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Pushing and overtaking others in a spatial game of exit congestion

Abstract and Figures

With self-driven particle models, like the social force model, most of the physics of moving crowds can be modeled. However, it has not been fully unraveled why large crowds evacuating through narrow bottlenecks often act against their self-interest. They form jams in front of the bottleneck, that slow down the evacuation, and fatal pressures build up in the crowd. Here, we take a novel approach, and model the local decision-making in an evacuating crowd as a spatial game. The game is coupled to the social force model, so that different strategies alter the physical parameters. With our integrated treatment of behavioral and physical aspects, we are able to simulate when, why and how typical phenomena of an evacuation through a bottleneck occur. Most importantly, we attain non-monotonous speed and kinetic pressure patterns, in contrast to the monotonous patterns predicted by the pure social force model. This is a result of impatient agents in the back of the simulated crowd pushing and overtaking their way forward. Our findings give insight into the origin of crowd disasters, since the build-up of kinetic pressure has been related to the risk of falling and crowd turbulence.
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Pushing and overtaking others in a spatial game of exit
congestion
Anton von Schantz1,, Harri Ehtamo1
Aalto University, Department of Mathematics and Systems Analysis, P.O. Box 111000,
FI-00076 Aalto, Finland
Abstract
With self-driven particle models, like the social force model, most of the physics
of moving crowds can be modeled. However, it has not been fully unraveled
why large crowds evacuating through narrow bottlenecks often act against their
self-interest. They form jams in front of the bottleneck, that slow down the
evacuation, and fatal pressures build up in the crowd. Here, we take a novel ap-
proach, and model the local decision-making in an evacuating crowd as a spatial
game. The game is coupled to the social force model, so that different strategies
alter the physical parameters. With our integrated treatment of behavioral and
physical aspects, we are able to simulate when, why and how typical phenomena
of an evacuation through a bottleneck occur. Most importantly, we attain non-
monotonous speed and kinetic pressure patterns, in contrast to the monotonous
patterns predicted by the pure social force model. This is a result of impatient
agents in the back of the simulated crowd pushing and overtaking their way
forward. Our findings give insight into the origin of crowd disasters, since the
build-up of kinetic pressure has been related to the risk of falling and crowd
turbulence.
Keywords: Crowd behavior, Spatial game, Multiagent model, Exit congestion
Corresponding author
Email address: anton.von.schantz@aalto.fi (Anton von Schantz)
Preprint submitted to Elsevier May 20, 2019
1. Introduction
When a crowd evacuates from a given space the individuals behave patiently
and the outflow is smooth, provided there is enough time. However, if time is
a scarce resource, the following physical phenomena are observed: the back
of the crowd pushing, arch formation, faster-is-slower effect, crowd turbulence
[1, 2, 3, 4, 5]. Moreover, the following behavioral characteristics are typical for
all humans: overtaking of other pedestrians, pushing, disregarding the personal
space of others [6, 7].
Most of the physical phenomena of crowd evacuations are captured by self-
driven particle models, of which best known are social force models [8, 9, 10, 11].
But, crowds do not only follow the laws of physics; pedestrians can change their
speed and direction at will [12, 13]. There are attempts to model this, too
[6, 14, 15]. In these, agents make decisions based on the behavior of their
nearest neighbors.
The most of the collective phenomena observed in evacuation situations have
been confirmed experimentally [1, 2]. Due to ethical and practical concerns the
effect of competitiveness on the evacuation has not been studied until recent
years. In this article, we are mainly referring to our own experiment [7]. In [7]
the competitive and non-competitive behavior of a crowd were studied. In the
competitive scenario pedestrians minimize their own evacuation time, whereas,
in the non-competitive scenario, pedestrians minimize the total evacuation time
of the crowd. The evacuation of a crowd of students under these behavioral
forms were observed by video-tracking. In the study, the following observations
were made: (a) In a non-competitive scenario, pedestrians are willing to give
space to others; they are careful not to push or even touch other pedestrians
around them. (b) In a competitive scenario, faster people overtake their prede-
cessors, whereas in a non-competitive scenario, people stick to their positions
within the crowd throughout the egress. (c) The number of overtakers is not
dependent on the initial positions of the pedestrians. (d) The number of over-
takers in the crowd increases with the distance to exit. (e) Overtaking others
2
makes an evacuation faster, because, if faster people do not overtake slower ones,
the whole crowd ends up moving in the speed of the slowest ones. We will refer
these items later on when we discuss similar features arising in our simulation
model.
So, the competitive scenario resulted in a faster evacuation than the non-
competitive scenario. However, if in the competitive scenario, the pedestrians
would be set to push even harder towards the exit, arch formation would in-
crease, which would result in a slower evacuation [16, 17]. If the situation would
get severe, crowd pressure would increase, and rare disastrous events like crowd
turbulence, could be observed [4, 5]. In that case, the crowd is so tightly packed
that individual decision-making no longer has an effect on the movement.
The obvious choice for modeling decision-making in an evacuating crowd
is game theory [18]. However, as shown in [7], not all crowd members are in
the same situation. The people in front of the exit escape fast, while the ones
in the back have to wait a long time. Thus, if time is a scarce resource, the
pedestrians at different locations are involved in completely different games.
Also, pedestrians only interact with their nearest neighbors, thus the decision-
making in a crowd should be modeled as a spatial game [19, 20].
In this article, our simulations are based on a spatial game, which is coupled
to the social force model [8], and was introduced in [21], where preliminary
results were computed mainly in a static environment, where the agents do not
move. We are able to model most commonplace phenomena of evacuations, and
also to predict speed and kinetic pressure profiles, which essentially differ from
the monotonous profiles predicted by the pure social force model.
2. The model
Our model describes the competitive evacuation of a crowd of agents from
a square room through an exit. An agent iNestimates his evacuation time
Tito equal the time it takes to queue to the exit, i.e., Ti=λi; here, λiis
the number of agents closer to the exit than agent i, and βis the capacity of
3
the exit. Here, N={1, ..., n}is the index set of agents in the room. In a
contest between two nearby agents iand j, their estimated evacuation times
are approximated Tij := (Ti+Tj)/2.
For each agent, there is a cost associated with his estimated evacuation time;
u(Ti;TASET ). Here, the available safe egress time TASET is a physical parameter
used in the fire safety literature, for the time it takes for the conditions to become
lethal [22].
The agents have two strategies to choose from: Patient and Impatient. It
is assumed that in an actual play of the game, these strategies correspond to
patient and impatient behaviors of the agents, respectively. Denote by Nithe
neighborhood of agent i; it will be specified later. In a contest between two
neighboring agents iand jNi, the agents assume the following outcomes:
(i) An impatient agent ican overtake its patient neighbor j, i.e., the cost of
agent idecreases by 4u(Tij)
=u0(Tij )4T, and the cost of jincreases by
the same amount.
(ii) Two patient agents do not interact with each other, i.e., their costs do not
change.
(iii) In the case of two impatient agents, neither can overtake the other. Instead
they will face a conflict with equal chance of getting injured, which results
in a cost of conflict C4T,C > 0, for both of them.
From the aforementioned assumptions, a 2×2symmetric game matrix is
constructed:
I mpatient P atient
Impatient C4Tu0(Tij )4T
P atient u0(Tij )4T0
.(1)
In the game in Eq. 1, agent iis a row player and agent ja column player.
When a particular pair of strategies is chosen, the cost for agent iis given in
the corresponding cell of the matrix, while for jthey are the same numbers,
4
but with the non-diagonal elements multiplied by -1. The agents minimize their
costs in the game.
Lets make some assumptions about the cost function. It is more costly, i.e.,
there is a larger risk to be exposed to fire-related harm, if the estimated evac-
uation time Tij is larger; hence, u0(Tij )>0. Also, the cost of being overtaken
u0(Tij )4Tis larger if the estimated evacuation time Tij is larger; u00(Tij )>0.
We finally assume, that C=u0(Tij ), when Tij =TASET . This means that the
agents then start to play the Prisoner’s Dilemma game, see below. An example
of a cost function fulfilling these conditions is u(Tij ) = CT 2
ij /(2TASET ).
In Eq. 1, we substitute u0(Tij ) = CTij /TAS ET , and divide the matrix by
u0(Tij )4T. This does not affect the equilibria of the game. Then, we get the
following game matrix
I mpatient P atient
Impatient TASET
Tij
1
P atient 1 0
.(2)
If 0< TASET /Tij 1, the game in Eq. 2 is Prisoner’s Dilemma game (PD),
and its only Nash equilibrium is (Impatient, Impatient). If TAS ET /Tij >1, the
game is Hawk-Dove game (HD), and it has two pure strategy Nash equilibria:
(I mpatient, P atient)and (P atient, Impatient). In addition, it has a mixed-
strategy equilibrium, where the strategies Impatient and P atient are played
with probabilities Tij /TASET and 1Tij /TASET . This is an evolutionary stable
strategy (ESS) [23].
An evacuation situation constitutes a spatial setting, where each agent up-
dates his strategy with a myopic best-response update rule. Maybe more real-
istic, would be to use fictitious play, where the agent takes into account also
the strategies of his neighbors in earlier encounters. However, numerical testing
showed no noticeable differences between the two update rules.
When agent ichooses his strategy, he observes the current strategies of his
neighboring agents within a skin-to-skin distance of 60 cm. Let Ni={jN|
kxixjk ≤ ri+rj+ 0.6}, where xiis the centre of mass for agent i, and rihis
5
radius. Agent ichooses the strategy Impatient, if the sum of costs for playing
Impatient is less than or equal to that of playing P atient
X
jIi
TASET
Tij
+|Pi| · (1) ≤ |Ii| · 1 + |Pi| · 0,(3)
where Iiand Piare the sets of impatient and patient agents, in the neighborhood
of agent i, respectively, and |Ii|,|Pi|denote the number of agents in these sets.
Note that the parameters Tij ,Ni,Piand Iichange with time according
to the dynamics presented below. The parameter TASE T depends on external
conditions, e.g., on fire-related conditions. It has a given value in the start of our
simulations and then it starts to decrease linearly with respect to time. Each
agent updates his strategy according to a Poisson process, frequently enough.
Hence, in a snapshot of the evacuation, the strategies of the agents should be
effectively in an equilibrium configuration; this was verified in [21] by extensive
numerical testing. Actually, in a static environment, where the agents do not
move, the game seems to converge to an equilibrium from any given strategy
configuration when each agent has updated its strategy only a few times.
It should be emphasized, that the game only describes what the agents
expect that will happen in different encounters. The agents’ decision-making
is based on these expectations. We do not assume that they can either take
into account the full complexity of a moving crowd affecting them. Hence,
when coupling the game with the social force model, the actual outcome of an
interaction is not necessarily exactly what the agents expected when selecting
their strategies.
The physical interactions between agents are modeled with the social force
equations [8]. It is assumed that a mixture of socio-psychological and physical
forces influence their motion in a crowd. At time t, agent iwith mass milikes
to move with a certain desired velocity v0
i(t), where v0
i(t) = v0
i(t)e0
i(t). Here,
v0
i(t)is his desired speed and e0
i(t)is a unit vector pointing towards the exit.
Agent iattempts to change his actual velocity vi(t)to v0
i(t)with a certain
characteristic reaction time τi. Agent ialso tries to keep a distance from other
6
agents jN, j 6=iand walls w, modeled by the repulsive social forces fsoc
ij (t)
and fsoc
iw (t), respectively. When agent iis in contact with another agent jor wall
w, the physical contact forces fsoc
ij (t)or fsoc
iw (t), respectively, arise. Additionally,
we assume that agent iis affected by a small random force ξi(t). For the exact
mathematical expressions of the aforementioned forces, see Appendix B. The
change of velocity in time tfor agent iis then given by the equation of motion:
mi
dvi
dt =mi
v0
ivi
τi
+X
j6=i
(fsoc
ij +fc
ij ) + X
w
(fsoc
iw +fc
iw) + ξi,(4)
The change of position xi(t)is given by the velocity vi(t) = dxi(t)/dt.
The strategy of agent ialters its individual movement parameters. Impatient
agents push their way to the exit and overtake others, which is modeled by a
higher desired speed v0
i, and smaller magnitude of the social force fsoc
ij between
iand other agents j. For impatient agents we set v0
i=5 m/s and the magnitude
of fsoc
ij to 1000 N, and for patient agents v0
i=1 m/s and the magnitude of fsoc
ij
we set to 2000 N. Not only does the game alter the physical parameters, but
also the other way round; the game parameters Tij,Ni,Piand Iichange with
position xi.
3. Results
Based on the above model assumptions, we now simulate several important
behavioral and physical phenomena of a crowd evacuating from a given space
through a narrow exit. These phenomena are insensitive to reasonable param-
eter variations. The parameter values for the social force model and the game,
and the numerical integration scheme are presented in the Appendices B, C,
and D, respectively.
3.1. The proportion of impatient agents during exit congestion
As we said above, at every snapshot of the evacuation dynamics, the strate-
gies are effectively in an equilibrium. In fact, along any semicircle centered at
7
the exit, the proportion of impatient agents approximates the ESS of the matrix
game of Eq. 2 at the distance in question (Fig. 1).
Figure 1: An evacuation of a large crowd; a snapshot of a simulation. The agents’ strategies
are effectively in an equilibrium, i.e., no agent can lower its cost by unilaterally deviating.
The proportion of impatient agents increases with distance to the exit. Among the semicircles
A, B, and C, the proportion of impatient agents are 0.75, 0.50, and 0.25, respectively. The
proportion of impatient agents along any semicircle approximates the evolutionary stable
strategy (ESS) of the game at that distance.
The proportion of impatient agents is larger when TASET is smaller or Tij
increases (Fig. 2). Larger Tij means that an agent is farther away from the
exit. Recall that in the game it was assumed that impatient agents overtake
patient neighbors. Thus, the equilibrium is in alignment with the experimental
result (d), that the number of overtakers increases with the distance to the exit.
It is essential that in our model the strategy of an agent is not a permanent
characteristic, but changes according to the agent’s location in the simulated
crowd and external conditions (cf. item (c)).
3.2. Faster-is-slower effect due to locally played game
If we fix the strategies of agents, impatient agents evacuate faster than pa-
tient agents (Fig. 3a). Thus, the assumption in the game that impatient agents
overtake their patient neighbors is realized in the simulations.
In our model, where agents’ strategies are not fixed, the more agents are
impatient, the lower is the outflow at the exit (Fig. 3b). So, the attempt of
8
Figure 2: Snapshots of simulations, where 200 agents evacuate from a room through a 1.2 m
wide exit, for different initial TASET . The agents’ strategies are effectively in an equilibrium.
The larger initial TASET is, the smaller the proportion of impatient agents is in the simulated
crowd. The proportion of impatient agents in a,b,c, and dare 1,0, 0.75, 0.50, and 0.25,
respectively. Moreover, the proportion of impatient agents increases farther away from the
exit.
agents to evacuate faster results in a slower evacuation for the whole simulated
crowd. This is a well-known social dilemma in evacuation research. Typically,
with the social force model [8] it has been simulated by altering the driving force
of agents in the simulated crowd. Here, we get it as a result of a locally played
exit congestion game.
The underlying physical cause of the faster-is-slower effect is the formation
of human arches in front of the exit. Human arches are structures that are
capable of interrupting the flow. They are a result of the combination of the
driving forces of the back of the crowd and the tangential friction forces between
the pedestrians in the arches. These arches are unstable and breakdown by
themselves, due to built-in fluctuations in the system. They may be more stable
when loaded, i.e., when the driving force of the back of the crowd is larger
[16, 17, 24]. An indicator of the formation of more stable arches is longer time
lapses between the passage of two consecutive pedestrians through the exit [16].
9
In our model, the more impatient agents there are in the simulated crowd, the
more probable longer time lapses are (Fig. 3c); hence the more stable arches of
agents form.
It is interesting to note that if we fix all agents to play patient, the simulated
crowd evacuates slower compared to the case when there is a small proportion
of impatient agents (Fig. 3b). If we consider a simulation with all agents pa-
tient to represent a non-competitive scenario, this result is comparable with
item (e), i.e., that agents evacuate faster in a competitive scenario compared
to a non-competitive scenario. This can be explained by the outflow in the
non-competitive scenario to be under the exit capacity. When there are some
impatient agents in the crowd pushing and overtaking, the outflow is increased
and the exit capacity is saturated. After this point, an increase in the propor-
tion of impatient agents will only increase arch formation, which starts to slow
down the evacuation.
3.3. Non-monotonous physical phenomena due to overtaking
It is in the benefit of agents in the back of the simulated crowd to be impa-
tient. Impatient agents in the back of the crowd try to overtake their patient
predecessors. If the crowd is sparse, impatient agents overtake their way to the
middle of the crowd, whereas in a dense crowd they merely push agents in front
of them. Thus, the speed in the back of the crowd is relatively high. When
agents reach the middle of the crowd, they turn patient. Also, in the middle of
the crowd, the density is high, thus, there is no room to move and the speed
drops down. Close to the exit, agents can move more freely, and the speed starts
to increase (Fig. 4a).
There is a constant transportation of momentum from the back of the crowd
forwards, since impatient agents with higher speed manage to overtake their
way to the middle of the crowd, where the average speed is low. This results in
kinetic pressure, which is defined as average density times the variance of speed,
where speed towards the exit is considered (see Appendix A.)(Fig. 4b, c). In
turn, the high kinetic pressure in front of the exit, is a result of arches of agents
10
Figure 3: Simulations of 200 agents evacuating from a 20 m×20 m room through a 1.2 m
exit, for different initial TASET . The results are averaged over 100 simulations. aNumber
of evacuated agents as a function of time. Here agents’ strategies are fixed throughout the
simulation. Initially, there are 100 impatient and 100 patient agents positioned randomly in
the room. Impatient agents leave the room faster than patient agents. bFlow at the exit
for different values of initial TASET ; the corresponding proportion of impatient agents in the
beginning of the evacuation is seen in the bottom x-axis. The error bars denote the standard
deviation of the flow. The flow is maximized when the proportion of impatient agents is
about 0.2, after which it starts to decrease, i.e., the faster-is-slower effect. cEmpirical survival
functions for the time lapse x between the passage of two consecutive agents through the
exit plotted in a log-log scale. The mean time lapses (s) are 0.90±1.01, 0.82±0.89, 0.68±0.68,
and 0.56±0.45 for initial TASET =0, 80, 150 and 500, respectively. The error given is the
standard deviation. The smaller initial TASET is, the higher the probability is for a longer
x. Long time lapses indicate that the agent flow at the exit is obstructed. This results from
more impatient agents pushing towards the exit, which causes intermittent arches of agents
to build up. This is the underlying cause of the faster-is-slower effect.
forming and breaking down, which causes a large variance in agent speed.
In a simulated crowd, where all agents behave similarly, e.g., in the pure
11
Figure 4: Simulation of 200 agents evacuating from a 20 m×20 m room through a 1.2 m
exit. The results are averaged over 100 simulations, from time intervals where the number
of agents in the room is in the interval [190,146]. aSpeed as a function of radial distance
to exit. The red curves correspond to simulations with the game for different initial TASE T ,
and the black curves to simulations without the game, where all agents iNhave the
same desired speeds v0
i=v0. The higher initial TASE T is or the lower v0is, the higher the
curve lies compared to other similarly colored curves. In simulations with the game, when
initial TASET is higher than 150, the curves show a non-monotonous pattern. Conversely, for
simulations without the game, the speed increases monotonically towards the exit. bKinetic
pressure field. Kinetic pressure is defined as average density times the variance of speed
hρ(y, t)it· h[vE(y, t)− hvE(y, t)it]2it, where speed towards the exit is considered. The kinetic
pressure is large in back of the simulated crowd, then decreasing, and increases again at the
exit. In the back of the simulated crowd, it is due to the overtaking behavior of impatient
agents, whereas in front of the exit, it is due to the formation and breakdown of arches of
agents. cKinetic pressure as a function of radial distance distance to exit. The kinetic pressure
exhibits two clear peaks. The larger initial TASE T is, the more apparent the second peak is.
dFor simulations without the game, the kinetic pressure increases monotonically towards the
exit. The method to calculate speed and kinetic pressure fields is explained in Appendix A.
social force model, all physical quantities increase monotonically towards to the
exit (Fig. 4a, d; for a more comprehensive depiction see Figs. A.1 and A.2).
12
On the other hand, if the simulated crowd were set to consist of impatient and
patient agents with fixed strategies, impatient agents would leave the room first,
and in an evacuation with worsening conditions, the last agents to leave the room
would be patient (Fig. 3a; see also movies in Supplementary Material). Such a
setting would not be in line with the experimental result (c), that the strategy of
an agent is not a permanent characteristic, but changes according to the agent’s
location in the simulated crowd and external conditions. In the setting, where
the agents’ strategies are fixed, the kinetic stress field would be similar to the
that of simulations with our model only initially, until the impatient agents have
left the patient agents behind.
4. Conclusions
So, we coupled the social force model with a spatial game, based on simple
assumptions on human behavior in competitive exit congestions. The agents
choose their behavior based on local conditions, getting them either trying to
overtake others or evacuate orderly. Not only were we able to simulate the
typical collective phenomena of an evacuation through a bottleneck, but our
model also explains the mechanism behind these phenomena. Most importantly,
we observed that the speed and kinetic pressure in the simulated crowd has a
non-monotonous dependence of the distance to exit. Quite intuitively, the agents
in the back of the simulated crowd are impatient and try to overtake their slower
predecessors. This transportation of high speed agents to areas with lower speed
results in kinetic pressure.
As stated earlier, kinetic pressure is related to the existence of velocity fluctu-
ations in the evacuating crowd. In addition to this, there exists contact pressure
in evacuating crowds [16, 25]. It is transmitted at contacts between pedestrians
during either short collisions or sustained contacts. In future work, it would be
interesting to calculate contact pressure in the simulated crowd, and study how
it related to the other calculated physical quantities.
It should be noted that the social force term is a totally artificial force,
13
which is known to cause some undesired effects. One is the increase of "social
pressure" at the exit, when the number of agents increase inside the room, even
if agents are not in contact. This is clearly unrealistic for non-touching agents.
Newer versions of the social force model neglect the social force term [11, 15].
It would be interesting to see how a more realistic description of the interaction
forces affects the calculated physical quantities.
In evacuation research literature, the performance of an evacuation has al-
most only been studied at the exit. And no wonder, judging by simulations
with the pure social force model [8], the descriptive physical quantities seem
to vanish as distance to exit increases. Our findings raises the question, what
other physical properties might be lost if self-driven particle models are used
to model evacuations? After all, to improve the performance of facilities and
reduce the risk of disasters, it is necessary to fully understand crowd dynamics
in situations where people gather and the skin-to-skin distance is close [26].
Acknowledgements
The study was funded by a grant from the Foundation for Aalto University
Science and Technology. The calculations in the study were performed using
computer resources within the Aalto University School of Science "Science-IT"
project.
Data Availability
For replication and validation of this study, all of our data are stored in
Zenodo (https://doi.org/10.5281/zenodo.1207077). The codes used in simula-
tion, data analysis and visual representation of the data are found in Github
(https://github.com/antonvs88
/crowddynamics-research), and are also available under the digital object iden-
tifier (DOI) 10.5281/zenodo.1207628.
14
Appendix A. Speed and kinetic pressure fields
Speed and kinetic pressure fields in Figs. 4, A.1 and A.2 are approximated
using a Voronoi method as in [27]. The Voronoi region Riassociated with agent
iNis the set of all points yin the room , whose distance to iis not greater
than their distance to the other agents jN, j 6=i. We define
Ri={y| kxiyk≤kxjykfor all j6=i}.(A.1)
Let agent iproduce a density distribution pi(y, t)defined by
pi(y, t) =
1
S(Ri)for yRi
0otherwise,
(A.2)
where S(Ri)is the area of Ri. Also, let agent iproduce a velocity distribution
Vi(y, t)defined by
Vi(y, t) =
vi(t)for yRi
0otherwise.
(A.3)
Next, we discretize the room into square cells ωof size 0.1 m ×0.1 m,
and calculate the values of the density and velocity fields in a cell ω. We
define αito be the area of the region ωRi, and the density in point yωat
time tis defined by
ρ(y, t) = Piαipi(y, t)
Piαi
,(A.4)
and the velocity is defined by
v(y, t) = PiαiVi(y, t)
Piαi
.(A.5)
Thus, the speed at time tin point yis defined by v(y, t) = kv(y, t)k.
To calculate kinetic pressure, we use the same gas-kinetic definition
hρ(y, t)it· h[vE(y, t)− hvE(y, t)it]2it,(A.6)
15
Figure A.1: Speed field hv(y, t)itfor different scenarios. The speed field is averaged over time
and over 100 simulations. In the simulations, 200 agents evacuate from a room through a 1.2
m wide exit. The upper four rows represent simulations without the game, where all agents
iNhave the same desired speeds v0
i=v0. The lower rows represent simulations with the
game for different initial TASET . From the leftmost to the rightmost column, the number of
agents in the room |N|are in the intervals [190,146], [145,101], [100,56] and [55,11].
where vE(y, t) = kvE(y, t)k, and vEis the component of vin the direction
of the exit. Thus, it is the average density times the variance of speed. If we
compare this to the "crowd pressure" quantity calculated in [15, 5, 4], then
note that there the variance of the velocity vector was used, i.e., hρ(y, t)it·
h[v(y, t)− hv(y, t)it]2it, not just the component pointing towards the exit. Also,
note that the sampling interval of the data points can have some affect on the
variance term of Eq. A.6. In our study, we collected data points every 0.1 s. A
larger sampling step seemed to give a somewhat larger variance.
As agents are discrete objects all around the space, density, speed and ki-
16
Figure A.2: Kinetic pressure field for different scenarios. Kinetic pressure is defined as average
density times the variance of speed hρ(y, t)it·h[vE(y, t)− hvE(y, t)it]2it, where speed towards
the exit is considered. The data is averaged over 100 simulations. In the simulations, 200
agents evacuate from a room through a 1.2 m wide exit. The upper four rows represent
simulations without the game, where all agents iNhave the same desired speeds v0
i=v0.
The lower rows represent simulations with the game for different initial TASET . From the
leftmost to the rightmost column, the number of agents in the room |N|are in the intervals
[190,146], [145,101], [100,56] and [55,11].
netic pressure are only mathematical abstractions. There is no "right" method
to approximate these quantities; each method has its advantages and disadvan-
tages. The Voronoi method assumes the distributions agents produce to be step
functions with adaptive width. Thus, in a situation considered homogeneous,
the resulting density from a group of agents will not show too much variation,
while inhomogeneities will not be masked by too large a width of the individual
density distribution [27]. One of the drawbacks of the Voronoi method is that
the approximation is poor on the boundary of the simulated crowd. However,
17
the speed and kinetic pressure should be close to zero in the back of the room, so
no essential information should be lost, even though the approximation there is
not most accurate. The other drawback is that the discretization affects results.
In this study, we set the discretization to be small enough that local information
is preserved. In [4, 5, 15, 25] a Gaussian distribution was used to describe the
individual density distributions. The main problem with this approach is that
the fixed extent of the Gaussian distribution masks inhomogeneities.
Appendix B. Social force model parameters
In simulations, we used social force model parameter values that have been
validated against data in [8, 28]. The phenomena presented in this article are ro-
bust to reasonable parameter variations. For agent iN, his mass is mi=80kg,
diameter 2riis uniformly distributed in the interval [0.5 m, 0.7 m], and reaction
time is τi=0.5 s. The desired speed v0
iof an impatient agent is 5 m/s and for
a patient agent 1 m/s. Note that pushing behavior of an impatient agent could
be modeled by decreasing τiinstead of increasing v0
i[21].
The repulsive social force between iand another agent jN, j 6=iis defined
fsoc
ij =Aie(rij dij )/Binij ,(B.1)
where the parameters Aiand Bidescribe the strength and the spatial extent of
the force, respectively. For impatient agents we set Ai=1000 N and for patient
agents Ai=2000 N. In simulations without game, we set Aito depend on the
desired speed Ai=2250-250v0
i. Thus, agents with higher desired speed also care
less about others’ personal space. For all agents Bi=0.08 m. The sum of the
agents’ iand jradii is rij =ri+rj, the distance between the agents’ centres
of mass dij =kxixjk, and nij = (n1
ij , n2
ij )=(xixj)/dij is the normalized
vector pointing from agent jto i. The psychological wall-agent interaction fsoc
iw
is treated similarly, but values Awand Bware used for the force constants. We
set Aw=2000 N, and Bw=0.08 m for all agents.
18
Agents iand jtouch each other if rij dij 0. In this case, we assume a
physical contact force
fc
ij =k(rij dij )nij +κ(rij dij )4vt
ji tij ,(B.2)
where tij = (n2
ij , n1
ij )is the tangential direction and 4vt
ji = (vjvi)·tij the
tangential velocity difference. The parameters k=1.2·105kg/s2and κ=2.4·105
kg/(m·s) are force constants. The first term in in Eq. B.2 represents a ’body
force’ counteracting body compression and the second term a ’sliding friction
force’ impeding relative tangential motion. The physical wall-agent interaction
fc
iw is treated similarly and same force constants are used.
The random force ξiin Eq. 4 is decomposed ξi=ξiηi, where the magnitude
ξiis drawn from a truncated Gaussian distribution with mean zero, standard
deviation of 0.1mim/s2, and it is truncated at three times of the standard
deviation. The components of the direction vector ηi= (η1
i, η2
i)are drawn
from uniform distributions on the intervals [cos(0),cos(2π)] and [sin(0),sin(2π)],
respectively. It was noticed that the random force is crucial to avoid gridlocks
by exactly balanced forces in symmetrical configurations.
Appendix C. Game parameters
The agents’ best-response strategy update times are determined from inde-
pendent Poisson processes with the Poisson parameter µ=0.001, meaning that
each agent updates its strategy on the average every 0.001 s.
In our model, agent iNestimates his evacuation time with the formula
Ti=λi, with the value β= 1.25. This means that agents approximate the
flow at the exit region to be 1.25 1/s throughout the whole evacuation. Of
course, in simulations, the flow varies with time, but its reasonable to assume
that the agents are not able to take that accurately into account in a stressful
situation.
19
Appendix D. Integration of the equations of motion
To integrate the equation of motion Eq. 4, we use the velocity Verlet al-
gorithm [29]. In the Algorithm D.1 below, fiis calculated by summing the
right-hand side terms from Eq. 4. Initially, when t= 0, the agents iN
are given positions xi(0) randomly in the room, and they are not moving, i.e.,
vi(0) = 0. With the time increment 4t, the algorithm for agent iis:
Algorithm D.1 velocity Verlet algorithm
1: procedure velocityverlet
2: Calculate fi(t)This step is only used initially.
3: vi(t+1
24t)vi(t) + 1
2
1
mi
fi(t)4t
4: xi(t+4t)xi(t) + vi(t+1
24t)4t
5: Calculate fi(t+4t)Here, the half-step velocity vi(t+1
24t)is used.
6: vi(t+4t)vi(t+1
24t) + 1
2
1
mi
fi(t+4t)4t.Go to step 3.
In the simulations we used 4t=0.001 s.
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23
... Step 2. The evacuation plans are simulated under different scenarios. More specifically, given an evacuation plan and a scenario ϑ k , the system defined by the constraints in Eq. (21) is simulated with a numerical integration scheme to obtain T k last (see, e.g., appendix of [25] for further details). This is done for all evacuation plans in the population and all scenarios. ...
... For example, in fire safety literature, a distinction is made between required safe egress time (RSET) and available safe egress time (ASET). RSET defines the time it takes to evacuate the crowd, and ASET the time before the conditions become lethal [25,41]. If RSET is less than ASET, the evacuation is efficient, and the evacuation is unacceptable if RSET is more than ASET. ...
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