Table Of ContentDynamics of Panic Pedestrians in Evacuation∗
Dongmei Shi†
Department of Physics, Bohai University, Jinzhou Liaoning, 121000, P. R. C
Wenyao Zhang
Department of Physics, University of Fribourg, Chemin du Musee 3, CH-1700 Fribourg, Switzerland.
Binghong Wang
Department of Modern Physics, University of Science and Technology of China, 230026, Hefei Anhui, P.R.C
(Dated: January 6, 2017)
Amodifiedlatticegasmodelisproposedtostudypedestrianevacuationfrom asingleroom. The
7 payoffmatrixinthismodelrepresentsthecomplicated interactionsbetweenselfishindividuals,and
1 themeanforceimposedonanindividualisgivenbyconsideringtheimpactsofneighborhoodpayoff,
0 walls, and defector herding. Each passer-by moves to his selected location according to the Fermi
2 function, and the average velocity of pedestrian flow is defined as a function of the motion rule.
n Two pedestrian types are included: cooperators, who adhere to the evacuation instructions; and
a defectors, who ignore the rules and act individually. It is observed that the escape time increases
J withthepaniclevel,andthesystemremainssmoothforalowpaniclevel,butexhibitsthreestages
5 for a high panic level. We prove that the panic level determines the dynamics of this system, and
the initial density of cooperators has a negligible impact. The system experiences three phases, a
] singlephaseofcooperator,amixedtwo-phasepedestrian,andasinglephaseofdefectorsequentially
G as the panic level upgrades. The phase transition has been proven basically robust to the changes
C of empty site contribution, wall’s pressure, and noise amplitude in the motion rule. It is further
shown that pedestrians derivethegreatest benefitfrom overall cooperation, butare trapped in the
.
n worst situation if they are all defectors.
li PACS numbers: 87. 23. Ge, 02. 50. Le, 87. 23. Cc, 89. 90. +n
n
[
I. INTRODUCTION such as fear, as well as the complicated forces among
1
v individuals. Helbing used social force to simulate such
6 Evacuationofpedestriansunderpanichasbeenexten- interactions [2], and Fukui avoided force concepts by
3 sively studied [1–20], because an understanding of the applying the CA model [20]. Game theory was pro-
2 dynamic features of this phenomenon can reduce the in- posedtostudythedynamicsincrowsevacuation[32–37].
1
cidence of injury and death. Escape panic can occur in Helio¨vaarawe presented a spatial game theoretic model
0
. different confinement sizes ranging from a rioting crowd for pedestrian behavior in situations of exit congestion
1 in a packed stadium to stunned customers in a bar. Es- [32], and Hao etal applied game theory to deal with the
0
capepanicischaracterizedbystrongcontactinteractions conflicts that arise when two pedestrians try to occupy
7
1 betweenselfishindividuals thatquicklygiverisetoherd- the same position[37].
: ing, stampede, and clogging [21–31]. Experimental ob- In this study, we modify the model [38] in which the
v
servationsandnumericalsimulationsaretwoapproaches effects of walls and defector herding are considered. A
i
X that have been widely applied to study this issue. Simu- defector herding unit is defined as a group of four indi-
r lationresultshaverevealedsomeinterestingdynamicfea- viduals who are all defectors in the same neighborhood.
a tures,suchaspedestrianarchformationaroundtheexit, The potential well energy or mean force imposed on an
herding,andinterferencebetweenarchesinmultiple-exit individual is defined by considering three aspects. An
rooms [2]. Disruptive interference, self-organized queu- individual will move to his selected location with a large
ing, and scale-free escape dynamics [26, 27] have also probability if his personal energy is higher than the po-
been observed. Experiments in genuine escape panic are tentialwellenergy. The velocityofa walkeris defined as
difficult to set up because of ethical and legal concerns. afunctionofthemotionrule,andaveragevelocityofthe
However, escape panic can be simulated by solving a pedestriansystemcanreflectdifferentdynamics,suchas
set of coupled differential equations [2, 3] or by applying smoothflow,andcongestion. Wefindthatthepaniclevel
the cellular automata (CA) technique [20, 26], whereby is crucial in determining the dynamics of the evacuation
the movement of confined pedestrians is tracked over systemandthecompetitionbetweencooperatorsandde-
time. An important task related to evacuation simula- fectors. Itisalsoobservedthatthreephasesofpedestrian
tions is how to describe the effects of subjective factors, are exhibited sequentially as panic level upgrades.
The remainderof the paper is organizedas follows. In
Section II, we describe the model. Our numerical simu-
∗ DynamicsofPanicpedestrians lationsandanalysisarepresentedinSectionIII. Conclu-
† [email protected] sions are drawn in Section IV.
2
II. MODEL
Pedestrian evacuation is studied on a model of square
lattice on which all the individuals are distributed uni-
formly. Two pedestrian types are considered: coopera-
tors, who adhere to the evacuation instructions; and de-
fectors, who ignore the rules and act individually. Each
pedestrianonlyinteractswithhis nearestneighbors,and
is deleted from the system if he arrives at the exit. We
apply game theory, and the payoff matrix representing
the interactions among the crowd is shown in Table 1.
To be specific, a cooperatorreceives a Reward(R) when FIG. 1. Schematic presentation of a defector herding unit in
he interacts with a collaborative neighbor, but suffers a which all the individuals are defectors who are circled in the
Sucker(S)ifhe encountersa defectorwhogainsa payoff figure. PD is the average payoff of the defector centered in
thecircle.
of Temptation (T). Two defectors receive the Punish-
ment (PM), respectively. In addition, a walker will gain
a payoff e (0 < e < R) if there is an empty site in his
a defector. Therefore, the potential well energy U will
x
neighborhood. Usually, T = 1+r, R = 1, S = 1−r,
includeU ifadefectorherdingunitexistsinadefector’s
du
PM = 0, where 0 < r < 1. r is cost-to-benefit ratio
neighborhood:
indicating the defector-cooperator payoff divide, which
describes the panic level during the escape. e = R/2 is
set in this model. 1 T
U = (n · )·s, (2)
du du
G 4
x
TABLEI. Payoff matrix where n is the number of defector herding units in the
du
neighborhoodofx. Theparameterstakesthevalues=0
if x chooses to cooperate, and s = 1 when x decides to
C D E
defect.
C (R, R) (S,T) (e, −) In addition, we consider the contribution of walls to
D (T, S) (PM, PM) (e, −) a Ux, so Ux will include Uw if walls exist in the neighbor-
hood of x:
a Cindicatesanindividualwhocooperates, andDindicates an
individualwhochooses todefect. Erepresentsanemptysite. 1
T >R>S>PM Uw = ·nw·Uw, (3)
G
x
where n is the number of walls in the neighborhood
P denotes the personal energy of individual x, and w
x
and U (0 < U < R) is the contribution of a wall to
is the average of cumulative payoff that x receives from w w
U . Combining all the quantities discussed above, the
all his neighbors. Each individual stays in a potential x
potential well energy U is given by
wellwhichisformedbytheneighboringpeopleandwalls x
(if walls exist in his neighborhood). U is the energy of
x
potential well x stays in, which consists of three parts.
U =U +U +U , (4)
Firstly, U isthe averagecumulativepayoffforallneigh- x p du w
p
bors of x, which represents the mean forces on x:
Itshouldbenotedthatx′spersonalenergyisexcludedin
U which actually reflects the pressures from x′s neigh-
x
1 borhood.
Up = X Pxj (1) At each time step, toward the exit’s position, each in-
G
x
j∈Gx dividualxrandomlyselectsacell y intheneighborhood,
and moves to y with probability W [38]:
where G is the scale of the interacting group centered
x
on x, which is defined as the number of directly linked
neighbors of x (excluding the empty sites). P is the 1
xj
W = , (5)
average of cumulative payoff that j receives from all his 1+exp(−(P −U )/κ)
x x
neighbors, where individual j belongs to the neighbor-
hood of x. where κ is the noise amplitude. κ = 0 indicates deter-
A defector herding unit is defined as a group in which minedoccupation,whereasκ=∞denotesstochasticoc-
fourdefectorsareinthesameneighborhood(Fig. 1). We cupation. It is apparent that the larger P is, the larger
x
consider that a defector herding unit contributes P = W is. The equation (5) means that if x owns more per-
D
T/4 to the energy of the potential well for a cooperator, sonal energy than the potential well energy, x will move
but contributes 0 to the energy of the potential well for to his selected location with a larger probability. If x
3
2000 r=0.1 2000 r=0.5
4800 IC=0.5 ors 1500 1500
ct
Time44460000 ors \ Defe 10500000 CDoefoepcetroartor 15000000
pe erat 1 10 100 1000 1 10 100 1000
a4200 p
Esc Coo 2000 r=0.65 2000 r=0.9
4000 of 1500 1500
er 1000 1000
b
3800 um 500 500
N
0.0 0.2 0.4 0.6 0.8 1.0 0 0
1 10 100 1000 1 10 100 1000
r Time step Time step
FIG. 2. Escape time as a function of the panic level r at the
initial density of cooperators ρIC =0.5.
2000 IC=0.1 2000 IC=0.3
ors 1500 1500
ct
takes the place of y, then the individual ever occupied y Defe 1000 CDoefoepcetroartor 1000
wilSlinbceepeuqsuhaetdiobna[c5k]ctoanthreefloercitgitnhaelmlooctaitoinonveolfoxci.tyofan ors \ 5000 5000
at
individual, v is defined as the system’s average velocity er 1 10 100 1000 1 10 100 1000
p
and given by, Coo 2000 IC=0.7 2000 IC=0.9
of 1500 1500
ber 1000 1000
1 um 500 500
v = X Wj, (6) N
NM 0 0
j∈NM 1 10 100 1000 1 10 100 1000
Time step Time step
where N is the number of individuals who have the
M
escape ability, and W is the value of W of individual
j
FIG. 3. Competition of cooperators and defectors for dif-
j representing its motion velocity. Different collective
ferent panic levels of r at ρIC = 0.5 (UP), and Competition
patterns of motion can be predicted according to v: for
between cooperators and defectors for different valuesof ρIC
v > 0.5, the system is realized to stay in the free flow;
at r=0.65 (DOWN).
when v → 0 the congestion forms. For 0 < v < 0.5, no
distinct collective patterns emerge.
All the individuals will update their strategies syn-
chronously by studying a more successful neighbor with
a probability W(s →s ),
x y
1
W(s →s )= , (7) 1.0
x y
1+exp(−(P −P )/τ)
y x
0.8
whereτ isthe noiselevelhavingthe identicalfunctionof 0.6 IC=0.5
κinequation[5],ands (s )isthestrategyx(y)adopts.
x y C
0.4
III. SIMULATIONS AND ANALYSIS 0.2
0.0
Thesimulationsarecarriedoutonasquarelatticerep- rd rc
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
resenting a single roomwith a scale of 50×50, and2000
r
pedestrians are considered. Only one exit exists, at a
site range of 23 − 26 on the lattice. Initially, all the FIG.4. Pedestrian phasetransition with paniclevel r when
pedestrians are distributed uniformly in the room with the initial cooperation density ρIC =0.5. ρC on the vertical
a cooperation density of ρIC = 0.5. Escape is realized coordinate is cooperation density among the pedestrians in
successfully when 95% of the pedestrians have escaped the room when 95% of pedestrians have escaped out of the
from the room. We set Uw = R/2 and κ = τ = 0.1. In room. rd is the critical value at which defectors emerge, and
all of the simulations, each data point is the average for rcisthecriticalvalueafterwhichcooperatorsbecomeextinct.
100 realizations.
4
Figure2showsthe variationinescapetime withpanic
level r at ρ =0.5. It is obviously seen that the escape
IC
0.4
time remains constant for small values of r (r < 0.5),
but increases with r for r ≥ 0.5. The results indicate 0.3
tehscaatpree,mwahinerineagsctaolmo misuachlwfaeyasrtlheaedbsetsot satrloanteggeyr desucrainpge Pd_bound 00..12
ttwimIenee.norcdoeorpetorautonrdsearsntdanddeftehcteodrsynaaremiincvs,esitnitgeartaecdt.ioDnsiffbeer-- -Pc_bound -00..10
-0.2
ent features are shown for different values of r (see it in -0.3 r=0.1 r=0.3
Fig. 3 (UP)): (1) for r =0.1 and 0.5, cooperators domi- r=0.63 r=0.67
-0.4 r=0.8 r=0.9
nate; (2) for r = 0.65, cooperators and defectors coexist
10 100 1000
andovertimetheirfrequenciesreachalmostidenticallev-
Time step
els; and (3) for r = 0.9, defectors dominate the system.
In Fig. 3 (DOWN), it is noted that cooperators coexist
FIG.5. (Coloronline)Thedifferencesbetweenmeanpayoffs
withdefectors,andevolvetoeventuallyreachalmostthe ofcooperatorsanddefectorsalongtheboundary(Pc−bound−
same mutual frequency, regardless of ρIC. Combining Pd−bound) at ρIC =0.5 for different panic levels of r.
this withthe resultinFig. 3(UP),wecanconcludethat
thepaniclevelrremarkablyaffectsthesystemdynamics,
whereas ρ has no distinct impact.
IC
The simulation results can be explained by analyzing
the panic level. It is known from the payoff matrix that
a high panic level induces a strong temptation to defect.
According to the definitions of the potential well energy
andmotionruleinEquations(1)−(5),alargeincrement
in the percentage of defectors will lead to a relatively t1 t2
largeincreaseinpotentialwellenergyorstrongpressure,
which accordingly reduces the escape velocity.
Moreover,pedestrianphasetransitionwithpanic level
r is discussed in Fig. 4. We observe that the system
experiences three phases sequentially: a single phase of
t3 t4
cooperator for r < 0.6, a mixed two-phase pedestrian
of cooperator and defector for 0.6(r ) ≤ r ≤ 0.7(r ),
d c
and a single phase of defector for r > 0.7. It has been FIG.6. (Coloronline)Spatialevolutionsofcooperatorsand
proven that as e increases from 0.1 to 0.9, rd = 0.6089, defectors at ρIC = 0.5, and r = 0.65. The black symbols
the standard deviation is σ(r ) = 0.0105; r = 0.7044, represent empty sites, the brown ones indicate the defectors,
d c
and σ(r ) = 0.0088. As U increases from 0.1 to and thewhite symbols denotecooperators.
c w
0.9, r = 0.5956, σ(r ) = 0.0133; r = 0.7111, and
d d c
σ(r )=0.0105. Itisobviouslyseenthatthephasetransi-
c
tionisbasicallyrobusttothechangesofeandU ,which itcanbe inducedthat cooperatorcluster’sboundarybe-
w
is consistent with the results shown in Fig. 4. Further- comes weaker as panic level r upgrades. Figure 6 shows
more, noise effect κ (in equation 5) was also studied: the spatialevolutionofpedestrians at r=0.65,inwhich
when κ = 0.001 (approaching determined occupation), the black symbols indicate the empty sites, the brown
r = 0.58, r = 0.7; for κ = 100 (approaching stochastic ones denote the defectors, and the white symbols repre-
d c
occupation), r = 0.54, and r = 0.72. It is concluded sent cooperative pedestrians. It is seen that since defec-
d c
thatphasetransitionisalsobasicallyrobusttothenoise, tors’ boundary is slightly firmer than cooperators’ (see
but the interval of mixed phase becomes broader when Fig. 5 in the phase of 0.6 ≤ r ≤ 0.7), many cooperator
noise becomes very large. clusters are formedbut don’t spreadalloverthe system.
To uncover the competition mechanism between the The evolution of the average velocity v for different
two types of pedestrians, mean payoffs of cooperators panic levels r at ρ =0.5 is shown in Fig. 7. It is seen
IC
and defectors along the boundary are studied in Fig. 5. that the average velocity decreases as the panic level in-
It is observed that in the phase of r < 0.6, P is creases, and declines sharply for r > 0.65. It is worthy
c−bound
significantly higher than P , whereas in the phase noting that v changes slowly with t for r < 0.65, but
d−bound
of r > 0.7, P is obviously higher. During the when r > 0.65, the velocity sharply decreases over time
d−bound
phase of 0.6 ≤ r ≤ 0.7, P keeps a little higher, until the system enters a uniform motion state. This
d−bound
but approachesto P as time evolves. Cooperator decrease in velocity is likely to be closely related to the
c−bound
cluster is proven crucial in cooperation spread because emergence of congestion, and the uniform-motion state
of its firm boundary. According to the results in Fig. 5, corresponds to free flow (moderate or relatively high ve-
5
patterns happen.
Finally,Figure8showstheescapetimeasafunctionof
0.5 tc v* ρIC for r=0.6. It is presentedthatpedestrians gainthe
greatest benefit for overall cooperation (ρ = 1), while
IC
0.4 the worst-case scenario occurs if all individuals defect
(ρ =0). The initial density ofcooperators(0<ρ <
IC IC
0.3 1) has no obvious influences on the escape time.
v I IV. CONCLUSIONS
0.2 r=0.1 III
r=0.5
0.1 r=0.65 Extending the model proposed in previous work, we
r=0.75 II
consideredtheeffectsofdefectorherdingandwallstoin-
0.0 r=0.9
vestigate pedestrian evacuation. Payoff represented the
10 100 1000
complicated interactions among individuals and was in-
Time step
cluded in the energy representation. We found that the
FIG. 7. Evolution of average velocity v at ρIC = 0.5 for escape time increased with the panic level, and three
different values of panic level r. v∗ = 0.5, I, II, and III phases of pedestrian were observedsequentially as panic
represent threestages a system experiences when r>0.65. level upgraded. Analysis of the results indicated that
a high panic level induced a strong temptation to de-
fect, which leaded to the emergence of widespread de-
fection. Furthermore, a large percentage of defectors in-
4600 duced a comparatively large increase in potential well
4500 energy, which reduced the average speed of pedestrian
flowaccordingtothemotionrule. Payoffsofcooperators
4400
e anddefectors alongthe boundary were investigated,and
m 4300
Ti theresultsrevealedthecompetitionmechanismsbetween
pe 4200 thetwotypesofpedestrian. Theaveragevelocitywasde-
a
sc 4100 finedasthemeanofcumulativemotionprobabilityinthis
E
4000 model, which revealed valuable and interesting dynam-
ics: the system was in the free flow when the panic level
3900
was low, but exhibited three stages - congestion forma-
3800
0.0 0.2 0.4 0.6 0.8 1.0 tion, congestion, and congestion evacuation - for a high
paniclevel. Weprovedthatthepaniclevelplayedanim-
IC
portant part in determining the dynamics for which dif-
FIG. 8. Variation of escape time with the initial density of ferent behaviors were observed for different panic levels.
cooperators ρIC at r=0.6. Phasetransitionexisted, andwasbasically robustto the
changes of empty site contribution, wall’s pressure and
noise effect. We also found that global cooperation was
locity)orcongestion(verylowvelocity). Therefore,from
the best strategy for the most efficient evacuation, and
the phase aspect we can conclude that the system gen-
the situation was worst for all defections. It was proven
erally remains in a free-flow state for phase of r < 0.6
that the initial density of cooperators had a negligible
when t < t (v > v∗), and then moves at a relative low
c impact on the escape efficiency during evacuation.
speed without distinct collective patterns. It exhibits
three behavior stages for phase of r > 7: in stage I,
local free flow and congestion coexist; in stage II, only
ACKNOWLEDGMENTS
congestionexists;andinstageIII,congestionevacuation
occurs. Congestionemergesbefore the systementersthe
uniform state owing to the small value for v in stage II. This work was supported by Specialized Foundation
It can be predicted from this analysis that the probabil- for Theoretical Physics of China (Grant No. 11247239),
ity of congestion is high for a high panic level. For the National Natural Science Foundation of China (Grants
phase of coexistence (0.6≤r ≤7), no distinct collective No. 11305017,11275186and 91024026).
[1] LF.Henderson,Thestatisticsofcrowdfluids.Nature229 [3] D. Helbing, Traffic and related self-driven many-particle
(1971), pp.381-383. systems, Rev.Mod. Phys., 73 (2001), pp.1067-1141.
[2] D.Helbing,I.Farkas,andT.Vicsek,Simulatingdynam- [4] A. Schadschneider, et al. Evacuation dynamics: Empir-
icalfeaturesofescapepanic,Nature407(2000),pp.487- ical results, modeling and applications. Encyclopedia of
490. ComplexityandSystemsScience,edMeyersR(Springer,
6
Berlin), (2009), pp.3142-3176. Sociol. 20 (1995), pp. 247-269.
[5] M. Moussa¨id, and J. D. Nelson, Simple heuristics and [22] N. Johnson and W. E. Feinberg, The impact of exit in-
themodellingofcrowdbehaviours.PedestrianandEvac- structions and number of exits in fire emergencies: a
uation Dynamics2012. Springer, (2014) pp.75-90. computersimulationinvestigation,J.Environ.Psych.17
[6] D.Helbing,A.Johansson, HZ.Al-Abideen,Dynamicsof (1997), pp.123-133.
crowd disasters: An empirical study. Phys. Rev. E 75 [23] R.Church,andR.Sexton,TestbedCenterforInteroper-
(2007), pp.046109. ability(FinalReport),TaskOrder3021(CaliforniaDept.
[7] G.Antonini,M.Bierlaire,andM.Weber,Discretechoice of Transportation, Sacramento), (2002).
models of pedestrian walking behavior. Transp. Res. [24] L. Chamlet, G.Francis, and P.Saunders,Networkmod-
Part. B: Methodol 40 (2006), pp.667-687. els for building evacuation, Management Sci. 28 (1982),
[8] S.Seriani, and R.Fernandez,Pedestrian traffic manage- pp. 86-105.
mentofboardingandalightinginmetrostations.Transp. [25] J. Bryan, Fire Protection Eng. 16 (Fall 2002), pp.4-10.
Res. Part C - Emerg. Technol. 53 (2015), pp. 76-92. [26] G.J.Perez,G.Tapang,M.Lim,andC.Saloma,Stream-
[9] W. Yu, A. Johansson, Modeling crowd turbulence by ing,disruptiveinterferenceandpower-lawbehaviorinthe
many-particle simulations. Phys. Rev. E 76 (2007), pp. exit dynamics of confined pedestrians, Physica A , 312
046105. (2002), pp.609-618.
[10] S. Hoogendoorn,(2004) Pedestrian flow modeling by [27] C. Saloma, C. J. Perez, G. Tapang, M. Lim, and C.
adaptive control. Transp Res Rec 1878 (2004), pp. 95- Palmes-Saloma,Self-organizedqueuingandscale-freebe-
103. haviorinrealescapepanic,PNAS,100(2003),pp.11947-
[11] M. Moussa¨id, D. Helbing, S. Garnier, A. Johansson, M. 11952.
Combe,andG.Theraulaz,Experimentalstudyofthebe- [28] M. Chraibi, A. Seyfried, and A. Schadschneider, Gener-
haviouralmechanismsunderlyingself-organizationinhu- alized centrifugal-force model for pedestrian dynamics.
mancrowds. Proc RoySocB276(2009), pp.2755-2762. Phys. Rev.E 82 (2010), pp.046111.
[12] D.Helbing,I.Farkas,andT.Vicsek,Simulatingdynam- [29] M. Moussaid, D. Helbing, G. Theraulaz, How simple
icalfeaturesofescapepanic,Nature407(2000),pp.487- rulesdeterminepedestrianbehaviorandcrowddisasters.
490. Proc. Natl. Acad. Sci. 108 (2011), pp. 6884-6888.
[13] Y. Qu, Z. Gao, P. Orenstein, J. Long, and X. Li, An [30] R.Y.Guo,H.J.Huang,andS.C.Wong,Routechoicein
effective algorithm to simulate pedestrian flow using the pedestrian evacuationunderconditionsofgood andzero
heuristicforce-basedmodel.Transp.B-Transp.Dynam. visibility: experimental and simulation results. Transp.
3 (2015), pp.1-26. Res. Part B C Methodol. 46 (2012), pp.669-686.
[14] K.W.Rio,MappingtheVisualCouplingbetweenNeigh- [31] A. Kneidl, D. Hartmann, and A. Borrmann, A hybrid
bors in Real and Virtual Crowds. Brown University, multi-scaleapproachforsimulationofpedestriandynam-
(2015). ics.Transp.Res.PartCCEmerg.Technol.37(2013),pp.
[15] M.Davidich,F.Geiss,H.G.Mayer,A.Pfaffinger,andC. 223-237.
Royer,Waitingzonesforrealisticmodellingofpedestrian [32] S.Helio¨vaara, H.Ehtamo, D.HelbingandT. Korhonen,
dynamics: acasestudyusingtwomajorGermanrailway Patient and impatient pedestrians in a spatial game for
stationsasexamples.Transp.Res.PartC-Emerg.Tech- egress congestion, Phys.Rev.E, 87 (2013), pp.012802.
nol. 37 (2013), pp.210-222. [33] R. Brown, Social Psychology (Free Press, New York,
[16] P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre, 1965).
andG.Theraulaz,Ahierarchyofheuristic-basedmodels [34] J. S. Coleman, Foundations of Social Theory (Belk-
of crowd dynamics. J. Stat. Phys. 152 (2013), pp. 1033- nap Press of Harvard UniversityPress, Cambridge, MA,
1068. 1990).
[17] D. C. Duives, W. Daamen, and S. P. Hoogendoorn, [35] H.Ehtamo,S.Helio¨vaara,T.Korhonen,andS.Hostikka,
2013. State-of-the-art crowd motion simulation models. Gametheoreticbest-responcedynamicsforevacuees’exit
Transp. Res. Part C C Emerg. Technol. 37 (2013), pp. selection, Adv.Complex Syst.13 (2010), pp. 113-134.
193-209. [36] SM.Lo,H.C.Huang,P.Wang,andK.K.Yuen,Agame
[18] F. Dietrich, and G. Koster, Gradient navigation model theory based exit selection model for evacuation, Fire
for pedestrian dynamics. Phys. Rev. E 89 (2014), pp. Saf. J. 41 (2006), pp.364-369.
062801. [37] Q. Y. Hao, R. Jiang, M. B. Hu, B. Jia, and Q. S. Wu,
[19] Y. Xiao, Z. Y. Gao, Y. C. Qu, and X. G. Li, A pedes- Pedestrian flow dynamics in a lattice gas model coupled
trian flowmodel considering theimpact of local density: with an evolutionary game, Phys. Rev. E 84 (2011), pp.
Voronoidiagram based heuristics approach. Transporta- 247-268.
tion Research Part C, 68 (2016), pp. 566-580. [38] D. M. Shi and B. H. Wang, Evacuation of pedestrians
[20] M. Fukui and Y. Ishibashi, Self-Organized Phase Tran- from a single room by using snowdrift game theories,
sitionsinCellularAutomatonModelsforPedestrians, J. Phys RevE, 75 (2013), pp. 022802.
Phys.Soc. Jpn. 68 (1999), pp.2861-2863.
[21] W.E.FeinbergandN.Johnson,FIRESCAP:Acomputer
simulation model of reaction to a fire alarm, J. Math.
