*. 2022; 14(13):7652.
https://doi.org/10.3390/su14137652

**Chicago/Turabian Style**

Sun, Xu, Kun Lin, Yu Wang, Shuo Ma, and Huapu Lu.
2022. "A Study on Pedestrian–Vehicle Conflict at Unsignalized Crosswalks Based on Game Theory" *Sustainability* 14, no. 13: 7652.
https://doi.org/10.3390/su14137652