Pedestrian flow dynamics and human behaviors in emergency are important topics for the prevention and control of disasters. They have been attracting great attention from various circles of researchers for several decades. In a pedestrian-particle system in emergency, plenty of agents crowd together and interact with each other, which may bring various self-organization phenomena such as arching, herding behaviors, “faster-is-slower” effect,^[1] etc.

Many physical models have reproduced those phenomena, which are conducive to understanding the mechanism of the phenomena and analyzing the properties of the crowd dynamics. In general, pedestrian evacuation models can be classified as microscopic models and macroscopic models. The macroscopic models are based on the network with modeling the pedestrians as fluid flow, with the shortage of the depiction about the heterogeneity and subjective interaction among pedestrians. In contract, microscopic models simulate agents as individuals, which can address the problems commendably. Social force model^[2–4] and cellular automata models including floor field models,^[5,6] lattice gas models,^[7,8] multi-grid models,^[9,10] agent-based models,^[11] potential field models,^[12] etc. are two kinds of representative microscopic models. The social force model is a multi-particle self-driven model based on newtonian mechanics, which is not very suitable for complex and large-scale movements of pedestrians due to the computational efficiency, while cellular automata models divide the walking space into grids and employ the time-step for evolution, thereby improving the computational efficiency greatly.

In an evacuation process, the movements of individuals and the crowd are all affected by environmental factors^[13–16] and human factors.^[17–21] To escape from the dangerous areas as fast as possible, pedestrians may schedule their paths in real time and change their behaviors constantly given by these factors. That is to say, the process of pedestrian evacuation is also a process of behavioral evolution. It is noted that as a social system of pedestrians, there exits a huge and complex network due to relationships and interdependence among pedestrians.^[22,23] Pedestrians learn from and interact with each other, especially with their neighbors. Interaction behaviors between pedestrians affect not only the evolution of their cooperation behaviors, but also their decision makings to path scheduling based on the facts that people intend to choose better neighbors and gain more benefits, which can have better advantages to arrive at the safe area. Game theory has been introduced into the evacuation models to investigate pedestrian dynamics and cooperation evolution, in which payoff matrix is used to calculate the interactions payoffs between pedestrians. Zheng et al.^[24,25] firstly integrated the game-theoretical method with static floor field evacuation model to study individual cooperative behaviors, and the interaction payoffs only between pedestrians who strive for the same position are focused. Zhang et al.^[26] further considered the compassion mechanism in the conflicts when more than one pedestrian tries to occupy the same position in the model, and its effects on evacuation dynamics and cooperation evolution are studied. Unlike previous study, Shi et al.^[27] and Guan et al.^[28] used the game theory model to deal with the interaction between neighboring pedestrians. Average interaction payoffs were proposed as inputs to calculate the probability of one pedestrian to enter into the targeted position and update his strategy according to different rules. However, the latter studies^[28] lost the sight of the effects of interaction between neighbors on the decision making how to move for one pedestrian, and as most previous study did, the former study^[27] addressed this but did not discussed comprehensive influences of interaction behaviors nor environment factors on pedestrian movement, and conflicts among pedestrians who compete for the same position were also neglected. Even though the existing researches have concerned themselves with the effects of interaction behaviors between pedestrians on pedestrian movements, how interaction behaviors and environment factors synthetically affect decision-makings of one pedestrian to path scheduling is still limited. Further study is necessary to reveal the influences of interaction behaviors as well as other subjective factors on cooperation evolution and movement dynamics of pedestrians in such a situation.

In this paper, we extend the cellular automaton model to the investgating of the evolutional strategy and the individual movement based on the game theory. In this model, the influence mechanism of the environment factor and interaction behaviors between neighbors synthetically on the decision-making of one pedestrian to path scheduling is focused on. The parallel update scheme is investigated and a new method to solve the conflicts when more than one pedestrian competes for the same position is proposed. The reminder of this article is structured as follows. In Section 2, the elaborated model and conflict rule are presented. In Section 3, the simulation scenario is introduced and a series of simulation results is analyzed and discussed. Finally, conclusions are drawn from the present study in Section 4.

Our model is defined on a squared domain divided into a number of discrete grids. Each grid can be either empty or occupied by sole one pedestrian. In our work, the size of a grid is set to be 0.4 m × 0.4 m, and an pedestrian can move to an empty grid or stay still at each step, here we take a simulation step length to be 0.3 s. Pedestrians make decisions consistently when scheduling their path from one grid to next one grid. To incorporate effects of interaction behaviors and environment factors into the decision-making process of one pedestrian, an extended model given by the exponential is proposed, namely, the transition probability P_ij of one pedestrian to a target cell can be formulated as follows:

Fig. 1.

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	Fig. 1. Moore neighbor and the transition probability for next step.

Table 1.

Table 1.

Table 1. Payoff matric of snowdrift game. . C D C (R,R) (S,T) D (T,S) (P,P)

Table 1. Payoff matric of snowdrift game. .

The parallel update scheme is utilized in this paper. One significant problem is to solve the conflicts when more than one pedestrian competes for the same position. The average payoffs of one pedestrian interacting between his neighbors are introduced to reflect his competitive ability. Therefore the target cell is occupied by the pedestrian whose position is (i, j) with the following probability

We also suppose that the pedestrians are rational. Pedestrians who cannot move during the conflicts will adjust their strategy with the probability according to Eq. (8), while the success will keep their strategy at the next step. In Eq. (8), is the average payoff of one pedestrian gained from his current interactive neighbors of the pedestrian with the strategy x while is the average payoff with the opposite strategy y. The k_c is the sense of judgement of one pedestrian, which reflects his degree of rationality, just like k_o.

The simulation procedure for the evacuation process is elaborated as follows: Step 1

Initialize the positions and strategies of pedestrians. All pedestrians are distributed randomly in the domain with the density of ρ = N/M, where N and M are the number of pedestrians and grids in domain respectively, and the initial fraction of cooperators is set to be ρ_co.

To provide trustworthy results, each test is simulated 50 times and the mean values are exhibited. The strategy of the pedestrian in the last step is recorded as the calculation of the final ρ_c.

The simulations are carried out in a room with the size of 10 m× 10 m. Only one door occupying one grid exists on the sites, it is located in the center of the bottom wall. Initially, all pedestrians are distributed uniformly with the density ρ = 0.6 and ρ_co = 0.5. We also set k_o = k_c = 2 in this paper. Figure 2 shows a snapshot of one simulation. As shown in this figure, square red symbols and square blue symbols donate cooperators and defectors respectively. Pedestrians gather at the door and the arching phenomenon which usually emerges at the bottleneck in real emergency evacuation is reproduced. The reliability and soundness of the model are verified.

Fig. 2.

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	Fig. 2. Arching phenomenon in one simulation.

Figure 3 presents the evacuation time with different values of k_s and k_u. It can be observed that the interdependence degree of pedestrians affects the evacuation time of pedestrians, but different features are shown for different values of parameters. In Figs. 3(a) and 3(b), larger values of k_u generally lead to longer evacuation time, while ccthe evacuation time is a little less when k_s > 3, k_u = 2 than that when k_s> 3 and there are no interaction behaviors among pedestrians, namely k_u = 0. In Figs. 3(c) and 3(d), it can be found that interaction behaviors between pedestrians can reduce the evacuation time obviously when k_s > 5. More importantly, in such a situation the bigger the value of k_u, the smaller the evacuation time is as shown in Fig. 3(d). It is the result of the combination of all parameters. Due to the influences of these parameters, interaction behaviors between people improve the possibility with which one pedestrian moves as described by Eq. (1). As a result, the more strongly a pedestrian wants to interact with his neighbors, namely, the lager the value of k_u is, the greater the chance for the pedestrian to move will be. Therefore, it can be concluded that the interaction behavior is of benefit to the evacuation efficiency to a certain extent, whereas it also depends on other factors, such as the knowledge of people to the exit and so on. Due to the small evacuation time with k_s = 2 and k_u = 3, the simulations for such a situation will be carried out later.

Fig. 3.

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	Fig. 3. Evacuation times with different values of ks and ku: (a) r = 0.3, λ = 1.2; (b) r = 0.7, λ = 1.2; (c) r = 0.3, λ = 1.6; (d) r = 0.7, λ = 1.6.

Figure 4 demonstrates the fraction of cooperators with different values of k_s and k_u. Overall, the fraction of cooperators declines with the increase of k_u, but increases when k_u is quite small as shown in Figs. 4(b) and 4(d), indicating that a small degree of interaction behaviors among pedestrians is conducive to the formation of cooperative behaviors, whereas too much interaction goes against cooperative behaviors of pedestrians.

Fig. 4.

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	Fig. 4. Fractions of cooperators with different values of ks and ku: (a) r = 0.3, λ = 1.2; (b) r = 0.7, λ = 1.2; (c) r = 0.3, λ = 1.6; (d) r = 0.7, λ = 1.6.

The panic degree r and conflict cost λ have an influence on the evacuation process of pedestrians, as illustrated in Figs. 5 and 6. It can be found that the large conflict cost λ can prolong the evacuation time. On one hand, large conflict cost probably leads the pedestrians to move unsuccessfully as shown in Eq. (7). When λ approaches to, we can find that no one can move to the target cell as long as more than one defector is involved in the conflict. On the other hand, large conflicts reduce the cooperative behaviors as shown in Fig. 6, and thus further reducing the possibility of moving according to Eq. (7). Even though the fraction of cooperator increases slightly with λ increasing in some situations, the influence of large value of λ is more prominent comparatively.

Fig. 5.

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	Fig. 5. Plots of evacuation time versus λ for various values of r.

Fig. 6.

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	Fig. 6. Plots of fraction of cooperators versus values of λ for various values of r.

The tendency of varying with λ depends on the value of r. The fraction of cooperators first decreases and then increases slightly when r < 0.7, otherwise goes up imperceptibly and falls down directly when r ≥ 0.7. This can be explained as the fact that larger value of λ makes smaller the possibility for pedestrians to move, pedestrians prefer to become defectors to obtain more payoffs from neighbors and more chances to move, but neighboring defectors compete for less payoffs and more defectors further reduce the possibility to move as shown in Eq. (7). When the value of r is relatively small, the effects of large λ and n_D on P_o are much stronger than that of r. Some defectors therefore change into cooperators again when λ is large enough. However, when the value of r is big, the payoffs, when defectors interact with cooperators, are quite considerable, pedestrians still intend to be defectors firmly in spite of large λ and n_D.

The pedestrians’s evacuation time also varies with r under different values of λ. When λ is relatively small, large values of r lead evacuation time to lessen otherwise the contrary results are obtained as shown in Fig. 5, indicating that the appropriate panic is conducive to the efficiency of the crowd evacuation. In fact, the bigger the value of r, the more the payoffs obtained by the defectors interacting with cooperators will be and the greater the possibilities for them to move. Even though large r makes more defective populations, adverse effects on the evacuation time are negligible when λ is small. However, the adverse effects are obvious and stick to the leading position when λ is large enough, this consequently lengthens the evacuation time. It also reflects the “fast is slow” phenomenon.^[1]

The fraction of cooperators decreases with the increase of r value, which obviously indicates that panic emotions of pedestrians are bad for their cooperative behaviors. This is because the bigger the value of r, the more the payoffs they obtain as a defector from a cooperator according to the snowdrift game theory, and therefore they are more likely to move as shown in Eq. (7). As a result, more pedestrians prefer to become defectors. Figure 7 further exhibits the evolution process of cooperators’ fraction with time at different values of r and λ. It can be seen that there are several differences among panels, and the value of ρ_c varies with r and λ at each point in the evolutionary process, but the cooperation fraction evolves to a consistent state much quickly and decreases after a period of time despite different values of parameters. This can be explained as a kind of self-organization effect. But the results are a little different from those reported previously^[28] due to the effects of interaction behaviors on the decision-making process in our model. It is also noticeable that the values of r and λ affect not only the final results but also the process of cooperative evolution. It is observed that the cooperator fraction drops sharply at beginning when r ≥ 0.6, but rises rapidly when r < 0.6. The evolution tendency of cooperator fraction ρ_c with value of λ changing is a little different, indicating that the panic degree and conflict cost of pedestrians both play an important role in decision makings of cooperative strategy.

Fig. 7.

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	Fig. 7. Evolution processes of cooperator fraction with time at different values of r for (a) λ = 1, (b) 1.2, (c) 1.4, and (d) 1.6.

In what follows, the evacuation dynamics and cooperation behaviors with different values of ρ_co are investigated as shown in Fig. 8. It can be found that the value of ρ_co has almost no influence on the evacuation time and the ultimate fraction of cooperators with all others holds equal. To further understand the result, the time-evolution processes of the fraction of cooperators with different values of ρ_co under different parameters are studied as presented in Fig. 9. It can be obviously observed from Fig. 9(a) to Fig. 9(e) that even though there are several differences among panels due to the influence of values of r and λ, the fraction of cooperator ρ_c in each situation reaches a consistent state very rapidly and then evolves with time synchronously despite different values of parameter ρ_co, that is to say, the evolution process of ρ_c lead to the same final result for different values of ρ_co. This phenomenon once again proves the self-organization effect mentioned above.

Fig. 8.

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	Fig. 8. Plots of evacuation time and fraction of cooperators versus ρco for different values of λ and r.

Fig. 9.

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	Fig. 9. Time-evolution process of cooperators’ fraction with different values ρco for (a) r = 0.2, λ = 1.2; (b) r = 0.4, λ = 1.2; (c) r = 0.6, λ = 1.2; (d) r = 0.6, λ = 1.4; and (e) r = 0.6, λ = 1.6.

In this paper, an extended evacuation model considering the effect of interaction behaviors between neighboring pedestrians and the evolution of cooperation behavior is proposed based on the CA model and the game theory model. In the model, one pedestrian’ decision-making to path scheduling is determined together by benefits from interaction behaviors and static environment. The parallel update scheme is adopted and a new method based on the average payoffs of one pedestrian interacting with his neighbors is presented to reflect his competitive ability. Fermi rule is used to update the cooperative or defective strategy of pedestrians. According to the model, the influences of interaction behaviors, panic index and conflict cost on the evacuation time and the cooperation evolution of pedestrians are further analyzed. The results indicate that interaction behaviors between neighboring pedestrians can reduce the evacuation time to a certain extent. When the interdependence degree is small, it is conducive to the formation of cooperation behaviors. Large conflict cost of defectors lengthens the evacuation time. Panic emotions of pedestrians are bad for their cooperative behaviors. When λ is relatively small, large values of r lead the evacuation time to lessen, otherwise the contrary results are obtained. In addition, a kind of self-organization effect is present in the evolution process of cooperation behavior, that is, the fraction of cooperators reaches a consistent state rapidly in spite of different values of parameters. This work offers a new insight into the understanding of the dynamic mechanism of pedestrian evacuation and also provides guidance for establishing the evacuation strategies.