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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 i∈Nestimates his evacuation time

Tito equal the time it takes to queue to the exit, i.e., Ti=λi/β; 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 j∈Ni, 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 C4T−u0(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 1−Tij /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={j∈N|

kxi−xjk ≤ 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

j∈Ii

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 j∈N, 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

i−vi

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 i∈Nhave 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

i∈Nis the set of all points yin the room Ω, whose distance to iis not greater

than their distance to the other agents j∈N, j 6=i. We deﬁne

Ri={y∈Ω| kxi−yk≤kxj−ykfor all j6=i}.(A.1)

Let agent iproduce a density distribution pi(y, t)deﬁned by

pi(y, t) =

1

S(Ri)for y∈Ri

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 y∈Ri

0otherwise.

(A.3)

Next, we discretize the room Ωinto square cells ωof size 0.1 m ×0.1 m,

and calculate the values of the density and velocity ﬁelds in a cell ω∈Ω. We

deﬁne αito be the area of the region ω∩Ri, and the density in point y∈ωat

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

i∈Nhave 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 i∈Nhave 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 i∈N, 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 j∈N, 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 =kxi−xjk, and nij = (n1

ij , n2

ij )=(xi−xj)/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 = (vj−vi)·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 i∈Nestimates his evacuation time with the formula

Ti=λi/β, 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 i∈N

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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