Game theoretic modeling of helping behavior in emergency evacuations with committed volunteers
Jaeyoung Kwak, Michael H. Lees, Wentong Cai, Ahmad Reza Pourghaderi, Marcus E.H. Ong
GGame theoretic modeling of helping behavior in emergency evacuations
Jaeyoung Kwak , ∗ Michael H. Lees , Wentong Cai , Ahmad Reza Pourghaderi , , and Marcus E.H. Ong , Complexity Institute, Nanyang Technological University, Singapore Informatics Institute, University of Amsterdam, The Netherlands School of Computer Science and Engineering, Nanyang Technological University, Singapore Health Systems Research Center (HSRC), Singapore Health Services, Singapore Department of Emergency Medicine, Singapore General Hospital, Singapore and Health Services and Systems Research (HSSR), Duke-NUS Medical School, Singapore (Dated: January 8, 2021)We study the collective helping behavior in a room evacuation scenario in which two volunteersare required to rescue an injured person. We propose a game theoretic model to study the evolutionof cooperation in rescuing the injured persons. We consider the existence of committed volunteerswho do not change their decision to help the injured persons. With the committed volunteers, allthe injured persons can be rescued depending on the payoff parameters. In contrast, without thecommitted volunteers, rescuing all the injured persons is not achievable on most occasions becausesome lonely volunteers often fail to find peers even for low temptation payoff. We have quantifiedvarious collective helping behaviors and summarized those collective patterns with phase diagrams.In the context of emergency evacuations, our study highlights the vital importance of the committedvolunteers to the collective helping behavior.
I. INTRODUCTION
Emergency evacuation simulations have received muchattention since Helbing et al. [1] characterized collectivephenomena in escape panic based on numerical simula-tions of self-driven many-particles. Some studies havestated that when people are escaping from the place ofdanger, they tend to move faster than their normal speedand behave in an individualistic manner, which may leadto people pushing each other [1–4]. In the majorityof studies, numerical simulations of emergency evacua-tion were developed based on such assumptions. In viewof this, various pedestrian emergency evacuation studieswere performed, for instance, predicting total evacuationtime in a class room [5] and preparing an optimal evacu-ation plan for a large scale pedestrian facility [6].However, other studies have presented evidence thatevacuees helped injured persons to escape the placeof danger, for instance, the WHO concert disaster oc-curred on December 3, 1979 in Cincinnati, Ohio, UnitedStates [7] and the 2005 London bombings in United King-dom [8]. Researchers have investigated helping behaviorin emergency evacuation by means of pedestrian simu-lations. For instance, von Sivers et al. [9, 10] appliedsocial identity theory to pedestrian simulation in orderto simulate helping behavior observed in the 2005 Lon-don bombings. In their studies, they assumed that allthe evacuees share the same social identify which makesthem willing to help others rather than be selfish.Although several studies focused on simulating helpingbehavior in emergency evacuations, little attention hasbeen paid to the strategic interactions among evacuees.When an evacuee is helping an injured person to escape,the evacuee’s helping behavior can be seen as an attemptto increase a collective good. This is especially true when ∗ [email protected] there are not enough dedicated rescue personnel, in suchcircumstances other evacuees may help one another. Atthe same time, helping an injured person can be a costlybehavior because the volunteering evacuee spends extratime and takes a risk to assist the injured person. If theevacuee feels that helping behavior is a costly behaviorfor him, he might not help the injured person.Game theory is a useful tool to predict how individu-als decide their strategy in response to others’ strategy.Under game theoretic assumptions, individuals are likelyto select a strategy in a way to maximize their own pay-off. Various game theoretical models have been appliedfor emergency evacuation simulations, for example, pris-oner’s dilemma game [11], snowdrift game [12], spatialgame [13, 14], and evolutionary game [15]. In our pre-vious study [16], we employed the volunteer’s dilemmagame [17, 18] to study the impact of volunteering cost oncollective helping behavior in emergency evacuations. Weobserved different patterns of collective helping behaviorsby changing volunteering cost parameter. Nevertheless,little is known about how the existence of volunteers canlead to behavioral changes of bystanders. One can imag-ine that, through strategic interactions, the presence ofcommitted volunteers may influence on other evacuees toalso help injured people. In this way one might observe aspread of helping behavior. On the other hand, in situa-tions where two volunteers are needed to help an injuredperson, an individual volunteer might give up helping theinjured person if he cannot find a potential partner.In this work, we propose an evolutionary game theo-retic model to examine the dynamics of evacuees behav-ior influenced by other evacuees. We perform numericalsimulations for a room evacuation scenario in which twovolunteers are required to move an injured person to theplace of safety. In our numerical simulations, each evac-uee updates his strategy after strategic interactions withother evacuees within his sensory range while the evac-uee moves in the room. Hence, the spatial formationof individuals and the strategic interactions between the a r X i v : . [ phy s i c s . s o c - ph ] J a n TABLE I. Payoff of a volunteer (C) and a bystander (D) ina two-player game. Here, R is the reward for mutual coop-eration, P is the punishment for mutual defection, T is thetemptation to defect, and S is the sucker’s payoff. The entitiesindicate the payoff for the row player.Volunteer BystanderVolunteer (C) R S
Bystander (D)
T P individuals are coupled together in our numerical study.We consider the existence of committed volunteers whodo not change their decision to rescue the injured per-sons. We characterize the collective helping behaviorby systematically controlling the game payoff parametersalong with the altruism strength. By means of numer-ical simulations, we observe different collective patternsof helping behavior in a room emergency evacuation sce-nario depending on the existence of committed volun-teers and the value of control parameters. We quantifydifferent collective helping behaviors based on the simu-lation results. We then study the well-mixed populationmodel to better interpret the numerical simulation re-sults. The evolutionary game theoretic model and itsnumerical simulation setups are explained in Section II.We then present and explain the simulation results withphase diagrams in Section III. Finally, we discuss thefindings of this study in Section IV.
II. MODEL
Our agent-based model consists of a game theoreticalmodel and a movement model. The game theoreticalmodel evaluates the probability that a bystander wouldturn into a volunteer helping an injured person. The evo-lutionary game model enables us to reflect the behavioralchanges of an individual based on the behavior of otherevacuees. The movement model calculates the sequenceof pedestrian positions for each simulation time step. Inthis section, we explain the details of our game theoreti-cal model and the numerical simulation setup.
II.1. Evolutionary game model
We employ an evolutionary game model to study thebehavioral change of players influenced by other players.As in our previous study [16], we consider two strategiesof ambulant pedestrians: volunteer (C) and bystander(D). A volunteer (C) helps an injured person to evacuatewhile a bystander (D) does not help the injured person.In line with the two-player game, the payoff of individ-ual i ’s strategy can be presented in a payoff matrix inTable I. When individual i meets individual j , there arefour types of payoff depending on strategy of each indi-vidual. If both the individuals are volunteers (C), theyreceive a reward for mutual cooperation, denoted by R . -1 0 1
0 1 2
H SDSH PD S T FIG. 1. Schematic representation of four different games on(
T, S ) space: the Harmony (H), the Snowdrift (SD), the Pris-oner’s dilemma (PD), and the Stag hunt (SH) games.
When both of them are bystanders (D), their payoff is P which is punishment for mutual defection. When by-stander i is a volunteer (C) and bystander j is a by-stander (D), bystander i receives the sucker’s payoff S which is associated with the unreciprocated cooperationcost [19]. At the same time, bystander j receives T whichreflects the temptation to defect. According to previouswork [20–22], the ( T, S ) space can be divided into fourareas: the Harmony (H), the Snowdrift (SD), the Pris-oner’s dilemma (PD), and the Stag hunt (SH) games [seeFig. 1].We assume that two volunteers are required to movean injured person from the room to the place of safety.A lonely volunteer cannot move the injured person byhimself, so he will try to find another volunteer who willmove the injured person together. A bystander near thelonely volunteer might decide to become a volunteer tohelp the lonely volunteer to move the injured person to-gether. In contrast, the lonely volunteer might give upfinding another volunteer and turn into a bystander.In the presented evolutionary game, player i considersnot only his own payoff but also player j ’s payoff. Assuggested by [21, 23], we express the effective payoff ofplayer i , u i , as a linear combination of player i ’s and j ’spayoffs: u i = (1 − Q ) π i + Qπ j , (1)where π i and π j denote the payoff of player i and j pre-sented in Table I, respectively. The altruism strength Q reflects how much player i regards player j ’s payoff,which is related to the degree of inequality aversion [24].For simplicity, all the players are assumed to have thesame Q value. Table II shows the effective payoff of avolunteer (C) and a bystander (D) according to Eq. 1.In each round, ambulant pedestrian i randomly selectsa neighbor j and adopts the neighbor j ’s strategy withthe probability [25–27]: p (∆ u ij ) = 11 + exp(∆ u ij /k ) , (2)where k reflects uncertainties in the game dynamics and∆ u ij = u i − u j is payoff difference. In case of k = 0, the TABLE II. The effective payoff of a volunteer (C) and a by-stander (D) in the presented evolutionary game. Here, Q isthe altruism strength reflecting how much player i regardsplayer j ’s payoff [21, 23, 24]. The entities indicate the payofffor the row player. Volunteer (C) Bystander(D)Volunteer (C) R e = R S e = (1 − Q ) S + QT Bystander (D) T e = (1 − Q ) T + QS P e = P
100 pedestrians
10m 2m10m 5m
FIG. 2. Schematic depiction of the numerical simulationsetup. 100 pedestrians are placed in a 10m ×
10m room in-dicated by a blue shade area. Pedestrians are leaving theroom through an exit corridor which is 5 m long and 2 mwide. The place of safety is set on the right, outside of theexit corridor. probability p (∆ u ij ) becomes 1 if u i < u j , and p (∆ u ij )is 0 when u i > u j . In case of k → ∞ , the probabil-ity p (∆ u ij ) becomes 0.5, indicating completely randomdecision making. II.2. Social force model
We describe the movement of pedestrians based on thesocial force model [28]. The position and velocity of eachpedestrian i at time t , denoted by (cid:126)x i ( t ) and (cid:126)v i ( t ), areupdated according to the following equations:d (cid:126)x i ( t )d t = (cid:126)v i ( t ) (3)and d (cid:126)v i ( t )d t = (cid:126)f i,d + (cid:88) j (cid:54) = i (cid:126)f ij + (cid:88) B (cid:126)f iB . (4)In Eq. (4), the driving force term (cid:126)f i,d describes the ten-dency of pedestrian i moving toward his destination. Therepulsive force terms (cid:126)f ij and (cid:126)f iB reflect his tendency tokeep certain distance from other pedestrian j and theboundary B , e.g., wall and obstacles. We refer the read-ers to Appendix A for further detailed description of thepresented social force model. II.3. Numerical simulation setup
Each pedestrian is modeled by a circle with radius r i = 0 . N = 100 pedestrians in a TABLE III. Game theoretic model parametersModel parameter symbol valuemutual cooperation payoff R P T [0, 1]sucker’s payoff S [-1, 1]altruism strength Q [0, 0.5]sensory range l s k ×
10m room indicated by a blue shaded area in Fig. 2.Pedestrians are leaving the room through an exit corridorwhich is 5 m long and 2 m wide. The place of safety isset on the right, outside of the exit corridor. The pedes-trian movement is updated with the social force modelin Eq. (4) [see Appendix A for more details].There are N i = 10 injured persons who need a helpin escaping the room and N = N − N i = 90 ambulantpedestrians who are either volunteers or bystanders. Inthe beginning of numerical simulations, one volunteer isselected for each injured person from the set of his neigh-boring pedestrians within the sensory range l s , thus thereare N c, = 10 initial volunteers. As described in Sec-tion II.1, we assume that an injured person needs twovolunteers, thus the maximum number of volunteers is20. We have performed numerical simulations for thecases without and with committed volunteers. In thecase without committed volunteers, the initial volunteerscan change their strategy in the simulations. In the othercase, where the initial volunteers are committed, the ini-tial volunteers do not change their strategy. The numberof volunteers evolves according to the result of strategicinteraction with other evacuees.All the ambulant pedestrians play the presented evolu-tionary game at each 0 . i randomly selects pedestrian j within the sensoryrange l s and then evaluates the probability of switchingto pedestrian j ’s strategy according to Eq. (2). Once abystander interacts with a lonely volunteer and decides tocooperate with the lonely volunteer to rescue an injuredperson, the bystander turns into a volunteer. Next, thenew volunteer shifts his desired walking direction vectortoward the position of injured person. After arriving atthe injured person, the new volunteer will evacuate theinjured person with the peer volunteer after a prepara-tion time of 5 s. On the other hand, if a lonely volunteerdecides to become a bystander, then he changes his de-sired walking direction vector toward the exit. To imple-ment the presented game theoretic model, the parametervalues in Table III are selected based on the previouswork [21, 26, 29]. Note that we control the game payoffparameters T and S along with the altruism strength Q . (a)
12 34 5 t = 0.55s
12 3 t = 0.6s(b) t = 0.75s t = 0.8s FIG. 3. Representative examples of individual strategychanges in the numerical simulations. The individuals chang-ing their strategy are indicated by red dotted circle lines.Open black circles indicate the injured persons and full darkcircles denote the volunteers helping the injured persons.Green circles represent the bystanders. The presented snap-shots are captured from pedestrian trajectories generatedfrom the case without committed volunteers with T = 0 . S = 0 .
1, and Q = 0. (a) A lonely volunteer (indicated by reddotted circle 1) at simulation time t = 0 .
55 s (left) turns intoa bystander at simulation time t = 0 . t = 0 .
55 s (left) become volunteers at simu-lation time t = 0 . t = 0 .
75 s (left) gives uprescuing an injured person and become a bystander at simu-lation time t = 0 . III. RESULTS AND DISCUSSIONS
The presented game theoretic model is studied bymeans of numerical experiments. We present the resultsfrom agent-based simulations. We then study a well-mixed population model to interpret the presented re-sults.
III.1. Agent-based simulations
From the numerical experiments, we can see individ-uals who change their strategy as a result of interactionwith their neighbors, refer to Fig. 3. As can be seenfrom Fig. 3(a), two bystanders become volunteers seem-ingly due to the influence of volunteers near them. At thesame time, a lonely volunteer turns into a bystander afterinteracting with a nearby bystander. In Fig. 3(b), sim-ilarly to the observation presented in Fig. 3(a), one cansee that a lonely volunteer gives up rescuing an injuredperson and becomes a bystander after interacting witha nearby bystander. In the case with committed volun- teers, one can also see the change of individual strategyas shown in Fig. 3To quantify the level of collective helping behavior,similarly to Ref. [11], we define a normalized fraction ofcooperators n c ∈ [ − ,
1] as n c = N c − N c, N c, , (5)where N c and N c, are the number of volunteers in thestationary state and in the initial condition, respectively.A positive value of n c suggests that the number of vol-unteers N c is greater than the initial one, N c, , inferringthat some lonely volunteers successfully find peer volun-teers. In case of n c = 1, all the lonely volunteers even-tually find their peers, thus all the injured persons arerescued. A negative value indicates that some lonely vol-unteers changed their mind to become bystanders. Incase of n c = −
1, all the lonely volunteers turn into by-standers, so none of the injured persons are rescued. Thestationary state values of n c in the ( T, S ) space are pre-sented in Appendix B.Representative time series of the fraction of cooper-ators n c in the case without committed volunteers arepresented in Fig. 4. The curves in Fig. 4(a) and Fig. 4(b)are generated with the same parameter combination of T , S , and Q but different sets of random seeds. As canbe seen from Fig. 4(a), the stationary state value of n c is positive but smaller than 1. In Fig. 4(b), the station-ary state value of n c reaches to 1, suggesting that all thelonely volunteers successfully find peer volunteers, thusall the injured are rescued. However, this complete rescueis not always possible in that the number of volunteers inthe stationary state is changing depending on the randomseeds. As shown in Fig. 4(c), the stationary state valueof n c is 0, inferring that the number of volunteers is notincreased in effect. In Fig. 4(d), n c becomes negative, im-plying that some lonely volunteers turn into bystanders,thus some injured persons are not rescued. Figure 4(e)presents that n c decreases to −
1, suggesting that none ofthe injured are rescued because all the lonely volunteersturn into bystanders.Figure 5 shows same graphs as Fig. 4(a)-(c) but forthe case with committed volunteers. As indicated byFig. 5(a), the stationary state value of n c becomes 1 when T is low and S is high for a given Q . That is, all thelonely volunteers successfully find peer volunteers, thusall the injured are rescued. For intermediate values of T and S , the n c curve increases as in Fig. 5(a) but the sta-tionary state value is smaller than 1. This indicates thatsome lonely volunteers successfully find peers while otherlonely volunteers fail to do that, see Fig. 5(b). When T is high and S is low, as can be seen from Fig. 5(c), thevalue of n c curve does not increase.In the context of emergency evacuations, it is inter-esting to examine whether all the injured persons arerescued. We characterize different patterns of collectivehelping behavior by means of P r which denotes the prob-ability of complete rescue. We measure the value of P r by counting the occurrence of complete rescue over 100 -1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (a) n c -1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (b)-1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (c) n c -1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (d) time (s) -1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (e) n c time (s) FIG. 4. Representative time series of the fraction of coopera-tors n c in the case without committed volunteers. (a) The n c curve (generated with T = 0 . S = 0 .
95, and Q = 0) reachesto 1, indicating that all the lonely volunteers successfully findpeer volunteers, thus all the injured are rescued. (b) The n c curve generated from the same parameter combination of T , S , and Q as (a), but with a different set of random seeds. Thevalue of n c increases in the course of time, but the stationarystate value is smaller than 1. This implies that some lonelyvolunteers successfully find peers while other lonely volunteersfail to o so, thus not all the injured persons are rescued. (c)The stationary state value of n c is 0, showing that the numberof volunteers is not increased in effect. The presented curveis generated with T = 0 . S = 0 .
15, and Q = 0. (d) The n c curve becomes negative, inferring that some lonely volunteersturn into bystanders, thus some injured persons are not res-cued. Here, the curve is generated with T = 0 . S = − . Q = 0. (e) The n c curve ( T = 0 . S = − .
75, and Q = 0)decreases to −
1, suggesting that none of the injured are res-cued because all the lonely volunteers turn into bystanders. independent simulation runs for each parameter combi-nation (
T, S ) along with a given value of Q . Based onthe value of P r , we define three phases: full cooperation,partial cooperation, and defection phases. The full co-operation phase is characterized by P r = 1, suggestingthat the complete rescue is always achievable regardlessof random seeds. One can identify the partial cooperationphase with 1 > P r >
0, in which the complete rescue canbe seen depending on random seeds. The defection phaseis characterized by P r = 0, implying that the completerescue is not observable from simulation results. In thecase without committed volunteers [see Fig. 6], the com-plete rescue probability P r is always 0, indicating thatthe full cooperation phase does not exist. In addition,the value of P r becomes smaller as Q grows for a given -1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (a) n c -1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (b) time (s) -1.5-1-0.5 0 0.5 1 1.5
0 1 2 3 4 5 (c) n c time (s) FIG. 5. Same graphs as Figs. 4(a)-(c) but for the case withcommitted volunteers. (a) When T is low and S is high for agiven Q , the stationary state value of n c is 1, suggesting thatall the lonely volunteers successfully find peer volunteers, thusall the injured are rescued. The presented curve is generatedwith T = 0 . S = 0 .
5, and Q = 0. (b) For intermediatelevel of T and S , the n c curve increases like in (a) but thestationary state value is smaller than 1, indicating that somelonely volunteers successfully find peers while other lonelyvolunteers fail to do that. The presented n c curve is obtainedwith T = 0 . S = 0 .
1, and Q = 0. (c) If T is high and S islow, the value of n c curve does not increase, denoting that thenumber of volunteers is not increased in effect. The presentedcurve is generated with T = 1 . S = − .
5, and Q = 0.
0 0.2 0.4 0.6 0.8 1 (a) P r S T = 0T = 0.2T = 0.4T = 0.6T = 0.8
0 0.2 0.4 0.6 0.8 1 (b) S T = 0T = 0.2T = 0.4T = 0.6T = 0.8
FIG. 6. Complete rescue probability P r in the case withoutcommitted volunteers as a function of S and T for differentvalues of Q : (a) Q = 0 (low level) and (b) Q = 0 . Q is not presented here because P r is always zero for the entire ( T, S ) space, implying thata complete rescue is not possible. For each parameter com-bination of (
T, S ), increasing Q yields lower P r . In the casewithout committed volunteers, P r = 1 is not realized. value of ( T, S ). In the case with committed volunteers[refer to Fig. 7], P r becomes 1 with high values of S when Q is at low level. This suggests the existence of full co-operation phase, see Fig. 7(a). For high level of Q , thevalue of P r tends to approach 1, but P r is still smallerthan 1. This implies that the full cooperation phase isnot observable when Q is high.Figure 8 summarizes numerical results of phase charac-terizations. The parameter space of ( T, S ) is divided intodifferent phases by means of the complete rescue proba-bility P r . In the case without committed volunteers [see -1 -0.5 0 0.5 1 (a) P r T = 0T = 0.2T = 0.4T = 0.6T = 0.8 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 (b) S -1 -0.5 0 0.5 1 (c) P r S T = 0T = 0.2T = 0.4T = 0.6T = 0.8
FIG. 7. Complete rescue probability P r in the case with com-mitted volunteers as a function of S and T for different valuesof Q : (a) Q = 0 (low level), (b) Q = 0 . Q = 0 . P r increases against S for a given T .One can see P r = 1 for high S with low T and Q , but it isnot observable if Q is above a certain value. Figure 8(a)], if Q is small, the partial cooperation phasecan be observed in a limited area of ( T, S ) but the coop-eration phase is not observed. The area of partial cooper-ation phase is getting smaller as Q increases and becomesinvisible when Q = 0 .
4. At the same time, the defectionphase is expanding in the (
T, S ) space as Q grows. FromFigure 8(b), when Q is small, one can observe the fullcooperation phase for high S and low T , correspondingto the Harmony (H) game. In addition, the defectionphase is widely observed in the Prisoner’s dilemma (PD)game and can be seen from some ( T, S ) combinations inthe Snowdrift (SD) and the Stag hunt(SH) games. As Q grows, the area of partial cooperation phase is expand-ing while the defection phase area is decreasing. At thesame time, surprisingly, increasing Q eventually leads tothe disappearance of full cooperation phase. This can beunderstood that increasing Q contributes to the spread-ing of cooperation, but some lonely volunteers still fail tofind a peer volunteer. This is apparently because the al-truism strength Q reduces the payoff difference betweenthe cooperator and defector. Consequently, the probabil-ity that a bystander switches to a volunteer is reduced.In terms of the two-player game, increasing Q makes allthe games (i.e., H, SD, SH, and PD games) result in thepartial cooperation phase, thus the collective helping be-havior becomes insensitive to the values of S and T . III.2. Interpretation in terms of well-mixedpopulation model
Some of the results presented in Section III.1 can beinterpreted in terms of the well-mixed population model.Based on the well-mixed population model, we study theevolution of the number of cooperators N C without con- sidering the spatial formation of individuals. The evolu-tion of N C is given as N C ( t + ∆ t ) = N C ( t ) + dN C ( t ) dt ∆ t, (6)where dN C ( t ) /dt is the time derivative of the number ofcooperators at time t and ∆ t is the time step. At t = 0, N C ( t ) is given as N C, denoting the initial number of co-operators. If dN c ( t ) /dt is positive, the number of coop-erators N C ( t ) is going to increase. Negative dN C ( t ) /dt leads to the decrement of N C ( t ), so N C ( t ) in the sta-tionary state is likely smaller than N C, . Note that, inour study, dN C ( t ) /dt is a function of payoff parameters T and S along with the altruism strength Q . We referthe readers to Appendix C for further details relate toEq. (6).In the case without committed volunteers, the major-ity of numerical simulations result in the defection phase,see Fig. 8(a). According to Eq. (6), this is seemingly be-cause of the negative value of dN C ( t ) /dt in the beginningof numerical simulations. Consequently, the number ofcooperators decreases in the course of time, leading tothe loss of volunteers.In the case with committed volunteers [see Fig. 8(b)],partial and full cooperation phases are frequently ob-served over a larger area of ( T, S ) space. This suggeststhat dN C ( t ) /dt is mostly positive in the beginning of nu-merical simulations. In addition, due to the committedvolunteers, dN C ( t ) /dt does not become negative irregard-less of payoff parameters T and S , implying that the sta-tionary state value of N C is equal or larger than N C, .It is generally accepted that the well-mixed populationmodel often fails to predict complex collective phenom-ena observed from numerical simulations due to its majorlimitation. The well-mixed population model is not ableto consider the spatial formation of individuals, thus theheterogeneity in spatial structure cannot be reflected inthe evolutionary game analysis. Previous studies [30, 31]have highlighted that the spatial structure is of great im-portance to the onset of collective cooperation behaviors.Unlike those previous studies, in this study, one can ob-serve that the results from the presented well-mixed pop-ulation approach is in good agreement with those fromnumerical simulations presented in Section III.1. This isseemingly because the evacuation scenario is studied fora small fixed square room and the mobility pattern ofindividuals is simple. IV. CONCLUSION
This paper presents a game theoretical model of help-ing behavior among evacuees. We have numerically in-vestigated the evolution of collective helping behavior ina room evacuation scenario. In our numerical simula-tions, injured persons were placed in the room and itwas assumed that each injured person needs the help oftwo volunteers during the evacuation. We assigned aninitial volunteer to each injured person in the beginning -1-0.5 0 0.5 1 0 0.5 1 1.5 2
PC D (a) S Q = 0 P r > 0 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 PC DQ = 0.1 P r > 0 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 PC DQ = 0.2 P r > 0 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 PC DQ = 0.3 P r > 0 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 DQ = 0.4 P r > 0 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 FC PC D (b) S T P r > 0P r = 1 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 FC PC DT P r > 0P r = 1 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 PC DT P r > 0P r = 1 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 PC DT P r > 0 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 PCT
FIG. 8. Phase diagrams summarizing the numerical results: (a) the case without committed volunteers and (b) the case withcommitted volunteers. The (
T, S ) space is divided into different phases by means of the complete rescue probability P r . Here,D, PC, and FC indicate the defection, partial cooperation, and full cooperation phases, respectively. Different symbols representthe boundaries between different phases: red circles ( (cid:13) ) for the boundary between FC and PC, and blue rectangles ( (cid:3) ) forthe boundary between PC and D. The full cooperation phase (i.e., P r = 1) is not observed in the case without committedvolunteers, but it can be seen in the case without committed volunteers for high S with low T and Q . In the case withoutcommitted volunteers, the area of partial cooperation (PC) phase shrinks as Q increases and is virtually disappeared when Q = 0 .
4. In the case with committed volunteers, in contrast, the area of partial cooperation phase is expanding as Q increases.In addition, the areas of full cooperation (FC) and defection (D) phases shrink and then disappear as Q increases. of numerical simulations. The initial volunteers who donot changing their decision to rescue an injured personare indicated as committed volunteers. In this study, weconsidered the cases without and with committed vol-unteers. The initial volunteers can change their strat-egy in the case without committed volunteers, but it isnot allowed in the case with committed volunteers. Thenumber of volunteers evolved as a result of strategic in-teractions between individuals.By systematically controlling payoff parameters andaltruism strength, we characterized different collectivehelping behaviors. In the case without committed vol-unteers, we observed the partial cooperation phase forlow temptation payoff T and high sucker’s payoff S withlow altruism strength Q . For high Q , the defection phasewas widely observed in a wide area of ( T, S ) space, evenin the Harmony game. However, different patterns wereobserved in the case with committed volunteers. For high S with low T and Q , the full cooperation phase was ob-served from the numerical simulations, in which all theinjured persons were rescued regardless of random seeds.As Q increases, it was observed that the area of par-tial cooperation phase was expanding in the ( T, S ) space.Consequently, when Q was high, the whole ( T, S ) spacebecame the partial cooperation phase, even in the Pris-oner’s dilemma game. It is suggested that the existenceof committed volunteers can contribute to the appear-ance of full cooperation phase in low Q and expansion ofpartial cooperation phase in high Q . Increasing the al- truism strength Q reduced the payoff difference betweenthe cooperator and defector, leading to the spreading ofdefection phase in the case without committed volunteersand partial cooperation phase in the case of committedvolunteers. Surprisingly, in the case with committed vol-unteers, increasing Q eventually also resulted in the dis-appearance of full cooperation phase. This is seeminglybecause increasing Q reduces the payoff difference be-tween the cooperator and defector, thus the probabilitythat a bystander switches to a volunteer is reduced.The numerical simulation results of this study can beunderstood in line with the well-mixed population model.In the case with committed volunteers, the time deriva-tive of the number of cooperators was mostly positive inthe beginning of simulations. Consequently, we could of-ten observe the partial and full cooperation phases. Onthe other hand, in the case without committed volun-teers, the time derivative of the number of cooperatorswas negative on most occasions, leading to the loss of vol-unteers. In contrast to previous studies [30, 31], the anal-ysis based on the well-mixed population model demon-strated a good agreement with our numerical simulationresults. This is seemingly due to the simple mobility pat-terns of evacuees in a small fixed square room.There are possible directions of future work address-ing the limitations of this study. To focus on the essentialfeatures of the strategic interactions in helping behavior,this study has considered a simple room evacuation sce-nario. The presented model needs to be tested with morecomplex geometry conditions to further examine com-plex nature of strategic interactions among individuals.In this study, the temptation payoff T and the sucker’spayoff S were assumed to be same for the every evacuee.It would be interesting to introduce different values of T and S to each evacuee. For instance, the values of T and S can be given depending on the distance to the exit,speed reduction factor α , or the estimated extra evacua-tion time incurred by helping the injured persons. In ad-dition, it was assumed that each individual interacts witha random neighbor within the sensory range. Accordingto Ref. [32], the emergence pattern of cooperation can beaffected by the selection of interaction rule for the evolu-tionary game. This is apparently because changing to an-other interaction rule is likely to make a difference in eval-uating the payoff difference with the interaction partners.One can consider other interaction rules, for instance, thestochastic interaction by which individuals can interactwith several different players in each time step. Anotherpossible direction of future work can be planned in linewith the number of committed volunteers. In this study,we observed that the partial and full cooperation phaseswere widely spreading in the ( T, S ) space as a result ofintroducing committed volunteers. This study presentedthe difference between without and with committed vol-unteers, but did not investigate under which conditionsthe committed volunteers contribute to the appearanceof system-wide cooperation. Previous studies [22, 33, 34]have examined under what conditions the committed mi-norities play a pivotal role in the emergence of coopera-tion in a well-mixed population and structured networks.In the context of pedestrian evacuations, the results andanalysis from our approach can be extended to quantifythe critical mass effect in collective helping behavior interms of the number of committed volunteers.
ACKNOWLEDGEMENTS
This research is supported by National Research Foun-dation (NRF) Singapore, GOVTECH under its Vir-tual Singapore Program Grant No. NRF2017VSG-AT3DCM001-031. We thank Mr. Vinayak Teoh Kan-nappan for his help in implementing the presented agent-based model with C++.
Appendix A: Details of the social force model
In Section II.2, we presented a general form of the so-cial force models [28, 35, 36]. This appendix providesfurther details of the presented social force model.The social force models describe the acceleration ofpedestrian i as a superposition of driving and repulsiveforce terms according to the following equation of motion: d(cid:126)v i ( t ) dt = (cid:126)f i,d + (cid:88) j (cid:54) = i (cid:126)f ij + (cid:88) B (cid:126)f iB . (A1) TABLE IV. Social force model parametersModel parameter symbol valueinterpersonal repulsion strength C p l p k n k t C b l b The driving force (cid:126)f i,d is given as (cid:126)f i,d = v d (cid:126)e i − (cid:126)v i ( t ) τ , (A2)where v d is the desired speed and (cid:126)e i is a unit vectorindicating the desired walking direction of pedestrian i .The relaxation time τ controls how fast the pedestrian i adapts its velocity to the desired velocity.The interpersonal repulsive force term (cid:126)f ij is specifiedaccording to the circular specification [35] which is themost simplistic form of the interpersonal repulsive inter-action. The explicit form of f ij can be written as (cid:126)f ij = C p exp (cid:18) r i + r j − d ij l p (cid:19) (cid:126)e ij , (A3)where (cid:126)e ij = (cid:126)d ij /d ij is a unit vector pointing from pedes-trian j to pedestrian i , and (cid:126)d ij ≡ (cid:126)x i − (cid:126)x j is the distancevector pointing from pedestrian j to pedestrian i . Thestrength and the range of repulsive interaction betweenpedestrians are denoted by C p and l p , respectively. In-stead of the circular specification, one might select a dif-ferent form of specifications such as elliptical specificationI (ES-1) [28] and elliptical specification II (ES-2) [36].In addition to the interpersonal repulsive force term (cid:126)f ij presented in Eq. (A3), the interpersonal elastic forceterm (cid:126)g ij is added when the distance d ij is smaller thanthe sum r ij = r i + r j of their radii r i and r j . We describe (cid:126)g ij as Helbing [35] suggested, (cid:126)g ij = h ( r ij − d ij ) (cid:8) k n (cid:126)e ij + k t [( (cid:126)v j − (cid:126)v i ) · (cid:126)t ij ] (cid:126)t ij (cid:9) , (A4)where k n and k t are the normal and tangential elasticconstants, respectively. A unit vector (cid:126)e ij is pointing frompedestrian j to pedestrian i , and (cid:126)t ij is a unit vector per-pendicular to (cid:126)e ij . The function h ( x ) yields x if x > x ≤ (cid:126)f iB is given as (cid:126)f iB = C b exp (cid:18) − d iB l b (cid:19) (cid:126)e iB , (A5)where d iB is the perpendicular distance between pedes-trian i and wall, and (cid:126)e iB is the unit vector pointing fromthe wall B to the pedestrian i . The strength and rangeof repulsive interaction from boundaries are denoted by C b and l b .To implement the presented social force model, theparameter values in Table IV are selected based on theprevious studies [28, 35, 37, 38].The ambulant pedestrians move with the initial de-sired speed v d = v d, = 1 . τ = 0 . v max = 2 . et al. [9, 10]. We applied speed reduc-tion factor α = 0 . v d = αv d, = 0 . (cid:126)v i ( t + ∆ t ) = (cid:126)v i ( t ) + (cid:126)a i ( t )∆ t, (A6) (cid:126)x i ( t + ∆ t ) = (cid:126)x i ( t ) + (cid:126)v i ( t + ∆ t )∆ t. (A7)Here, (cid:126)a i ( t ) is the acceleration of pedestrian i at time t which can be obtained from Eq. (A1). The velocity andposition of pedestrian i is denoted by (cid:126)v i ( t ) and (cid:126)x i ( t ),respectively. The time step ∆ t is set as 0.05 s. Appendix B: Exploration of the (T, S) space
In Section III.1, we presented the representative timeseries of the fraction of cooperators n c : the case withoutcommitted volunteers [Fig. 4] and the case wit committedvolunteers [Fig. 5]. To complement the results presentedin Figs. 4 and 5, we evaluate the stationary state valuesof n c in terms of the average and standard deviation of n c , i.e., µ ( n c ) and σ ( n c ). The values of µ ( n c ) and σ ( n c )are measured over 100 independent simulation runs for aparameter combination of ( T, S ) with a fixed value of Q .In the case without committed volunteers, µ ( n c ) growsas S increases for a given value of Q , indicating that high S leads to the increment of volunteers. The σ ( n c ) curvesincrease and then decrease if Q is at low and intermediatelevels, but the change of σ ( n c ) against S is not significantwhen Q is high. In Fig. 9(a), µ ( n c ) is almost 1 when S is high and T is low, but σ ( n c ) is not zero. This sug-gests that the complete rescue of all the injured personsis not always achievable, which is consistent with our ob-servation from Fig. 4(a) and Fig. 4(b). In addition, if weincrease Q , we can observe that the minimum value of µ ( n c ) is increasing while its maximum value is decreas-ing for given value of T . This is seemingly due to theincrement of σ ( n c ). In Fig. 10, we present the averageand standard deviation of the normalized fraction of co-operators in the ( T, S ) space. In line with the results inFig. 9, the difference between minimum and maximumvalues of µ ( n c ) is decreasing while the value of σ ( n c ) isgrowing as we increase Q .In the case with committed volunteers, the µ ( n c )curves show an increasing trend as S increases for a givenvalue of Q , similarly to the case without committed vol-unteers. However, one can notice different patterns of µ ( n c ) and σ ( n c ) curves when Q is high, see Fig. 11(c).It can be seen that µ ( n c ) is virtually 1 but smaller than1. In addition, σ ( n c ) is decreasing for increasing Q , but σ ( n c ) does not become 0. This implies that the station-ary state values of n c is smaller than 1 even for high S , -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 (a) µ ( n c ) T = 0T = 0.5T = 1T = 1.5 -1 -0.5 0 0.5 1 σ ( n c ) -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 (b) µ ( n c ) T = 0T = 0.5T = 1T = 1.5 -1 -0.5 0 0.5 1 σ ( n c ) -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 (c) µ ( n c ) S T = 0T = 0.5T = 1T = 1.5 -1 -0.5 0 0.5 1 σ ( n c ) S FIG. 9. Stationary state values of n c in the case withoutcommitted volunteers: (a) Q = 0 (low level), (b) Q = 0 . Q = 0 . µ ( n c ) and σ ( n c ), are presented in the leftand right columns, respectively. Different symbols representthe different values of T . Results are evaluated over 100 in-dependent simulation runs. For each given Q , µ ( n c ) grows as S increases. The σ ( n c ) curves increase and then decrease if Q is at low and intermediate levels, but the change of σ ( n c )against S is not significant when Q is high. thus the complete rescue is not always possible. This iscompatible with the results presented in Fig. 8(b). Fig-ure 12 presents the numerical results of µ ( n c ) and σ ( n c )in the ( T, S ) space.
Appendix C: Details of the well-mixed populationmodel description
In Section III.2, we interpreted the numerical simula-tion results from the cases without and with committedvolunteers. This appendix provides more details of ourinterpretation, especially how the well-mixed populationmodel can explain the numerical simulation results.Based on previous studies [22, 33, 34], we considera well-mixed population model with a finite populationconsisting of N individuals. For the case without com-mitted volunteers, the evolution of the number of co-operators, dN C ( t ) /dt , can be described in terms of theincrease rate f + and decrease rate f − , dN C ( t ) dt = f + − f − . (C1)The increase rate f + is equal to the product of the pairselection probability p s and the strategy adaption prob-0 Q = 0 (a)
0 0.5 1 1.5 2-1-0.5 0 0.5 1 S Q = 0.1
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.2
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.3
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.4
0 0.5 1 1.5 2-1-0.5 0 0.5 1 -1-0.5 0 0.5 1 (b)
0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 S
0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 FIG. 10. The stationary state values of n c for the case without committed volunteers in the ( T, S ) space: (a) average µ ( n c )and (b) standard deviation σ ( n c ). Each panel corresponds to different values of altruism strength Q . Results are averaged over100 independent simulation runs. -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 (a) µ ( n c ) -1 -0.5 0 0.5 1 σ ( n c ) T = 0T = 0.5T = 1T = 1.5 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 (b) µ ( n c ) -1 -0.5 0 0.5 1 σ ( n c ) T = 0T = 0.5T = 1T = 1.5 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 (c) µ ( n c ) S -1 -0.5 0 0.5 1 σ ( n c ) S T = 0T = 0.5T = 1T = 1.5
FIG. 11. Same graph as Fig. 9 but for the case with com-mitted volunteers: (a) Q = 0 (low level), (b) Q = 0 . Q = 0 . Q , the µ ( n c ) curves show an increasing trend as S increases.However, µ ( n c ) is virtually 1 while σ ( n c ) is small when Q is high, which is different from the case without committedvolunteers. ability p (∆ u DC ): f + = p s p (∆ u DC ) = N C N D N ( N −
1) 11 + exp[( T e − S e ) /k ] , (C2)where N is the number of ambulant pedestrians and N C and N D = N − N C are the number of cooperators (i.e., volunteers) and defectors (i.e., bystanders), respectively.With the pair selection probability, we select a pair ofplayers with different strategies, i.e., one player is a co-operator ( C ) and the other one is a defector ( D ). Here,∆ u DC = u D − u C is payoff difference obtained by sub-tracting from u D to u C . Note that the payoffs u C and u D are given as S e and T e , which are given in Table 1,respectively. Based on the strategy adaption probability,player i ( D ) updates his strategy to player j ’s one ( C ),i.e., switching from D to C .Likewise, the decrease rate f − is given as f − = p s p (∆ u CD ) = N C N D N ( N −
1) 11 + exp[( S e − T e ) /k ] . (C3)Next, we check the trend of the number of cooperators N C ( t ) in the course of time by looking into the sign of itschange rate dN C ( t ) /dt . As described in Section III.2, apositive value of dN C ( t ) /dt possibly leads to the full orpartial cooperation phases, while a negative value resultsin the defection phase. Based on Eqs. (C2) and (C3), werewrite Eq. (C1) as dN C ( t ) dt = N C N D N ( N − (cid:26)
11 + exp[( T e − S e ) /k ] −
11 + exp[( S e − T e ) /k ] (cid:27) . (C4)One can notice that the first term on the right-handside is always positive for N > N C >
0, thusthe sign of dN C ( t ) /dt exclusively depends on the sign of p (∆ u DC ) − p (∆ u CD ). Fig. 13(a) presents the value of dN C ( t ) /dt in the ( T, S ) space for different values of Q ,comparable to the results presented in Fig. 10(a). As Q increases, the value of dN C ( t ) /dt tends to decrease fora given value of T and S , suggesting the disappearance1 Q = 0 (a)
0 0.5 1 1.5 2-1-0.5 0 0.5 1 S Q = 0.1
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.2
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.3
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.4
0 0.5 1 1.5 2-1-0.5 0 0.5 1 -1-0.5 0 0.5 1 (b)
0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 S
0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 0 0.5 1 1.5 2 T -1-0.5 0 0.5 1 FIG. 12. The stationary state values of n c for the case with committed volunteers in the ( T, S ) space: (a) average µ ( n c ) and(b) standard deviation σ ( n c ). Each panel corresponds to different values of altruism strength Q . Results are averaged over 100independent simulation runs. of partial cooperation phase and spreading of defectionphase.Now we focus on the case with committed volunteers.Like the case without committed volunteers, dN C ( t ) /dt is described in terms of the increase rate f + and decreaserate f − , i.e., dN C ( t ) /dt = f + − f − [Eq. (C1)]. Notethat the committed volunteers do not update their strat-egy but they can induce defectors to become cooperators.The increase rate f + is same as Eq. (C2), while the de-crease rate f − has to be modified. The sampling rate p s is given as p s = ( N C − N C, ) N D N ( N − , (C5)where N C, is the initial number of volunteers. In the casewith committed volunteers, all the initial volunteers arecommitted volunteers. Accordingly, ( N C − N C, ) denotesthe number of volunteers who are not initial volunteers,indicating the number of volunteers who can change theirstrategy depending on the outcome of strategic interac-tions. The decrease rate f − is modified as f − = ( N C − N C, ) N D N ( N −
1) 11 + exp[( S e − T e ) /k ] . (C6)In the beginning of our numerical simulations, N C issame as N C, , implying the decrease rate f − becomeszero. The time derivative of N C should read dN C ( t ) dt = N C N D N ( N −
1) 11 + exp[( T e − S e ) /k ] − ( N C − N C, ) N D N ( N −
1) 11 + exp[( S e − T e ) /k ] . (C7)Note that, for N C, >
0, Eq. (C7) is always larger thanEq. (C4), meaning that dN C ( t ) /dt of the case with com- mitted volunteers is higher than that of the case with-out committed volunteers. Furthermore, the value of dN C ( t ) /dt is positive in the beginning of our numeri-cal simulations, likely leading to the increment of N C .Figure 13(b) shows dN C ( t ) /dt in ( T, S ) space for differ-ent values of Q , showing qualitatively the same results asthose presented in Fig. 12(a). As Q increases, the valueof dN C ( t ) /dt tends to decrease for high S and low T butincrease for the rest of ( T, S ) space. This implies thatthe area of partial cooperation phase is expanding.2
Q = 0 (a)
0 0.5 1 1.5 2-1-0.5 0 0.5 1 S Q = 0.1
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.2
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.3
0 0.5 1 1.5 2-1-0.5 0 0.5 1
Q = 0.4
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