^{1}

^{2}

^{1}

^{3}

^{1}

^{1}

^{2}

^{3}

The conflict between pedestrians and vehicles plays a significant role in influencing the efficiency of intersections. In turn, the effectiveness of intersections greatly affects the entire network. Statistical data indicates that up to 70% of people move in groups (such as friends, couples, or families walking together). The pedestrian group-crossing behavior, as well as an analysis of the dynamics between groups of pedestrians and vehicles at unsignalized intersections, deserves a thorough study. In this paper, a model based on the multidimensional dirty faces game is proposed to analyze the crossing behavior of pedestrians and vehicles as “rational people.” Computer simulations were performed to investigate the effect of the group size on crossing behavior and conflict risks. The relationship between heterogeneity of waiting time and walking speed is also investigated. These findings can be used to advance understanding on the “Chinese style road crossing” phenomenon and elucidate the dynamics involved.

Pedestrians are among the most vulnerable road users. Traffic accidents involving pedestrians have become a major problem, especially in developing countries [

With a population of more than 1.37 billion, China has her traffic characteristics. First, the existing road traffic planning may not provide adequate safety for pedestrians in some cases. Hence, mixed traffic consisting of motorized vehicles and pedestrians is ubiquitous. Second, unsignalized crosswalks widely exist in small cities and suburbs. Third, motorists are supposed to give way to pedestrians at unsignalized crosswalks, but these motorists do not always comply due to a general lack of safety consciousness [

Besides motion characteristics, behavioral characteristics like the cognitive, social, or even psychological behavior need to be properly accounted for in the model, to investigate the crossing behavior of pedestrians and vehicles thoroughly. Hence, modeling the pedestrian crossing behavior from the perspective of behavioral science is an essential process. Game theory is a subject of coordination of conflicts, suitable for studying crossing behavior. Although some studies have utilized game theory to investigate the crossing behavior [

It is worth noting that it has long been recognized that group behavior is not simply the sum of individual behaviors [

In reality, pedestrians and vehicles have to compete for limited road resources in a road-crossing scenario and this can be modeled as a “game” between pedestrians and vehicles. Through an in-depth analysis of the interference mechanism between

The dirty faces game is well-studied and usually found in the literature on iterated reasoning. It plays a central role in most discussions of common knowledge [

In our study, we consider pedestrians and vehicle drivers as the two players of the game, where each player has two choices: strategy 1 is that the vehicle lets the pedestrian go first, and strategy 2 is that the vehicle goes first. The strategy chosen by the two players in areas of potential conflict will directly lead to the changes in intersection traffic conditions. By simple deduction, the following scenarios can be inferred.

(i) Both players choose strategy 1 (the vehicle lets the pedestrian go first); then the pedestrian decides to pass the intersection.

(ii) Both sides choose strategy 2 (the pedestrian lets the vehicle go first); then the vehicle decides to cross the intersection.

(iii) The pedestrian adopts strategy 1 and the vehicle adopts strategy 2; then the pedestrian and vehicle collide (this situation is very dangerous).

(iv) The pedestrian adopts strategy 2, while the vehicle adopts strategy 1; then both pedestrian and vehicle give way to the other (this situation is especially inefficient).

Through the above analysis, it is straightforward to see that both pedestrian and vehicle adopting strategy 1 or strategy 2 at the same time are the kind of situation that we would like to have. The key question is, how do we let pedestrian and vehicle adopt the same strategy simultaneously? The best approach is to make strategy 1 or strategy 2 becomes the common knowledge for both sides. Since both sides cannot communicate directly when they cross the road; therefore, they do not know which strategy his/her opponent will choose. At this moment, the choice between vehicle allowing the pedestrian to go first or the pedestrian allowing the vehicle to go first is not the common knowledge for both parties. Therefore, some factors that drive towards the formation of common knowledge must exist, and the signal lights in the intersection are exactly this driving factor. When the traffic light in the crosswalk turns green, the pedestrian sees that and adopts strategy 1; the vehicle also sees that and adopts strategy 1. The pedestrian also knows that the vehicle will adopt strategy 1, while the vehicle knows that the pedestrian will adopt strategy 1. The pedestrian further knows that the vehicle is aware that the pedestrian will adopt strategy 1; the vehicle also knows that the pedestrian is aware that the vehicle will adopt strategy 1 as well. At this time, “the vehicle lets the pedestrian go first” becomes common knowledge for both sides. When the traffic light turns red, the analysis is the same as what we have just discussed. The only difference is that “the pedestrian lets the vehicle go first” becomes common knowledge for both sides at this time.

Since unsignalized crosswalks are ubiquitous in small cities in China, we now focus on the formation processes of common knowledge at these unsignalized crosswalks. Bayer et al. concluded that three elements are needed to form the common knowledge: (i) each player should generate a dominant strategy first, (ii) each player can assess and analyze the dominant strategy of the others, and (iii) each player can announce the dominant strategy of their own immediately. In order to obtain common knowledge at unsignalized intersections, pedestrian and vehicle usually produce and announce the dominant strategy of their own first according to the traffic condition and then assess his/her opponent’s dominant strategy. The announcement and assessment cannot be explained in words and can only be shown in acceleration, deceleration, horn honking, or gesturing. The interactive announcement and assessment may be more than one round. Sometimes, several rounds of the game are needed to achieve common knowledge, and common knowledge may not even be fully achieved in some instances.

The “Chinese style road crossing” refers to the behavior of Chinese pedestrians crossing an intersection, not by following the traffic signals but whether there is a sufficient number of pedestrians who feel safe to cross [

For a pedestrian group which consists of

The probability that the vehicle driver selects “the pedestrian gives priority to vehicle” as the dominant strategy is given by

Generate

When both pedestrians and driver enter the one-step mode, they further announce the dominant strategy of their own via acceleration or deceleration, and they may also recognize the dominant strategy of the other. As such, we define two parameters

Now, we generate

Generate

Figure

Schematic diagram to illustrate the multidimensional dirty faces game of pedestrian group-crossing behavior.

Due to differences in arrival times and traffic conditions, different pedestrians may experience different waiting times. When the waiting time increases, pedestrians become impatient and try to cross the street impetuously [

Other parameters remain unchanged. Simulations were performed to investigate the effect of the waiting time on the crossing behavior of pedestrians. In the simulation model, it is assumed that each pedestrian in the group has the same theoretical passing time (i.e., they have the same walking speed) and estimation accuracy, where the estimated passing times satisfy the same normal distribution.

The simulation was performed under different combinations of

The effect of the group size on the conflict probability in consideration of the heterogeneity of the waiting time (the number of pedestrians is 1, 2, 3, 5, 10, and 20, from (a) to (f) graph).

In order to further explore the effect of the group size on the decision mechanism of crossing behavior, we tracked the number of different crossing results for two combinations of

Number (out of the sample of 1 million) of pedestrians/vehicles passing through the intersection under different modes (

Number of pedestrians | Zero-step mode | One-step mode | Two-step mode | Failure mode | ||||
---|---|---|---|---|---|---|---|---|

Pedestrians pass | Vehicles pass | Pedestrians pass | Vehicles pass | Pedestrians pass | Vehicles pass | Stagnation | Collision | |

1 | 104861 | 66740 | 320947 | 319672 | 40807 | 36828 | 99449 | 10696 |

2 | 205403 | 32584 | 464945 | 139396 | 45129 | 23831 | 79336 | 9376 |

3 | 293856 | 19750 | 510476 | 65228 | 36746 | 13827 | 53544 | 6573 |

5 | 445324 | 9261 | 477980 | 16943 | 19625 | 4588 | 23455 | 2824 |

10 | 691310 | 2405 | 291102 | 1362 | 4914 | 534 | 7907 | 466 |

20 | 906194 | 322 | 91674 | 52 | 1044 | 26 | 672 | 16 |

Number (out of the sample of 1 million) of pedestrians/vehicles passing through the intersection under different modes (

Number of pedestrians | Zero-step mode | One-step mode | Two-step mode | Failure mode | ||||
---|---|---|---|---|---|---|---|---|

Pedestrians pass | Vehicles pass | Pedestrians pass | Vehicles pass | Pedestrians pass | Vehicles pass | Stagnation | Collision | |

1 | 10360 | 239795 | 37148 | 652438 | 10692 | 14407 | 31766 | 3394 |

2 | 21910 | 179492 | 74377 | 617891 | 19958 | 23732 | 56776 | 5864 |

3 | 47495 | 166282 | 94645 | 569087 | 23235 | 28254 | 65019 | 5983 |

5 | 80325 | 134685 | 149506 | 472117 | 32868 | 34685 | 88424 | 7390 |

10 | 141327 | 85536 | 221891 | 265069 | 39024 | 32741 | 200580 | 13832 |

20 | 268657 | 54876 | 303924 | 115707 | 38807 | 23802 | 183627 | 10600 |

Table

As shown in Table

In Section

Figure

The crossing probability for pedestrians for different group sizes (

Figure

The crossing probability for pedestrians for different theoretical times of vehicles (the number of pedestrians is 20; the total sample size is 1,000,000).

Figure

The collision probability

Using the multidimensional dirty faces game, coupled by Monte-Carlo simulations, we have proposed an original and novel method to analyze the crossing behavior of pedestrians group-behavior and vehicles as “rational people.” Our model is able to capture the multistage dynamic interactions between pedestrians and vehicles.

By considering the heterogeneity of waiting times, our results show that, with an increase in the group size, the size of the parameter space that may cause a conflict decreases, while the conflict probability also decreases. When the theoretical passing times of both pedestrians and vehicle are equal, the crossing probability of pedestrians is higher than that of the vehicle in all modes. This phenomenon becomes more pronounced with an increase in the group size. When the theoretical passing time of the vehicle is smaller than the theoretical passing times of pedestrians, the probability that the pedestrians cross the intersection increases, with an increase in the group size. At the same time, the probability that both sides enter the failure mode or encounter a collision increases. Considering the heterogeneity of walking speeds and an increase in the group size, the crossing probability for pedestrians becomes larger, and the size of the parameter space that may cause a conflict decreases. However, the probability that both sides enter the failure mode or have a collision increases. Through an in-depth analysis of the theoretical passing time of the vehicle corresponding to the peak of the collision probability, we observe that the theoretical passing time of the vehicle is smaller than that of the average pedestrian.

Our findings have provided a theoretical framework to advance understanding of the illegal pedestrian

The probability that the vehicle driver selects “the pedestrian gives priority to vehicle” as the dominant strategy is

See Pseudocode

Set

The authors declare that they have no conflicts of interest.

This project was supported by the National Natural Science Foundation of China (Grant no. 61375068), Ministry of Education, Humanities and Social Sciences research projects (13YJAZH106 and 15YJCZH210), Anhui Provincial Natural Science Foundation (1708085MF164), and Talent Project for Higher Education Promotion Program of Anhui Province.