# Drivers’ Skills and Behavior vs. Traffic at Intersections

^{*}

^{†}

## Abstract

**:**

**CO**(cooperative) and

**DE**(defective), the tendency of drivers from each of these groups to deviate from compliance with traffic rules is established. The effective driver behavior translates into disrupting traffic by slowing it down. Participant interactions are described using game theories that provide information for simulations algorithms based on cellular automata. Three different ways of using this combination of descriptions of traffic participants to examine the impact of their behavior on the traffic dynamics are shown. Directions of the further, detailed analysis are indicated, which requires basic research in the field of game theory models.

## 1. Introduction

#### 1.1. Driver Decision Models

**Regulatory drivers**are abbreviated as**CO**(Cooperator).**Non-compliant drivers**are abbreviated as**DE**(Defector).

#### 1.2. Intersections, Drivers and Traffic

**NaSch**, v. [30]) described in the Section 1.3. This is a proven method that allows testing the impact of changes in driver behavior on vehicle flow parameters. However, in the description of the dynamics of traffic at the intersection, three elements can be distinguished, the specification of which is important for aspects of interest to us. Those are:

- (i)
**Identifying road elements**boils down to the rules of right-hand traffic. This means that priority is given to the one on the main road when meeting at the intersection of two vehicles, i.e., the one who sees the second vehicle on its left (priority of the road on the right-hand side). This objective determination is transformed by decision makers. It is known that the main reasons for perturbation in the stream of vehicles are driving behaviors that do not comply with traffic rules (v. [16,31]). Earlier studies by Mesterson-Gibbons [32] have found various quantifications of driver behavior; however, two categories of drivers have been adopted for the purposes of this study. By convention, they are those who follow the rules of the road (**CO**drivers) and those who do not comply with them (**DE**drivers). With the approach used to analyze the phenomenon, the proportions of the types of drivers present in the population in the analyzed area are significant.- (ii)
**Determining the behavior of drivers**is based on the fact that at each meeting, before the settlement of traffic in the next step (nearest second), the type of driver is identified, and, on this basis, his decision is determined, which translates into vehicle behavior. There are several ways to identify the types of individual drivers in the considerations. In the models selected for detailed analysis, the method of determining the driver’s behavior is different and depends on the assumptions made earlier in relation to the rules functioning in everyday life.- (iii)
**Priority assignment**at the intersection results from setting their priorities based on the types described above and assigned to drivers. Costs (in units of increasing or decreasing speed) related to strategies adopted by drivers are determined. Depending on the adopted model, the payout values are different.

#### 1.3. Nagel–Schreckenberg Model

- (i)
**Acceleration/Braking.**The car increases its speed by one, if it is not higher than the maximum speed and the number of free cells in front of it. When the distance to the car ahead is less than the current speed, the vehicle slows down to a value equal to the empty space in front of it. In the mathematical notation it looks like this:$${v}_{i}(t+1)=min({v}_{i}\left(t\right)+1,{v}_{max},{d}_{i}).$$- (ii)
**Random event.**A car with a certain probability decreases its speed by 1, provided it is not less than zero. The equation for the described situation is as follows:$${v}_{i}(t+1)=\left\{\begin{array}{cc}max({v}_{i}(t+1)-1.0),\hfill & \mathrm{with}\mathrm{probab}.p\hfill \\ {v}_{i}(t+1),\hfill & \mathrm{with}\mathrm{probab}.1-p.\hfill \end{array}\right.$$- (iii)
**Update position.**The car moves as many cells as its current speed, according to the formula:$${x}_{i}(t+1)={x}_{i}\left(t\right)+{v}_{i}(t+1).$$

**NaSch**model reliably reflects the movement of vehicles on the road and the mutual interactions of drivers. One example that is noticed when analyzing the results of simulations is the occurrence of start–stop waves, showing how sudden braking of one driver affects other road users.

#### 1.4. Manuscript Organization

## 2. Simulation Analysis and Discussion of Generalizations for Model I

#### 2.1. Model Description

**four scenarios**are possible:

- (i)
- Both drivers are cooperators (CO).
- (ii)
- The driver driving on the right, i.e., the one with priority, is a non-compliant driver.
- (iii)
- The driver on the left, i.e., the one who should give way, is a driver who does not comply with the rules.
- (iv)
- Both drivers do not follow traffic rules.

**DE**and the right

**CO**is a simplification. In fact, it is more complex and usually

**CO**delay more than

**DE**. As a possible extension, the authors suggested ${d}_{DC}^{rd}=c>a+1$. The further discussion of the issue is given in Section 2.4, where the type of driver is interpreted in terms of their strategies.

#### 2.2. Simulations

**NaSch**model, with a low probability of random events and a maximum speed of 1. According to the recommendations of the model creators, updating position on the road should be asynchronous, thus, with each iteration of the program, we drew a different order of road updates. Roads 1–4 are in the horizontal direction and Roads 5–8 the vertical direction. Drivers on even roads followed the natural turn, and drivers on odd went in the opposite direction. By performing a single move for each street, the obtained results were placed in the appropriate positions on the network. The next stage was the analysis of behavior at intersections. The first 50 steps were skipped to allow the entire system to be filled with cars. The areas before and at the intersection were taken into account. As per the authors’ recommendations, the intersection results were also updated in a random order. In the event of a meeting of two drivers at the intersection, the individual waiting time for each of them was set. It was calculated according to the values in Table 1, minus 1, because the time needed to cross the intersection was not taken into account. Waiting time was then used in the previously mentioned algorithms for updating road positions. A vehicle ordered to wait could not increase its speed until the designated number of steps had elapsed. Simulations were carried out for various driver relations on the road.

#### 2.3. Outcome of Simulations

**CO**drivers—that is, those who comply with traffic rules in each case—are higher than for the other groups. It can be concluded that the movement of cooperating drivers is faster and smoother than that of

**DE**drivers who are more exposed to greater penalties when interacting at intersections. In addition, for high load of drivers on the road, with low probability of DE drivers, the average speeds are lower, but the presence of DE drivers improves traffic.

#### 2.4. Extension of Model Interpretation

**the estimation of driver type ratio**. There are three types of meetings in the model under consideration: (

**CO**vs.

**CO**;

**CO**vs.

**DE**; and

**DE**vs.

**DE**). Each of these types of meetings consequently gives one of the three effects of stream modification over the main road. Let ${\eta}_{ji}$ be a random variable equal to 1 when in the ith meeting of drivers the driver on road j is

**CO**and 0 if he is

**DE**, and ${\xi}_{i}={\eta}_{ai}+{\eta}_{bi}$. If the random variables ${\eta}_{ji}$, $j\in \{a,b\}$ are independent, identically distributed with $\mathbf{P}({\eta}_{ji}=1)=p=1-\mathbf{P}\left({\eta}_{ji}\right)$, then parameter p can be estimated by one of the methods described by Bickel and Doksum [27] or Berger [28]. Based on n meetings of drivers with ${\sum}_{\xi}={\sum}_{1}^{n}{\xi}_{i}$, the minimax estimate is ${\widehat{p}}_{minmax}=\frac{a+{\sum}_{\xi}}{b+2n}$ (cf. [34,35]).

**modeling drivers strategies**is crucial in the topic under consideration.

## 3. Model II as an Extension of the First Model

#### 3.1. Description of the Model

**core**. The driver type is updated every $\tau $ simulation steps. After this time, the probability of imitating each strategy is calculated, informing about the chance to change the current type of a driver to another with the probability of imitating it.

**CO**to

**DE**(the road-complying driver to non-complying driver) occurs with the probability of ${P}_{D}$, and in the reverse with the probability of ${P}_{C}$. These probabilities are described by the formula:

#### 3.2. Simulation and Results

**core**) of drivers who comply with traffic rules and resist attempts to force them to break these rules. In the second and third models, we assumed that there was a core of resistant and law-abiding drivers of 10% and 30% in total, respectively.

**DE**drivers to all participants in situations where the initial probability of DE driver occurrence was $0.25$, $0.5$, and $0.75$, respectively, as shown by different line colors. It is noticeable that the values converge and remain at a similar level. It can be stated that the system is stabilizing in terms of the distribution of individual types of drivers. A similar situation is presented in Figure 6b showing simulation results with a driver core of 10%. The system stabilizes at a lower level than before. In addition, the case with a high probability of occurrence of DE driver stands out more from the others. The situation after increasing the driver core to 30% is presented in Figure 6c. It is important that this time the driver attitude stabilizes more quickly than in the other cases, and the level of stabilization is even less than in the case of the core of 10%. In addition, the charts for the different initial probabilities of

**DE**are more similar. This presents an important fact resulting from the above analysis: the core of drivers significantly affects the level of stabilization of number of DE and CO drivers. The greater is the core, the lower is the stabilization level for DE drivers, which, when combined with the results of the previous model, gives a better flow and efficiency of the movement system. Thus, to ensure better quality of traffic, the emphasis should be on generating a larger core of drivers so that as many of them as possible are resistant to the negative influence of other road users.

- (i)
- The case with a lack of the drivers’ core complying with the regulations.
- (ii)
- The case where the drivers’ core was 10%.
- (iii)
- The case when the drivers’ core was 30%.

## 4. Model III with Impatient Drivers

#### 4.1. Problem Formulation

#### 4.2. Generating Driver Behavior

#### 4.3. Driver Strategies for Prioritization

- (i)
- The driver who should step down complies with the rules. By following them, it gives way regardless of the type of a driver at the cross-roads.
- (ii)
- The driver who should give way is a driver who does not comply with the rules. He will try to impose priority, thus forcing the compliant driver to give way to him/her to ensure his/her safety.
- (iii)
- Both drivers are non-compliant drivers, so both can try to cross the intersection at the same time. Because of their abnormal behavior, both should stop and then the driver on the road can pass first.

#### 4.4. Simulation

**CO**-type, always gives way. He waits two time steps, which is as much as his opponent needs to enter and leave the intersection. Then, if he is not moving, he can start the maneuver, and, if another opponent arrives, the situation repeats. Waiting times before the intersection of each driver are counted. Additionally, drivers whose stopping does not result directly from waiting before the intersection, but is caused by its earlier blocking, are also included. We consider cars standing in a traffic jam to those whose average of the previous five speeds is less than or equal to 0.2. Based on the received waiting times, the probability of change for each driver is determined.

- (i)
- Average system speed
- (ii)
- The number of a driver type changes that have occurred at each time step
- (iii)
- The number of conflicts between drivers that occurred at each time step
- (iv)
- The attitude of DE drivers, i.e., those who changed their type to non-compliant
- (v)
- Average waiting times for drivers before intersections

## 5. Model Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CO | Regulatory drivers (Cooperator) |

DE | Non-compliant drivers (Defector) |

NaSch | Nagel–Schreckenberg |

CP, LCP | Complementarity Problem, Linear Complementarity Problem |

$\mathrm{NE}(A,B)$ | the set of Nash equilibria |

ק | The rate of non-compliant (jamming, difficult) drivers |

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**Figure 7.**Box plots for the various initial probabilities of

**DE**drivers vs. different

**core**of

**CO**drivers.

**Figure 8.**Box plots for various initial probabilities of

**DE**drivers vs. the size of the

**CO**drivers’

**core**.

Right | CO | DE | |
---|---|---|---|

Left | |||

CO | $(2,\phantom{\rule{0.277778em}{0ex}}1)$ | $(2,\phantom{\rule{0.277778em}{0ex}}1)$ | |

DE | $({d}_{DC}^{ld},\phantom{\rule{0.277778em}{0ex}}{d}_{DC}^{rd})$ | $({d}_{DD}^{ld},\phantom{\rule{0.277778em}{0ex}}{d}_{DD}^{rd})$ |

Right-h. s. | CO | DE | |
---|---|---|---|

Left-h. s. | |||

CO | $(2,\phantom{\rule{0.277778em}{0ex}}0)$ | $(2,\phantom{\rule{0.277778em}{0ex}}0)$ | |

DE | $(0,\phantom{\rule{0.277778em}{0ex}}2)$ | $(3,\phantom{\rule{0.277778em}{0ex}}1)$ |

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
---|---|---|---|---|---|---|---|---|---|

Total number of driver type changes | 6 | 66 | 380 | 7220 | $16,351$ | $19,873$ | $21,056$ | $22,457$ | $23,128$ |

Frequency of driver type changes | 0.0001 | 0.0009 | 0.0051 | 0.0963 | 0.2180 | 0.2650 | 0.2807 | 0.2994 | 0.3084 |

The total number of conflicts | 2 | 0 | 0 | 25 | 218 | 248 | 376 | 337 | 388 |

Frequency of conflicts | 0.00002 | 0 | 0 | 0.0003 | 0.0029 | 0.0033 | 0.0050 | 0.0045 | 0.0052 |

Average ratio of DE drivers | 0.0073 | 0.0001 | 0.0004 | 0.0071 | 0.0197 | 0.0262 | 0.0184 | 0.0244 | 0.0231 |

Average wait time | 1.4527 | 1.3126 | 1.3759 | 2.4733 | 3.1016 | 3.3106 | 3.3708 | 3.4362 | 3.4878 |

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Szajowski, K.J.; Włodarczyk, K.
Drivers’ Skills and Behavior vs. Traffic at Intersections. *Mathematics* **2020**, *8*, 433.
https://doi.org/10.3390/math8030433

**AMA Style**

Szajowski KJ, Włodarczyk K.
Drivers’ Skills and Behavior vs. Traffic at Intersections. *Mathematics*. 2020; 8(3):433.
https://doi.org/10.3390/math8030433

**Chicago/Turabian Style**

Szajowski, Krzysztof J., and Kinga Włodarczyk.
2020. "Drivers’ Skills and Behavior vs. Traffic at Intersections" *Mathematics* 8, no. 3: 433.
https://doi.org/10.3390/math8030433