Effects of individual attitudes and motion decisions in room evacuation models
EEffects of individual attitudes and motion decisions inroom evacuation models
V. Dossetti a,b,c , S. Bouzat d,e , M. N. Kuperman d,e a CIDS-Instituto de Ciencias, Benemérita Universidad Autónoma de Puebla, Av. SanClaudio esq. 14 Sur, Edif. 103D, Puebla, Pue. 72570, Mexico b Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Postal J-48,Puebla 72570, Mexico c Consortium of the Americas for Interdisciplinary Science, University of New Mexico,Albuquerque, NM 87131, USA d Consejo Nacional de Investigaciones Científicas y Técnicas e Centro Atómico Bariloche (CNEA), (8400) Bariloche, Río Negro, Argentina
Abstract
In this work we present a model for the evacuation of pedestrians from anenclosure considering a continuous space substrate and discrete time. Weanalyze the influence of behavioral features that affect the use of the emptyspace, that can be linked to the attitudes or characters of the pedestrians. Westudy how the interaction of different behavioral profiles affects the neededtime to evacuate completely a room and the occurrence of clogging. We findthat neither fully egotistic nor fully cooperative attitudes are optimal fromthe point of view of the crowd. In contrast, intermediate behaviors providelower evacuation times. This lead us to identify some phenomena closelyanalogous to the faster-is-slower effect. The proposed model enables for theintroduction of Game Theory elements to solve conflicts between pedestrianswhich try to occupy the same space. Moreover, it allows for distinguishingbetween the role of the attitudes in the search for empty space and theattitudes in the conflicts. In the present version we only focus on the first ofthese instances.
1. Introduction
During the last decade there has been a multiplication of mathematicalmodels for pedestrian dynamics and enclosure evacuation aiming at unveilingthe most relevant aspects affecting such processes. Despite the fact that all
Preprint submitted to Physica A November 8, 2018 a r X i v : . [ n li n . C G ] S e p hese models represent similar processes (or even the same), they implementdifferent methodologies with their corresponding strengths and limitations.The consideration of behavioral aspects related to the interactions be-tween individuals has produced some notable contributions in the area [1, 2].In particular, using a mechanistic approach, the so called social-force model[2, 3, 4, 5] captures the fact that the motion of an agent is governed by thedesire of reaching a certain destination and by the influence it suffers fromthe environment, which includes the other agents. The model links the col-lective motion of the individuals to self-driven many particle systems (see [6]and references therein). In order to simplify the calculations, advances havealso been maid with cellular automaton models [7, 8].Recently, some attention was drawn upon the fact that a full descriptionof the behavioral aspects involved in an evacuation process needs to considerthe disposition of the individuals as internal states which may influence theresponses of the pedestrians and, ultimately, their motion (see, for exam-ple, [9]). The idea is that not only the environment but also the internalmotivations govern their reactions [10]. Thus, some of the interactions be-tween two or more individuals in a moving crowd can be thought as conflictswhich the pedestrians can tackle using different strategies, according to theircharacters or attitudes. These considerations have naturally led to the in-clusion of Game Theory elements in the models for pedestrian dynamics.The first works considering Game Theory within evacuation models [11, 12]analyzed the emergence of pushing behavior in evacuation situations. Sub-sequently, several authors analyzed various aspects of the complex subject oflinking individual nature to crowd dynamics [13, 19, 14, 15, 16, 17, 18] usingGame Theory. All these works considered two available strategies for eachpedestrian. Namely, a cooperative one and a defecting one. The coopera-tive strategy is normally associated to kind and patient behavior consistingin not looking for taking advantage from others, and, specially, not rushingto an empty space when a favorable move is available [13, 17]. Conversely,the defective strategy can represent an impatient pedestrian, who rushes toconquer gaps, disrespectful to others.In a slightly different context, it was suggested recently [21] that, despitethe success of the models based on the Social forces paradigm, more sim-ple assumptions could lead to highly satisfactory predictions of pedestrianbehavior. In [21], the authors consider that the reaction of an individual ismostly governed by the visually perceived environment, and that a pedes-trian, instead of being repelled by the neighbors, actively looks for empty2pace or free paths through the crowd. Hence, from different points of view,the attitude of the individuals toward the empty space (i.e. the way in whichpedestrians use and let use the available space) arise as a character-dependentkey element influencing the dynamics.Clearly, a strategy of motion reflecting the complex behavior of a pedes-trian cannot be determined by a unique feature within a model. Severalaspects should be included in order to define the use of space and the at-titude of the pedestrian in conflicts or competitions for space. Part of theobject of this work is to understand how some of the different elements whichmay conform a strategic profile affect the dynamics of an evacuation processwhen considered separately. This means, to disentangle such different com-ponents, and move towards a better understanding of the emerging results.With this in mind we develop a spatially-continuous discrete-time modelfor pedestrian dynamics that enables for the analysis of the influence of theindividual attitudes at different stages of the evacuation dynamics. Thesestages include the decisions made at the presence of available space that pro-motes a favorable move, and the strategy adopted if confronted with conflictsthat may derive from these decisions. More specifically, the model considerstwo different stages at which the attitudes of the pedestrians can be intro-duced. First, a stage at which each pedestrian decides its attempt of motionaccording to the preferred direction, the available free space perceived, andreflecting its degree of rationality, patience, cooperativeness or aggressive-ness. Second, a stage at which the conflicts among individuals that attemptto move to overlapped positions are solved. This second stage could in princi-ple include Game Theory elements to reflect the character of the pedestriansand their fitnesses to win the competitions. However, as a first approach, inthis work we solve the conflicts at random and focus on the analysis of theaspects involved in the first stage. At such a first level, the actual behavioralpatterns of pedestrians, which may include cooperative or egotistic attitudesand rational or impulsive decision making, are incorporated in the modelthrough different deterministic or noisy ways of deciding the attempts of mo-tion. The selfish or cooperative attitude is mimicked at this level throughan aggressive or coordinated management of the available space to move. Inparticular, we focus on the analysis of the effects of the randomness and the rationality in the use of empty space to move, with a connotation that willbecome clear later. We also study the influence of the minimal separationthreshold considered by the pedestrians for attempting a step. This parame-ter is related to the inclination of the individuals to avoid collision with other3edestrians and, thus, to their degree of cooperativeness or politeness.In agreement with the suggestions in [21], the proposed model startsfrom very simple assumptions and a small number of parameters. Moreover,given that it considers finite-size pedestrians in a continuous-space, it can bethought as more realistic than most cellular automaton models. Furthermore,as mentioned before, it enables for the inclusion of Game Theory formalismto solve the conflicts which emerge in the competition for space. Due to theseproperties, we believe that the model can be relevant for the analysis of manyaspects of pedestrian dynamics beyond those studied in this work. Moreover,by considering appropriate adaptations and more complex geometries, it mayconstitute a useful tool for the design of actual spaces and buildings.
2. The model
We consider a spatially continuous model of N pedestrians, representedby rigid disks of radius r = d/ , occupying a square room of size L with asingle door. In ( x, y ) -coordinates, the walls are located at x = ± L/ , y = 0 and y = L , while the exit is in the middle of the y = 0 wall, running from ( − l d / , to ( l d / , . The time is discretized in steps of duration δt = 1 (in arbitrary units) while the space is continuous so that the position of thecenter of the disks can be any point in the room. We analyze two versionsof the same basal model, that will be referred to as rational and stochastic .In both cases, the individuals head towards the door in order to leave theroom. More precisely, a walker located at position (cid:126)r = ( x, y ) heads towardsthe point (cid:126)r E at the exit, which is defined as (cid:126)r E = (0 , if | x | ≥ l d / or as (cid:126)r E = ( x, if | x | < l d / .The formulation for the rational model is the following. At each timestep, each walker selects a desired position to which he/she will attempt tomove. For this, the walker first explores the possibility of moving in thedirection of the exit, and if it is not possible, he/she explores the possibilityof making a lateral motion. This is done with the following algorithm. Let (cid:126)r i be the position of the walker i at time t , which is represented by the center ofthe circle in the sketch in Fig. 1. We define ˆ u E = ( (cid:126)r E − (cid:126)r i ) / | (cid:126)r E − (cid:126)r i | (i.e. thedirection towards (cid:126)r E ). As a first attempt, the walker selects a direction ofmotion ˆ u , which is randomly chosen within an angle of amplitude η around ˆ u E (i.e. η/ to each side), as shown in Fig. 1. Note that this impliesthat we consider an uncertainty or randomness in the way the pedestrianspoint toward the exit, which is characterized by the parameter η . Then,4he walker analyzes what is the maximum distance he/she can move in thedirection ˆ u without overlapping another walker (for this we consider all thewalker’s positions at time t ). If such a distance is larger than a threshold µ d ( < µ < ), the desired position for the walker i is set as (cid:126)r i + λ ˆ u . Here, λ > µ d is the size of the desired step, which is set as long as possible upto a maximum distance d (i.e. µ d < λ < d ). Conversely, if the maximumdistance allowed in the direction ˆ u is smaller than µ d , the walker attemptsfor a lateral motion (see Figure 1). For this, he/she randomly selects oneof the two directions perpendicular to ˆ u , here referred to as ˆ u ⊥ . Then, thewalker checks the maximum distance allowed for a motion in the direction ˆ u ⊥ . If this results larger than µ d , the desired position is set as (cid:126)r i + λ ˆ u ⊥ .Instead, if there is a blockage, the walker will not attempt for moving in theconsidered time step, and he/she will remain at position (cid:126)r i at time t + δt .Once all the individuals have defined the positions for their motion at-tempts, we have to check for the possibility of conflicts. Note that a conflictmay arise when the desired positions of two or more walkers overlap. Insuch a case, the conflict is solved by selecting at random one of the walkersinvolved, which will be finally allowed to move. The rest of the individualsin the conflict will remain at their original positions. Note that, accordingto what was indicated in the introduction, this is the stage of the modelat which Game Theory could be included to solve the conflicts, as done forinstance in [17].It is worth remarking that the rational model includes two main parame-ters defining the stepping dynamics. Namely, the angle η characterizing therandomness in the pointing toward the exit by the walkers, and µ ( < µ < ),which defines the threshold distance ( µd ) for attempting a step.Now we introduce the stochastic model, whose main difference with therational one is that the attempt for a lateral motion is decided at randominstead of being a consequence of the impossibility of making a forward mo-tion. The algorithm is the following. At each time step, for each walker, thedirections ˆ u and ˆ u ⊥ are defined using ˆ u E and η , just as in the deterministiccase. Then, each walker is set to attempt for a lateral motion (direction ˆ u ⊥ )with probability α or for a forward motion (direction ˆ u ) with probability (1 − α ) . If the available distance in the chosen direction is larger than µd ,the walker will attempt for a step of length λ in such direction. In contrast,if there is a blockage, the walker will not attempt for moving. After all theindividuals have defined their desired positions, the conflicts are solved inthe same way as for the rational model.5 igure 1: Scheme showing the direction of frontal and lateral movements and the uncer-tainty introduced by η η and µ of the rational model, the stochastic model has a third parameter α ( ≤ α ≤ ), which defines the probability of attempting a lateral movement. Second,the random choice between the directions ˆ u and ˆ u ⊥ reflects an irrational behavior of the pedestrian. Note that the available distance is checked onlyin the randomly chosen direction.For both models, as initial condition we consider random positions forthe N pedestrians within the room, taking care of avoiding overlapping.The parameters µ , η , and also α for the stochastic case, define the wayin which the pedestrians decide their attempts of motion. Thus, they can beinterpreted as dependent on the characters of the pedestrians. For instance, arelatively large value of µ could be related to a more respectful (cooperative)attitude than a low value of µ , since a walker with larger µ is more carefulin avoiding physical interaction with the pedestrians in the direction he/shemoves. Similarly, larger values of η or α indicate a lower propensity formoving straight to the exit, so that this may be also attributed to a morecooperative attitude. The predisposition of the individuals can be naturallyintroduced in the model to define the strategy of motion, that affect boththe decision making process when searching for empty space to move (ruledby the parameters µ , η and α ), and the attitude or strategy during a conflictresolution. As indicated in the introduction, in this work we focus on theeffects observed only during the first case. Our definition of cooperative anddefective attitudes alludes to the observed behavior during the choice of thenext move and not to deeper internal motivations that the individuals mayhave. In this sense, walking straight to the exit ignoring the presence ofthe others is the most intuitive and obvious interpretation of an egotisticattitude.
3. Results
We begin by analyzing the dependence of the mean evacuation time (orexit time) on η within the two models. For this we consider a fixed value µ = 0 . and different values of α in the case of the stochastic model. Theeffects of varying that threshold µ will be studied later. The results are shownin Figure 2.First we observe that, for any value of α , the stochastic model resultsin longer exit times than the rational one. As we show later, this result is7ndependent of the values of the rest of the parameters with the exceptionof very small values of µ . We can affirm that, in most cases, the rationalbehavior is strategically more effective than the stochastic one (both for eachindividual and for the crowd).Figure 2 also show us that the evacuation time has a non monotonicbehaviour, adopting a minimal value for an intermediate value of η . Thisindicates that the presence of certain limited amount of noise improves theoverall performance of the pedestrians, but the effect is reversed once η ex-ceeds a critical value, i.e., with an increase in the uncertainty when choosingthe moving direction. This can be understood by considering that for verysmall η , the desired directions of movement of several pedestrians can pointto similar targets, increasing the possibility of blockage, while large uncer-tainties can lead to unnecessary winding of the trajectories increasing theexit time. We have verified the robustness of this result against changes in µ , as shown later, in the sizes of the door and the room and in the numberof pedestrians.Additionally, we observe that for the stochastic model there is an optimalnon-negligible value of α that minimizes the exit time. To observe this moreclearly we present in Figure 3 the information about exit times for differentvalues of η as a function of α . Qualitatively, the dependence of the exit timeon α has the same behavior for any value of η . There is always an optimalvalue that minimizes the exit time and that is almost constant for a givensize of the door. For example, for l d = 6 d , the optimal value is α ≈ . with only small variations (less than 7.5%) as a function of η , as shown inthe inset of Figure 3.For that optimal situation, the resulting exit time is of the order of the onefound for the rational model, and even the dependence on η is qualitativelythe same, as shown in Figure 2 for stochastic and rational agents. However,this is not the case for lower and higher values of α . On the one hand, when α is very low, the whole group of pedestrians might end up in a cloggedconfiguration or, with some intermittences, stay blocked until a noisy eventhelps to clear the blockage. We have observed these two effects in severalsimulations.As mentioned before, the increase in the exit time is mainly due to a noisywandering behavior (more evident for the stochastic version of our model) orto the occurrences of blockages due to the lack of space for the pedestriansto move. The first can be tuned by the amount of noise considered in thedynamics (characterized by the parameters η and α ). This, in turn, also8 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 03 0 01 0 0 01 0 0 0 0 exit time h p - 1 a =0.02 a =0.1 a =0.55 a =0.7 r a t i o n a l exit time h p - 1 Figure 2: Mean evacuation time for the rational and stochastic models as a function of η/π . Here each curve corresponds to a different choice of α with µ = 0 . , while the solidblack curve with white circles corresponds to the rational version of the model, zoomedin the inset. The results were obtained by averaging over 1000 realizations, considering aroom of size L = 100 with a door of size l d = 6 d filled to a 40% of its capacity, containing N = 1000 agents. .01 0.1 1 a e x i t t i m e h = 0 h = p /8 h = 2 p /8 h = 3 p /8 h = 4 p /8 Figure 3: Mean evacuation time for the stochastic model as a function of α with µ = 0 . .We consider different values of η . The results were obtained by averaging over 1000realizations, considering a room of size L = 100 with a door of size l d = 6 d filled to a40% of its capacity, containing N = 1000 agents. The inset shows a zoom of the curvecorresponding to the rational case η π − e x i t t i m e µ = 0 µ = 0.01 µ = 0.1 µ = 0.4 µ = 0.6 µ = 0.7 µ = 0.8 Figure 4: Mean evacuation time for the rational model as a function of η/π for differentvalues of µ . The results were obtained by averaging over 1000 realizations, considering aroom of size L = 100 with a door of size l d = 6 d filled to a 40% of its capacity, containing N = 1000 agents. E-3 0.01 0.1 1400450500550600650700750 rational = /8 =2 /8 =3 /8 =4 /8 e x i t t i m e stochastic, =0.55 Figure 5: Mean evacuation time as a function of µ . Here each curve corresponds to adifferent choice of η . We show the rational and the stochastic cases, for the last one, thecase when α is optimum. The horizontal axis is in units of d − . The results were obtainedby averaging over 1000 realizations, considering a room of size L = 100 with a door of size l d = 6 d filled to a 40% of its capacity, containing N = 1000 agents. .001 0.01 0.1 1 m e x i t t i m e l d = 2d l d = 4d l d = 6d l d = 8d l d = 10d l d d -1 m a e x i t t i m e l d d -1 a (a)(b) rationalstochastic Figure 6: Mean evacuation time a) for the rational case and as a function of µ , b) forthe stochastic case as a function of α . The results were obtained by averaging over 1000realizations, considering a room of size L = 100 and different door sizes, as shown in b).The room contained N = 1000 agents. in = 1000 m = . m = . m = . t = 88t = 71 t = 435t = 308t = 194t = 163 t = 270 t = 374t = 327t = 228t = 142t = 63N in = 800 N in = 600 N in = 400 N in = 200 Figure 7: Snapshots for different instances of the evacuation process with rational agents(black circles) for different values of µ and η = π/ . The red dots represent the conflictsbetween two or more agents that are solved randomly with just one them advancing tothe desired position. The green square represents de position and width of the door atthe bottom wall of the room with L = 100 and l d = 6 d . The initial condition on theleft column is the same in all cases, with the room filled to a 40% of its capacity with N = 1000 agents, while the blue curve corresponds to the trajectory of one of the agentsuntil it leaves the room. affects the distribution of the available space and consequently the numberof blockages. Still, there is something else we can do to prevent clogging.Namely, to increase the parameter µ , which characterizes the minimal emptyspace that a given individual has to observe for attempt a motion in certaindirection.In Figure 4 we show the exit-time vs. η curves for several values of µ within the rational model. We observe that the effect of preserving certaindistance from the other pedestrians favors the evacuation speed unless thisdistance is too large and so the restriction has a completely different effect:the effective free space to move is reduced to the point that no one can move.As a consequence, the dependence of the exit time on µ shows that there is14 in = 1000 α = . α = . α = . t = 173t = 116 t = 1099t = 689t = 427t = 265 t = 418 t = 585t = 512t = 357t = 222t = 98N in = 800 N in = 600 N in = 400 N in = 200 Figure 8: Snapshots for different instances of the evacuation process with stochastic agents(black circles) for different values of α , µ = 0 . , and η = π/ . The red dots representthe conflicts between two or more agents that are solved randomly with just one themadvancing to the desired position. The green square represents de position and width ofthe door at the bottom wall of the room with L = 100 and l d = 6 d . The initial conditionon the left column is the same in all cases, with the room filled to a 40% of its capacitywith N = 1000 agents, while the blue curve corresponds to the trajectory of one of theagents until it leaves the room.
15n optimal value ( µ ≈ . for a door of size l d = 6 d ). When µ > . thereis a considerable proportion of realizations that show transient or permanentblockages. Surprisingly, this phenomenon is not replicated when consideringthe stochastic model, for which increasing the value of µ results always in alonger exit time, as shown in Figure 5. This fact might be indicating thatthe randomness in the lateral motion (measured by α ) is already efficient increating empty spaces and adding the effect of µ results also in an effectivereduction of the available space as µ increases. Although Figure 5 only showsthe exit time for α = 0 . , we have studied what happens for several othervalues of α observing in all the cases the same behavior.We must mention that the optimal value for µ in the case of rationalagents (see Figure 5), or for α in the case of stochastic ones (see Figure 3),depends on the size of the door l d as shown in Figure 6. In particular, theinsets in the figure shows the dependence of the position of the minimal, µ and α , as a function of l d for the rational and stochastic cases, respectively.As apparent from the figure, these results suggest that, in order to optimizethe flow of pedestrians out of the room in an evacuation situation, one needsonly to control the mean distance among the pedestrians in relation to thesize of the door itself (and the mean diameter of the agents): the narrowerthe door, the more distance is required among agents to keep an optimal flowthrough the door.Now let us go back to the interpretations of large and small values ofthe parameters µ, η and α as signatures of cooperative or egotistic behavior,respectively. From this point of view, the fact that we find non vanishingvalues of these parameters minimizing the exit time indicates that neitherthe most egotistic nor the most cooperative behavior of individuals is optimalfrom the perspective of the crowd. In the following section we analyze theconnection of this results with a well known effect observed in evacuationprocesses. faster-is-slower effect The so called faster-is-slower effect, originally predicted in [2], is asso-ciated to the fact that the optimal evacuation time through a narrow dooris minimized at a given critical value at which the pedestrians try to move[5, 22]. Hence, for desired velocities above the critical one, we have that thelarger the velocity, the slower the evacuation process. This effect has beenrecently confirmed experimentally for many different systems [23], suggest-ing that this phenomenon as a universal behavior for active matter passing16hrough a narrowing.As we now indicate, some of the simulations results presented before haveclear analogies with the faster-is-lower effect. In the present model, the speedof movement is not a parameter as in [2], it is always d though the movesmight be limited due to the lack of free space. The haste of the pedestrianto leave the room is captured in different ways by the model parameters. Forinstance, a small value of µ indicates an impatient or hurried behavior, sincethe pedestrians try to take advantage of every empty space available, withoutcaring about the distance to other individuals. Meanwhile, low values of η and α ( only in the stochastic model) indicate a propensity to walk straight tothe exit, and a low disposition to perform lateral motions or temporal turnoffs which may benefit the whole group. Thus, for instance, the increasein the exit time with decreasing µ obtained for the rational model at lowvalues of µ resemble the faster-is-slower effect. The same can be said aboutthe increasing of the exit time obtained for decreasing α observed within therandom model at small α , or the increasing with decreasing η for both modelsat small η . It is worth saying that in the three cases there is a region where theexit time decrease monotonically, until reaching the critical values at whichthe faster is slower effect starts to be observed. This monotonic behaviourhas been observed when the the pedestrians only show a moderated haste[24].To understand the dynamical properties that lead to the faster-is-slower- like effect at small values of the parameters, we analyze in more detail theevolution of the populations and individual position for various simulations.In Figures 7 and 8 we present snapshots of different instances of the evacua-tion process for the rational and stochastic models, respectively, consideringthe same initial condition. The bottom row presents the optimal situation ineach case for the value of η and l d (and µ for the stochastic case) considered.Notice how the generation of available space between the agents, throughthe threshold for the rational case (associated with the parameter µ ) or byperforming random lateral movements for the stochastic one (associated withthe parameter α ), speeds up the evacuation process.Moreover, these results put in evidence an interesting fact regarding theeffects of µ on the rational agents. As mentioned before, if a rational agentcannot advance towards the door a distance of at least µ , it will opt for a lat-eral movement. In contrast, a stochastic agent will randomly chose a lateralmovement depending only on the value of α . In both cases the agent willstay still if it cannot advance a distance of at least µ in the chosen direction.17lease notice that, for the stochastic agents, this threshold has been takeninto account with µ = 0 . in the figure. Nonetheless, the rational agentsrequire a larger value of µ to optimize their exit time, as their decision mak-ing process regarding the implementation of frontal or lateral movements istied to this parameter, contrary to what happens with the stochastic agents.Therefore, this points to the fact that it is the implementation of a sizableamount of lateral movements in both cases, with the consequent generationof space among agents, the most important feature to optimize the exit timeof the whole group.In each of the plots in the first columns of Figs. 7 and 8, we also show thetrajectory for one of the agents (blue solid curves). It can be seen that thetrajectories for the stochastic model are more winding that those from therational one. This leads to the larger exit times observed in the stochasticmodel.The faster-is-slower -like effects reminds us of social dilemmas . This means,situations in which there is a conflict between individual and collective in-terests. In a social dilemma, each individual receives a higher payoff for asocially defecting choice than for a socially cooperative choice, no matterwhat the other individuals do, but all individuals are all better off if all co-operate than if all defect. The payoff to be selfish is higher than the payoffto be non-selfish, but group members are worse off if everyone is selfish thanif everyone is not [27, 28]. In an evacuation scenario, the individuals wouldin principle always profit from moving in a faster way to the exit, but ifeverybody does the same, the result is an increase in the evacuation time, acollective disadvantage.
4. Final remarks and conclusions.
Recently it was suggested that formalisms with more simple assumptionsthan the Social Force model could lead to satisfactory predictions of pedes-trian behavior [21]. In that paper, the authors assume that the reactions ofthe pedestrians are mostly governed by the visually perceived environmentand the active search for empty spaces. The present work points in the samedirection. In the models here proposed, the movement of the pedestrians isdriven by the wish of reaching the exit and the seek for empty space to move,and limited by the competition for space with other pedestrians.Due to their relevance, the behavioral aspects involving the characterand internal motivations of the pedestrians has call the attention of scientist18nterested in modeling evacuation processes. The most simple but powerfulapproximation is to consider that there are two types of characters, promptedsometimes as defectors and collaborators or patient and impatient. Those twoarchetypes represent the usual conflict presented in any social dilemma: thecompromise between the collective and the individual interest and the choiceto favor one or the other. A given behavioral profile or strategy of motion,however, result from a combination of several factors that are reflected inand affect the pedestrian dynamics in different ways. The present work triesto disentangle these effects to promote a better understanding of the results.First, our model helps us to distinguish between the behavioral featuresaffecting the decision making processes associated to the selection of the nextmove, and those associated to the behavior at conflicts in the competitionfor space. To emphasize the relevance of the distinction between these twolevels of behavior, we can imagine two extreme but possible attitudes of apedestrian. First, we can think on an individual who is inpatient or unkindin his/her search for space, but who avoid physical contact and, thus, acts asa cooperative person in conflicts. This behavior could correspond to that ofa disrespectful but coward individual. On the other hand, we can think onan individual who is gentle in his/her search for possibilities of motion, andtries to avoid conflicts but who becomes a strong competitor in any conflictin which he/she is forced to enter.Our studies focus on the analysis of the influence of different behavioralfeatures arising on the first stage, i.e. on the use of empty space. At this level,the recognition of available spaces and the decision processes for steppingis affected by the internal moods and character of the individuals. Theconsequences of these internal states on the stepping behavior are modeledthrough a set of independent parameters defining the dynamics. Namely, µ , which characterizes the tendency to keep distance from others, η , whichmeasures the fluctuations in the definition of the preferred direction, and α ,which defines the probability of a lateral movement (only for the random i.e. no rational model). We have argued that low values of these parameters canbe associated to impatient or defective behaviors while relatively large valuescorrespond to more cooperative attitudes. Our results show that, in mostcases, the exit time is minimized at intermediate values of the parameters,indicating that neither the most cooperative nor the most defective attitudesare optimal from the point of view of the crowd. We have then argued that theexistence of regions of the parameters for which the exit time grows with thehurry or impatience of the pedestrians indicates global behaviors analogous to19he faster is slower effect. We have shown that a thorough explanation of theslowing down of an evacuation process is due to an amalgam of several effectsbut ultimately connected to the reduction of the available empty space.As part of our studies we have shown that a distance preserving attitudefavors the evacuation by inhibiting the clogging. Perhaps one of the most in-teresting implication of this results is the possibility that, by communicatingsimple basic instructions to the pedestrians, the evacuation process can beoptimized. If people could overcome panic by having clear instructions, thebehavior associated to the rational walker with the addition of maintainingcertain distance from others would clearly help.As a final remark, we want to emphasize the claim given in the intro-duction concerning the utility that the model here developed may have forstudying general features of pedestrians dynamics, as well as for particularapplications.
5. Acknowledgments
The authors gratefully acknowledge the computing time granted on thesupercomputers THUBAT-KAAL (CNS-IPICyT) and MIZTLI (DGTIC-UNAM).VD is grateful to V.M. Kenkre (UNM) for his hospitality. The authors ac-knowledge partial financial support from CONACyT and from VIEP-BUAP(Mexico) and CONICET (Argentina).
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