Modeling Helping Behavior in Emergency Evacuations Using Volunteer's Dilemma Game
MModeling Helping Behavior in EmergencyEvacuations using Volunteer’s Dilemma Game
Jaeyoung Kwak , Michael H. Lees , Wentong Cai , and Marcus E. H. Ong , Nanyang Technological University, Singapore 639798, Singapore { jaeyoung.kwak, aswtcai } @ntu.edu.sg University of Amsterdam, Amsterdam 1098XH, The Netherlands [email protected] Singapore General Hospital, Singapore 169608, Singapore Duke-NUS Medical School, Singapore 169857, Singapore [email protected]
Abstract.
People often help others who are in trouble, especially inemergency evacuation situations. For instance, during the 2005 Londonbombings, it was reported that evacuees helped injured persons to escapethe place of danger. In terms of game theory, it can be understood thatsuch helping behavior provides a collective good while it is a costly behav-ior because the volunteers spend extra time to assist the injured personsin case of emergency evacuations. In order to study the collective effectsof helping behavior in emergency evacuations, we have performed numer-ical simulations of helping behavior among evacuees in a room evacuationscenario. Our simulation model is based on the volunteer’s dilemma gamereflecting volunteering cost. The game theoretic model is coupled witha social force model to understand the relationship between the spatialand social dynamics of evacuation scenarios. By systematically changingthe cost parameter of helping behavior, we observed different patterns ofcollective helping behaviors and these collective patterns are summarizedwith a phase diagram.
Keywords:
Emergency Evacuation · Helping Behavior · Game Theory · Volunteer’s Dilemma Game · Social Force Model
Pedestrian emergency evacuation is a movement of people from a place of dangerto a safer place in case of life-threatening incidents such as fire and terrorist at-tacks. Numerical simulation has been a popular approach to perform pedestrianemergency evacuation studies, for instance, predicting total evacuation time ina class room [1] and preparing an optimal evacuation plan for a large scalepedestrian facility [2].Based on numerical simulations, it has been identified that evacuees are oftenin conflict with others when more than two evacuees try to move to the sameposition [3]. Game theory has been used to model strategic interactions amongevacuees in such a conflict. Under game theoretic assumptions, each evacuee a r X i v : . [ phy s i c s . s o c - ph ] J un J. Kwak et al. has his own strategies and selects a strategy in a way to maximize his ownpayoff. Various emergency evacuation simulations have been performed based ondifferent game theory models including evolutionary game [4], snowdrift game [5],and spatial game [6,7].Although those game theory models successfully modeled evacuees egress, es-pecially from a room, other aspects of evacuees behavior such as helping behaviorhave not been sufficiently studied. In the context of emergency evacuations, ithas been reported that evacuees help injured evacuees to evacuate from the placeof danger, for instance the WHO concert disaster occurred on December 3, 1979in Cincinnati, Ohio, United States [8] and 2005 London bombings in UnitedKingdom [9].A few studies have investigated helping behavior in emergency evacuationby means of pedestrian simulation. Von Sivers et al. [10,11] applied social iden-tity and self-categorization theories to pedestrian simulation in order to simulatehelping behavior observed in 2005 London bombings. In their studies, they as-sumed that all the evacuees share the same social identify which makes them bewilling to help others rather than be selfish. Lin and Wong [12] applied the vol-unteers dilemma game [13,14] to model the behavior of volunteers who removedobstacles from the exit. Their work can be considered as a helping behaviormodeling study in that some evacuees were voluntarily removing the obstaclesso they helped others in the same room to evacuate faster.One can observe that such a helping behavior provides a collective good incase of emergency evacuations. This is especially true when there are not enoughrescuers, more injured persons can be rescued with the help of other evacueesthan by only the rescuers. In order to study the collective effects of helpingbehavior in emergency evacuations, we have developed an agent-based modelsimulating such helping behaviors among evacuees. Based on the agent basedmodel, we represent individual behaviors with a set of behavioral rules and thensystematically study collective dynamics of interacting individuals. In our agentbased model, we assumed that helping an injured person can be a costly behaviorbecause the volunteer spends extra time and take a risk to assist the injuredperson in the evacuation. If individuals feel that helping behavior is a costlybehavior for them, they might not turn into volunteers. Thus we implementedthe volunteers dilemma game model [13,14] to reflect the cost of helping behavior.Pedestrian movement is simulated based on social force model [15].The remainder of this paper is organized as follows. The simulation modeland its setup are explained in Sec. 2. We then present its numerical simulationresults with a phase diagram in Sec. 3. Finally, we discuss the findings of thisstudy in Sec. 4.
We employ the volunteers dilemma game model to study helping behavior ofpassersby in a room evacuation [13,14]. A passerby is an evacuee who is not odeling Helping Behavior in Emergency Evacuations 3 injured and can play the volunteers dilemma game model. According to thevolunteers dilemma game, two types of players are considered. Passerby i can beeither a volunteer (C) who helps an injured person to evacuate or a bystander (D)who does not help the injured person. Once a passerby decides to be a volunteer,he approaches to and then rescues the injured person. We can express the payoffof player i in terms of collective good U and volunteering cost K < U , see Table 1.The payoff of a bystander (D) is U if there is at least one volunteer, 0 otherwise.It can be understood that, bystanders are benefited by the volunteer. However,if nobody volunteers, the collective good U cannot be produced because all theplayers are bystanders. The collective good U can be produced by volunteerswhen they rescue injured persons. For simplicity, we assume that the value of U is constant if there is at least one volunteer. The payoff of a volunteer (C)is always U − K , indicating that his payoff is constant regardless other players’choice. Table 1.
Payoff of a volunteer (C) and a bystander (D) for the different number ofother players choosing C (based on Refs. [13,14]). Here, U is the collective good, K < U is the volunteering cost, and N ≥ Player i ’s choice The number of other players choosing C0 1 2 ... N − U − K U − K U − K U − K U − K Bystander (D) 0
U U U U
Actor i ’s expected payoff E i is given as: E i = q i − N (cid:89) j (cid:54) = i q j U + (1 − q i )( U − K ) . (1)Here, q i is the probability that player i chooses D and 1 − q i for choosing C.The number of players is indicated by N . The probability that all players j (cid:54) = i choose D is denoted by (cid:81) q i and 1 − (cid:81) q i indicates the probability that at leastone actor j (cid:54) = i chooses C. The first term on the right hand side reflects thepayoff of player i when he selects D but benefited when there is at least onevolunteer. The second term on the right hand side indicates the payoff of player i if he selects C.We assume that player i adopts the mixed-strategy which is the best strategyfor him. In a mixed-strategy equilibrium, every action played with positive prob-ability must be a best response to other players mixed strategies. This impliesthat player i is indifferent between choosing C and D, so a small change in thepayoff E i with respect to q i (i.e., the probability of choosing D) becomes zero: dE i dq i = − U N (cid:89) j (cid:54) = i q j + K = 0 . (2) J. Kwak et al. p i N β = 0.01 β = 0.1 β = 0.5 β = 0.7 p * N β = 0.01 β = 0.1 β = 0.5 β = 0.7 Fig. 1.
Bystander effect on helping behavior: (a) p i , the probability that player i vol-unteers to rescue an injured person and (b) p ∗ , the probability that an injured personis rescued. After assuming q i = q j , we can obtain probability that player i chooses D q i = (cid:20) KU (cid:21) N − = β N − , (3)where β = K/U is cost ratio, which can be interpreted as the risk of volunteering.Accordingly, the probability that player i chooses C is given as p i = 1 − q i = 1 − β N − . (4)The probability that at least one player selects C is denoted by p ∗ , i.e., p ∗ = 1 − q Ni = 1 − β NN − . (5)Equations 4 and 5 show good agreement with the bystander effect, see Fig. 1.Figure 1(a) shows a decreasing trend of p i as the number of players N increases,inferring that players are less likely to volunteer seemingly because they believeother players will volunteer. Note that the social pressure from other players isnot considered here, so the existence of volunteers does not affect on players’behavior. Figure 1(b) presents the trend of p ∗ which reflects the chance that aninjured person is rescued. As the number of players N increases, the value of p ∗ approaches to a certain value, 1 − β . According to the social force model [15], the position and velocity of each pedes-trian i at time t , denoted by x i ( t ) and v i ( t ), evolve according to the following odeling Helping Behavior in Emergency Evacuations 5 equations: d x i ( t )d t = v i ( t ) (6)and d v i ( t )d t = f i,d + (cid:88) j (cid:54) = i f ij + (cid:88) B f iB . (7)In Eq. (7), the driving force term f i,d = ( v d e i − v i ) /τ describes the tendencyof pedestrian i moving toward his destination. Here, v d is the desired speed and e i is a unit vector indicating the desired walking direction of pedestrian i . Therelaxation time τ controls how quickly the pedestrian adapts ones velocity tothe desired velocity. The repulsive force terms f ij and f iB reflect his tendencyto keep certain distance from other pedestrian j and the boundary B , e.g., walland obstacles. A more detailed description of the social force model can be foundin previous studies [15,16,17,18].
100 pedestrians
Fig. 2.
Schematic depiction of the numerical simulation setup. 100 pedestrians areplaced in a 10m ×
10m room indicated by a yellow shade area. Pedestrians are leavingthe room through an exit corridor which is 5 m long and 2 m wide. The place of safetyis set on the right, outside of the exit corridor.
Our agent-based model consists of helping behavior model and movementmodel. The helping behavior model computes the probability that a passerby
J. Kwak et al. would help an injured person based on the volunteer’s dilemma game. The move-ment model calculates the sequence of pedestrian positions for each simulationtime step. Our agent-based model was implemented from scratch in C++.Each pedestrian is modeled by a circle with radius r i = 0 . N = 100pedestrians are placed in a 10m ×
10m room indicated by a yellow shade area inFig. 2. Pedestrians are leaving the room through an exit corridor which is 5 mlong and 2 m wide. The place of safety is set on the right, outside of the exitcorridor. There are N i injured persons who need a help in escaping the roomand N = N − N i passersby who are ambulant. Some passersby might turn intovolunteers who are going to approach to and then rescue the injured persons.The number of volunteers is determined based on the volunteer’s dilemma gamepresented in Sec. 2.1.The volunteer’s dilemma game is updated for each second. We assumed thatthe volunteer’s dilemma game is a macroscopic behavior like goal selection andpath navigation patterns [19]. In line with Heli¨ovaara et al. [6], each passerbycan play the volunteer’s dilemma game a few times during the whole simulationperiod. With the update frequency of one time per second, most of passersbyplay the volunteer’s dilemma game up to ten times before they leave the room. Apasserby can decide whether he will volunteer to rescue an injured person withina range of 3 m. Once the volunteer decides to rescue the injured person, thenhe shifts his desired direction walking vector e i toward the position of injuredperson. Once the volunteer reaches the injured person, he will flee to the placeof safety with the injured person after a preparation time of 5 s.The pedestrian movement is updated with the social force model in Eq. (7).The passersby move with the initial desired speed v d = v d, = 1 . τ = 0 . v max = 2 . et al. [10,11]. We applied speedreduction factor α = 0 . v d = αv d, = 0 . v i ( t + ∆t ) = v i ( t ) + a i ( t ) ∆t, (8) x i ( t + ∆t ) = x i ( t ) + v i ( t + ∆t ) ∆t. (9)Here, a i ( t ) is the acceleration of pedestrian i at time t which can be obtainedfrom Eq. (7). The velocity and position of pedestrian i is denoted by v i ( t ) and x i ( t ), respectively. The time step ∆t is set as 0.05 s. Figure 3 shows snapshots of our agent-based model simulations. Open blackcircles indicate injured persons and full dark circles show volunteers helping theinjured persons. Gray circles represent the passersby. If the helping cost is low, odeling Helping Behavior in Emergency Evacuations 7 (a) (b)
Fig. 3.
Snapshots of helping behavior in a room evacuation scenario: (a) all the injuredpersons are rescued in case of N i = 15 and β = 0 .
1, and (b) some injured personsare not rescued (in the red dotted circle) in case of N i = 15 and β = 0 .
2. Open blackcircles indicate injured persons and full dark circles show volunteers helping the injuredpersons. Gray circles represent the passersby. J. Kwak et al. all rescuednot all rescued β N i Fig. 4.
Schematic phase diagram of collective helping behavior in the room evacuationscenario. all the injured persons are likely to be rescued, as shown in Fig. 3(a). However,if the helping cost is high (i.e., high β ), then some injured persons might not berescued, see the red dotted circle in Fig. 3(b). By systematically changing thevalue of N i and β , we observed different patterns of collective helping behaviorssummarized in the schematic phase diagram (see Fig. 4). For each parametercombination ( N i , β ), we performed 30 independent simulation runs.We also looked into the impact of cost ratio β and the number of injuredpersons N i on the total evacuation time T . The total evacuation time T isdefined as the length of period from the start of evacuation to the moment whenthe last evacuee leaves the exit corridor. We measured the average and standarddeviation of the total evacuation time, i.e., T avg and T sd , based on the values oftotal evacuation time T obtained over 30 independent simulation runs for eachparameter combination ( N i , β ). Figure 5 indicates that change in the value of β does not make a noticeable different to the average total evacuation time T avg .This is seemingly because β only affects the probability that a passerby turnsinto a volunteer. As indicated in Fig. 6(a), the average total evacuation time T avg increases as the number of injured persons N i grows. Having more injuredpersons indicates that there are more volunteers who move in the reduced desiredspeed, so the total evacuation time increases due to the volunteers rescuing theinjured persons. In addition, the standard deviation of total evacuation time T sd increases as the number of injured persons N i grows, in that the difference inevacuation time among evacuees gets larger. odeling Helping Behavior in Emergency Evacuations 9 T a v g ( s ) β N i = 15N i = 12N i = 9N i = 6N i = 3 Fig. 5.
Average total evacuation time T avg as a function of the number of injuredpersons N i and β . (a) T a v g ( s ) N i (b) T s d ( s ) N i Fig. 6.
Total evacuation time in case of β = 0 . N i : (a) average T avg and (b) standard deviation T sd . We have numerically investigated helping behavior among evacuees in a roomevacuation scenario. Our simulation model is based on the volunteer’s dilemma et al. game reflecting volunteering cost and social force model simulating pedestrianmovement. We characterized collective helping behavior patterns by systemati-cally controlling the values of cost ratio β and the number of injured pedestrians N i . For low cost ratio values, one can expect that all the injured pedestriansare rescued by volunteers. For high cost ratio values, on the other hand, it wasobserved that not all the injured persons can be rescued. When the number ofinjured persons is large, a low value of cost ratio yields a result that all theinjured pedestrians are rescued. A schematic phase diagram summarizing thecollective helping behavior patterns is presented.A very simple room evacuation scenario has been used in order to study thefundamental role of helping behavior in the evacuation especially the numberof evacuated pedestrians. In this study, the severity of injuries are assumed tobe the same for all the injured persons, so each injured person can be rescuedby a volunteer. According to patient triage scale in Singapore [23,24], patientscan be categorized based on the severity of injuries and the desired number ofvolunteers are different for various types of injuries. Future work can reflect theimpact of patient injury levels in collective helping behavior by assuming differ-ent number of required volunteers for each patient. This study can be extendedfrom the perspective of game theory. As stated in Diekmann’s study [14], it canbe interesting to introduce different values of collective good U and volunteeringcost K to each passerby. By doing that, we can reflect personal difference inwillingness to volunteer in emergency evacuations. In addition, one can imaginethat the value of U can be changed depending on the number of injured personsand volunteers. For instance, the values of U are different when there are oneinjured person, one volunteer and two injured persons, three volunteers. Evolu-tionary game [4] can be also introduced in order to reflect behavioral changes ofpassersby influenced by the existence of volunteers, which might be observablein emergency evacuations. Acknowledgements
This research is supported by National Research Foundation (NRF) Singapore,GOVTECH under its Virtual Singapore program Grant No. NRF2017VSG-AT3DCM001-031.
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