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Twotype Multiagent Game for Egress Congestion

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Our starting point is a recently introduced spatial multiagent game for egress congestion. We present a twotype extension of the game. In the game, the agent chooses its strategy by observing its neighbors’ strategies. The agent’s reward structure depends on its distance to the exit and available safe egress time (TASET ). Although TASET is a well-defined physical quantity, it is assumed that the agents interpret it subjectively: it is assumed that there are high TASET and low TASET agent types. Also, we apply the game to a cellular automaton (CA) evacuation model. We show that high TASET agents are on average able to overtake low TASET agents. However, the more there are high TASET agents in the crowd, the more the evacuation becomes inefficient for the whole crowd.
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Twotype multiagent game for egress congestion
Anton von Schantz, Harri Ehtamo Ilmari Pärnänen
Aalto University School of Science Capgemini Finland Oy
{anton.von.schantz, harri.ehtamo}@aalto.fi ilmari.parnanen@capgemini.com
Abstract
Our starting point is a recently introduced spatial multi-
agent game for egress congestion. We present a twotype
extension of the game. In the game, the agent chooses
its strategy by observing its neighbors’ strategies. The
agent’s reward structure depends on its distance to the exit
and available safe egress time (TASET ). Although TASET
is a well-defined physical quantity, it is assumed that the
agents interpret it subjectively: it is assumed that there
are high TASET and low TASET agent types. Also, we apply
the game to a cellular automaton (CA) evacuation model.
We show that high TASET agents are on average able to
overtake low TASET agents. However, the more there are
high TASET agents in the crowd, the more the evacuation
becomes inefficient for the whole crowd.
1. Introduction
In the faster-is-slower effect, the attempt of individuals to
evacuate faster results in a slower evacuation for the whole
crowd. In [5] the phenomenon was studied experimen-
tally. A group of people were set to evacuate in two dif-
ferent scenarios, one where people tried to evacuate faster
by pushing their way to the exit, and another where push-
ing was forbidden. It was shown that when the individuals
tried to evacuate faster, the evacuation for the whole group
become slower. Contradicting experimental results have
been presented in [8]. However, the results of these two
experiments are not directly comparable with each other
due to differences in incentive systems as well as in the
exit types of the test geometry.
In [6] Helbing gave a physical explanation to the faster-
is-slower effect. When a crowd tries to evacuate through a
bottleneck, the back of the crowd push the agents in front
of them. The driving force of the back of the crowd, com-
bined with the frictional force between the agents, causes
human arches to build up. The arches break down when
there are large enough fluctuations in the forces. These
arches not only slow down the evacuation, but people are
squeezed and suffocated in them.
An indicator of the severity of the arch-formation is the
distribution of time lapses between consecutive evacuat-
ing people. The stronger the human arches are, the more
probable are longer time lapses between consecutive evac-
uating people. In [5] it was shown that in the scenario,
where people tried to evacuate faster, the probability for
longer time lapses was higher. It was also speculated, that
if the situation is competitive enough, the expected value
for the time lapse distribution might not converge. This
means that rare disastrous events are possible.
A common explanation for the self-destructive behav-
ior of a crowd is that people are panicking. However,
sociologists have been unanimous for decades that panic
doesn’t occur in crowds, and that panic is merely a term
for a poorly understood phenomenon [10]. Because
of ethical concerns, there is little experimental research
on the decision-making of people in fire emergencies.
Nonetheless, there are experiments, where a fire emer-
gency has been simulated by using an analogous reward
structure to that in competitive fire egress [11,13, 14]. In
the experiments, it was shown that the faster-is-slower ef-
fect is a result of rational behavior. If individuals in a fire
emergency are considered rational, their decision-making
can be modeled with game theory [2, 3, 7, 9].
Heliövaara et al. [7] use the spatial game approach to
model the decision-making of agents in an egress conges-
tion. The agents have two different behaviors, or strate-
gies to choose, Patient and Impatient. They observe the
strategies of the agents in their immediate neighborhood
and choose their strategy according to the myopic best re-
sponse rule. In the game, an agent’s reward structure de-
pends on its distance to the exit and available safe egress
time (TASET ). TASET is a widely used measure in the fire
evacuation literature [4]. It describes the time it takes for
the fire conditions to become dangerous.
The game model has been coupled to a cellular automa-
ton (CA) evacuation model in [16, 17]. The CA model is
based on the model by Kirchner et al. [12]. In the CA,
the agents’ positions are updated with simple stochastic
update rules. Many emergent phenomena in evacuations,
e.g., faster-is-slower effect, clogging at bottlenecks, and
herding, can be realistically simulated with a CA [12,16].
Also, since the model is discrete in time and space it is
computationally light.
The model in [16] is the starting point of our article. In
[7,16], it was assumed that all agents have the same value
for TASET . The contribution of this article is to extend the
1328
Proceedings of the 50th Hawaii International Conference on System Sciences | 2017
URI: http://hdl.handle.net/10125/41311
ISBN: 978-0-9981331-0-2
CC-BY-NC-ND
game to allow two types of agents in the crowd. Although
TASET is a well-defined physical quantity, it is assumed
that the agents interpret it subjectively: it is assumed that
there are high TASET and low TASET agent types. In Sec. 2
the new game is presented, and in Sec. 3 we examine its
equilibria.
Also, we modify the CA model in Sec. 4. We exclude
the herding effect, since it is not relevant when we are
simulating the evacuation through a single exit. In [16],
the probability that none of the agents are allowed to move
in a situation, where several agents try to move to the same
cell, was modeled with the friction parameter
µ
. It was
held constant throughout the simulation. Here, we set
µ
to depend on the density of the crowd and proportion of
impatient agents in the crowd.
We are interested in how the agent’s subjective inter-
pretation of the fire threat, i.e., the agent’s type affects the
efficiency of the evacuation on both individual and crowd
level. Thus, in Sec. 5, we study the evacuation efficiency
of high TASET and low TASET agents. We are interested
in their efficiency on both individual and crowd level. We
examine both evacuation time, and the distribution of time
lapses between consecutive agents.
2. Game-theoretical model
Next, we present a twotype spatial multiagent game for
egress congestion. It should be mentioned, 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 the
agents can take into account the full complexity of all in-
teracting agents. Hence, when coupling the game with the
CA model in Sec. 4, the actual outcome of an interaction
is not necessarily exactly what the agents expected when
selecting their strategies.
The game models a competitive egress from a room
with a single exit. In the game, there are nagents, in-
dexed by i,iI={1,...,n}. The agents are located in
a room, which is discretized into a square grid, so that a
single agent occupies a single cell of the grid.
Each agent has an estimated evacuation time Ti. Agent
iestimates it by calculating
Ti=
λ
i
β
.(1)
Here,
λ
iis the number of agents closer to the exit than
agent i, and
β
is the capacity of the exit. Since the agents
are packed in front of the exit, the walking time to the
exit is assumed to be much smaller in comparison to the
queuing time. Thus, the agents assume Tiequals only the
time it takes to queue to the exit. In a contest between two
nearby agents iand j, their estimated evacuation times are
approximated Tij := (Ti+Tj)/2.
Each agent has a cost function u(Tij ), for which
u0(Ti j )0. The derivative condition describes the fact
that it is more costly, i.e., there is a larger risk to be ex-
posed to fire related harm, if the estimated evacuation time
Tij is large. Assuming that 4Tis small we can approxi-
mate,
4u(Ti j ) = u(Ti j − 4T)u(Ti j )
=u0(Ti j )4T.(2)
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:
1. In an impatient vs. patient agent contest, the impatient
agent can overtake the patient agent. The impatient
agent reduces its estimated evacuation time by 4T.
Thus, the reduction in its cost is 4u(Tij ). Because the
patient agent’s evacuation time is increased by 4T, its
cost is increased by 4u(Ti j ).
2. In a patient vs. patient agent contest, the patient agents
do not compete with each other, they keep their posi-
tions and their costs do not change.
3. In an impatient vs. impatient agent contest, neither
agent can overtake the other, but they will face a con-
flict and have an equal chance of getting injured. The
risk of injury is described by a cost C>0, which af-
fects both agents. The constant Cis called the cost of
conflict.
From the aforementioned assumptions, a 2×2game
matrix is constructed
Impatient Patient
Impatient C,C−4u(Ti j ),4u(Ti j )
Patient 4u(Ti j ),−4u(Ti j )0,0.
In the game matrix, agent iis a row player and agent
ja column player. When a particular pair of strategies
is chosen, the costs for the two agents are given in the
corresponding cell of the matrix. The cost to agent iis
the first number in a cell, followed by the cost to agent j.
Because this is a cost matrix, the agents want to minimize
their outcome in the game.
1329
2.1. Cost function
In the fire safety literature, available safe egress time
TASET is a widely-used physical measure for the time it
takes for the fire conditions to become lethal [4].
Often computer simulations are needed to assess TASET
accurately [4]. Thus, it is hardly conceivable that hu-
mans, that are under time pressure and have limited cog-
nitive abilities [15] could accurately estimate TASET . In
other words, it is very unlikely that humans have a real-
istic perception of the threat of the situation. Probably,
some of them consider the situation more threatening than
others. In our game model, we assume the agents to have
a subjective measure of TASET . We assume that there are
two types of agents in the crowd, high TASET agents and
low TASET agents. We denote their types by TH
ASET and
TL
ASET , respectively. Thus, for the type variable it holds
TASET ∈ {TH
ASET ,TL
ASET }.
Let us now go back to Eq. (2). If we for simplicity as-
sume 4T=1, we have 4u(Tij )
=u0(Ti j ). So, the cost of
being overtaken is approximately u0(Tij ). Let’s make an-
other assumption about u(Tij ). Recall, that u0(Tij )0.
We additionally assume that u00(Ti j )0, i.e., an agent
considers the cost of being overtaken larger if the esti-
mated evacuation time Tij is larger. Also, we assume that
when Tij =TASET , the agent is indifferent between being
overtaken or risking an injury in a conflict against another
impatient agent, i.e., u0(TASE T ) = C.
Now we can give an explicit form to the cost function
u(Ti j ). An example of a cost function that fulfills these
conditions is
u(Ti j ) =
CT 2
i j
2TASET
,ifTij 0,
0,ifTij <0.
(3)
In Fig. 1, the cost functions for high TASET agents
(TH
ASET =1000), and low TASET agents (TL
ASET =400) are
depicted.
For both cost functions, the cost of conflict parameter is
set C=3. From the figure, it can be seen that for the low
TASET agent, the cost function grows faster. Now, substi-
tute 4u(Ti j ) = u0(Ti j )in the game matrix, and divide it
by u0(Ti j ). This does not affect the equilibria of the game.
Finally, substitute u0(Ti j) = Ti j/TASET . Then, we get the
following game matrix
Notice how the game now only depend on the parame-
ter TASET /Ti j . Whether agents iand jare of same or dif-
ferent type, the game is either symmetric or asymmetric,
respectively. In a symmetric game, the costs only depends
on the strategies chosen by the agents, not on which agent
is playing them. In an asymmetric game, the costs also
depend on which type of the agents is playing the strate-
gies. For general comments on 2 player 2×2games see
Figure 1: Cost functions for high TASET agents (TH
ASET =1000),
and low TASET agents (TL
ASET =400).
Impatient Patient
Impatient TASET
Tij
,TASET
Tij
1,1
Patient 1,1 0,0.
the appendix in [7].
In the symmetric case, when 0<TASET /Tij 1, the
game played is Prisoner’s Dilemma, PD, and the only
Nash equilibrium (NE) is (Impatient, Impatient). The only
NE of PD is an evolutionary stable strategy (ESS) [18].
And when TASET /Ti j >1, the game played is Hawk-Dove,
HD, and there are two pure strategy Nash equilibria (Im-
patient, Patient) and (Patient, Impatient). There is also a
mixed strategy equilibrium, where the strategy Impatient
is played with probability Tij /TASE T , and the strategy Pa-
tient with probability 1Tij /TASE T . The mixed strategy
equilibrium is an ESS.
In the asymmetric case, it holds for agent i,TASET =
TH
ASET , and for agent j,TASET =TL
ASET , or vice versa:
if 0<TH
ASET /Ti j,TL
ASET /Ti j 1, the game played is an
asymmetric PD, and the only NE is (Impatient, Impa-
tient), which is an ESS; and, when TH
ASET /Ti j >1and
0<TL
ASET /Ti j 1, the game played is less-known in the
game theory literature, but it is sometimes called the game
of Samson and Delilah. From hereon, we will call this
instance of the game Samson. In Samson, the only NE,
which is an ESS, is where the high TASE T agent plays Pa-
tient and the low TASET agent plays Impatient. Lastly, if
TH
ASET /Ti j,TL
ASET /Ti j >1, the game played is an asymmet-
ric HD, and there are two pure strategy NE (Impatient, Pa-
tient) and (Patient, Impatient). There is also a mixed strat-
egy equilibrium, where the high TASET agent plays Impa-
tient with probability Tij /TL
ASET , and Patient with proba-
1330
bility 1Tij /TL
ASET , and the low TASET agent plays Impa-
tient with probability Tij /TH
ASET , and Patient with proba-
bility 1Tij /TH
ASET . The pure strategy NE’s are ESS’s.
2.2. Update of strategies
Next, we present how the agents update their strategies.
The agents play the game with the agents in their Moore
neighborhood, i.e., the agents in the 8 nearest cells. We
denote the set of agents in the Moore neighborhood of
agent iby Ni. Notice that when we later set the agents
to move, the neighboring agents will change as the agents
move.
The game is played over periods t=0,1,.... Agent
i’s strategy on period tis denoted by st
i. Here, st
iS
and S={Patient, Impatient}. The agents are assumed
to update their strategies according to the shuffle update
scheme, i.e., during nperiods each of the agents updates
its strategy once. The order in which the agents update
their strategies is randomly chosen.
The agents’ objective is to minimize the sum of costs
against their neighbors. The agents are assumed to up-
date their strategies with the myopic best-response rule,
i.e., agent iobserves the current strategies of its neigh-
bors in Ni, not considering past or future play, and chooses
its best-response strategy. The best-response strategy of
agent iis denoted by BRi. Note that the types of the neigh-
boring agents doesn’t affect BRi. The reason for this is
that the agent’s type is its private information, which we
assume the other agents can’t observe.
Let us assume the initial strategy profile of the agents to
be s0= (s0
1,...,s0
n). On period tagent ichooses its strategy
as follows
s(t)
i=BRi(s(t1)
i;Ti,Ti)
=argmin
s0
iS
jNi
vi(s0
i,s(t1)
j;Tij ).(4)
Here, the notation s(t1)
iis used to denote the strategies
of all other agents than agent iat period t1, and Tiin-
cludes the estimated evacuation times of these agents. The
function vi(s0
i,s(t1)
j;Tij )gives the cost defined by the pre-
viously introduced game matrix to agent i, when it plays
strategy s0
i, and agent jhas played strategy s(t1)
jon period
(t1)
The simplicity of the myopic best-response rule reflects
the limited cognitive abilities that humans are assumed to
have in a stressful situation [15]. Also, it is unrealistic to
assume that the agents updated their strategies simultane-
ously. In many real social systems the agents update their
strategies independent of each other [19]. Thus, the shuf-
fle update scheme is more suitable for strategy update.
3. Equilibrium analysis
In a Nash equilibrium of the spatial game, none of the
agents can lower its cost by unilaterally deviating from its
equilibrium strategy. In [7, 16] the equilibria of the spa-
tial multiagent game for egress congestion with a single
agent type was analyzed. There, given any initial configu-
ration of strategies, the game always converges to an equi-
librium, when all the agents have updated their strategies
about 10 times with the shuffle update scheme. For a cer-
tain sized crowd, with a specific value for TASET , there are
multiple equilibria, i.e., the strategies can be distributed in
many ways in the square grid, so that no agent can lower
its cost by unilaterally deviating. However, the proportion
of impatient agents is the same in each of these equilibria.
Next, we analyze the equilibria of the twotype spa-
tial multiagent game for egress congestion. where 1498
agents have been set to play the game until equilibrium is
reached. In an egress congestion, people typically orga-
nize into a half-circle-like formation in front of the exit.
Thus, the agents are set into a half-circle in front of the
exit. We set TH
ASET =1000 and TL
ASET =400. In Fig. 2a all
agents are high TASET , in Fig. 2b all agents are low TASE T ,
and in Fig. 2c, we consider a crowd with 50 % high TASET
agents and 50 % low TASET agents. In Fig. 2, the equi-
librium configurations of these three scenarios are shown.
Black squares represent impatient agents and gray squares
represent patient agents. The half-circles divide the area
into subareas, where different games are played.
As expected, the strategies of the agents in Figs. 2a and
2b converged to an equilibrium. In Fig. 2a, we can distin-
guish between two areas. They are separated by a black
curve. In the area inside the black curve, all the agents
play HD. Outside the black curve, the estimated evacua-
tion time Tij is so large in comparison to TASET that the
agents play PD. Thus, all agents outside the black curve
are impatient. This area is denoted with C. In Fig. 2b,
TASET is lower in comparison to Fig. 2a. Thus, the agents
are more threatened by the situation, and a higher propor-
tion of agents are impatient, in comparison to Fig. 2a. This
results in the agents playing PD much closer to the exit.
Thus, the black curve is much lower here than in Fig. 2a.
The area in Fig. 2b, where the agents play HD is denoted
by A.
Interestingly, also in Fig. 2c, where there are two type
of agents, the strategies converge to equilibrium, when the
agents have updated their strategies about 10 times with
the shuffle update scheme. In the equilibrium, we can dis-
tinguish between 3 different areas. In area A, agents of the
same type play HD against each other, and agents of dif-
ferent type asymmetric HD against each other. In area C,
agents of the same type play PD against each other, and
agents of different type asymmetric PD against each other.
1331
(a)
(b)
(c)
Figure 2: Equilibrium configurations, where (a) all agents are
high TASET , (b) all agents are low TASET , (c) 50 % are high TASET
agents and other 50 % low TASET agents. Black squares repre-
sent impatient agents and gray squares represent patient agents.
The half-circles divide the area into subareas, where different
games are played.
In between these two areas is area B, where high TASET
agents play HD against each other, low TASET agents play
PD against each other, and agents of different type play
the game Samson against each other.
A surprising effect can be noticed, by comparing the
proportion of impatient agents in the different scenarios
of Fig. 2. In the single type crowd, in Fig. 2a, ca. 60 % of
high TASET agents are impatient, whereas, in the twotype
crowd, in Fig. 2c, only 40 % of high TASET agents are im-
patient. However, the opposite effect does not happen to
low TASET agents, in both the single and twotype crowds
ca. 90 % of the low TASET agents are impatient.
This can be explained by examining the agents’ best-
response strategy from Eq. (4), which can be expressed as
an inequality. Recall the costs associated with interactions
of different strategists. Now, agent ishould play Impa-
tient, if the cost of playing Impatient against its neighbors
is less than or equal to the cost of playing Patient:
jNi
TASET
Tij
+ (|Ni|−|NImp
i|)≤ |NIm p
i|,(5)
where |Ni|is the number of agents in the neighborhood
of iand |NIm p
i|is the number of impatient agents in the
neighborhood of i. To make the analysis simpler, let us
approximate Ti=Tj,jNImp
i. Then the above inequality
can be written as
|NIm p
i|
|Ni|Ti
TASET
.(6)
Here, |NIm p
i|/|Ni|is the proportion of impatient agents
in the Moore neighborhood of agent i. In a spatial equi-
librium of the game, Eq. (6) has to hold for all agents
iN. We can infer that in an equilibrium a low TASET
agent plays Impatient with a higher proportion of impa-
tient agents in its neighborhood compared to a high TASET
agent. Thus, in a twotype crowd, in an equilibrium, low
TASET agents are going to fill most of the space for impa-
tient agents.
3.1. Sensitivity analysis
Next, the same simulation setup is used as in Fig. 2. We fix
the value for TH
ASET and alter the value for TL
ASET . Then,
we study the proportions of impatient agents in the spa-
tial equilibrium of the game. The simulations for each
fixed TH
ASET are run for 100 different values of TL
ASET ,
equally distributed in the interval [0,TH
ASET ]. The inter-
polated curves of the data points are shown in Fig. 3.
For example, see Fig. 3, the curve in the top, there
TH
ASET =500. The simulations show that when the value
for TL
ASET is increased, the proportion of impatient agents
in the spatial equilibrium decreases.
In Fig. 4 we examine how altering the proportion of low
TASET agents in a crowd changes the proportion of impa-
tient agents in the equilibrium. The simulations are run
for 100 different proportions of impatient agents, equally
1332
Figure 3: Proportion of impatient agents in the spatial equilib-
rium, when TH
ASET is fixed and TL
ASET is altered.
distributed between [0,1]. Otherwise, same settings are
used as in Fig. 2, i.e., TH
ASET =1000 and TL
ASET =400.
Figure 4: Proportion of low TASET agents in the crowd vs. the
proportion of impatient agents in the equilibrium of the crowd.
As the proportion of low TASET agents in the crowd in-
creases, the proportion of impatient agents in the equilib-
rium of the game increases.
It should be noted that the size of the crowd also has an
effect on the proportion of impatient agents in the equilib-
rium. Since, when the agents update to their best-response
strategy, they consider the ratio TASET /Tij . Recall Eq. (1);
Tij is calculated for an agent by considering the amount of
agents closer to the exit than the agent in question. Thus,
if the crowd size is increased, the agents in the back of the
crowd will have higher values for Tij .
4. Simulation model
The interactive decision-making situation of individuals
in an egress congestion can be modeled with a spatial
game as above. Next, we present the CA model, to which
the game is coupled. The CA is a modified version of
the model used in [16]. In the CA, the agents move in
the cells of a discrete square grid. At the beginning of
a time step, the agents are allowed to move one cell in
orthogonal directions. The dimensions of a cell are as-
sumed to be 0.4m×0.4m, and the length of a time step
0.3s. Movement in diagonal directions is not allowed,
since it accounts to a higher velocity. The agents’ move-
ment directions are determined by transition probabilities.
These transition probabilities are proportional to the static
floor field SF. The closer the cells are to the exit, the
higher values SF attains. In [16] a more complex CA
was used, where the agents movement also depended on
the so-called dynamic floor field DF. The dynamic floor
field is omitted here, since its main purpose is to model
the herding effect of agents, which is not relevant phe-
nomenon in the evacuation geometries we use in this arti-
cle.
So, in our model the probability pl m for an agent to
move to a neighbor cell (l,m)is calculated as follows
plm =1
ZekSF SFlm(1
ξ
lm ),(7)
where
ξ
lm =1for forbidden cells (walls and occupied cells)
0else
and the normalization
Z=
(l,m)
ekSF SFlm(1
ξ
lm ).
Here, kSF [0,)is the agent’s coupling parameter to
SF. Basically, the larger kSF is, the more straight the agent
is moving towards the exit, or the more assertive they
are. For a thorough analysis on how the floor fields and
coupling parameters affect the movement of the crowd,
see [1, 12, 16].
The agents movement is updated with the parallel up-
date scheme, i.e., all the agents desired movement direc-
tions are updated simultaneously. Now, there are situa-
tions where several agents desire to move to the same cell,
i.e., conflict situations. In [12] a friction parameter
µ
was
introduced, which describes the probability that none of
of the agents in a conflict situation are able to move to the
cell. The friction parameter works as a kind of local pres-
sure between the agents [12]. In [12] it was shown that
these conflict situations are not just an artifact of the par-
allel update scheme, but an important feature to describe
evacuation dynamics correctly.
4.1. Friction parameter
In [12, 16],
µ
is assumed to be fixed throughout the sim-
ulation. However, if
µ
is to work as a kind of local pres-
sure between agents, it should vary based on the size of
1333
the crowd, and on the proportion of impatient agents in it.
We define
µ
as
µ
=b1
ρ
a
ρ
Im p +b2
ρ
a+b3
ρ
Im p.(8)
Here, bi,i=1,2,3are coefficients for which b1+b2+
b3=1, and
ρ
Im p is the proportion of impatient agents in
the crowd.
ρ
ais calculated by dividing the current size
of the crowd with the initial size of the crowd. Thus, in
the beginning of the evacuation simulation
ρ
a=1, and as
agents are able to evacuate the room,
ρ
adecreases. This
is a reasonable approximation for the effect of crowd size,
if the initial crowd is large and the agents evacuate from
a room with a single exit. However, for more complex
geometries
ρ
ashould be calculated differently.
Since
ρ
a,
ρ
Im p [0,1], it holds that
µ
[0,1]. The
interaction term
ρ
a
ρ
Im p captures the interaction between
crowd size and proportion of impatient agents. We as-
sume the interaction term has the largest impact on
µ
.
For simplicity, we assume that the impact of
ρ
aand
ρ
Im p
equally large. This leads us to use the parameter values
b1=0.6,b2=0.2,b3=0.2in simulations in our arti-
cle. We are mainly interested in qualitative phenomena
in evacuations. For quantitative accuracy, the correct co-
efficient values should obviously be estimated from ex-
perimental data.
Let us examine how altering the value of a single coef-
ficient bi,i=1,2,3, affects the value of
µ
. If the value for
b1is increased,
µ
decreases. On the other hand, if b1is
decreased,
µ
increases. The effect of parameters b2and b3
is not so straightforward. If b2is increased,
µ
decreases
when
ρ
a>
ρ
Im p, and
µ
increases when
ρ
a<
ρ
Im p. For the
parameter b3the opposite occurs, if b3is increased,
µ
in-
creases when
ρ
a>
ρ
Im p, and
µ
decreases when
ρ
a<
ρ
Im p.
As
ρ
aand
ρ
Im p change during the simulation of an evac-
uation, altering the coefficients b2and b3can increase
µ
at some stages in the simulation and decrease it in oth-
ers, compared to a simulation done with the values for
bi,i=1,2,3used in this article. Thus, altering parame-
ters b2and b3results in nonlinear effects on the evolution
of
µ
during the simulation.
4.2. Spatial game coupled with a CA evacu-
ation model
The spatial game is coupled to the CA evacuation model.
For technical purposes, the movement of the agents is up-
dated in parallel. Without parallel update, there would
be no well-defined time scale. On the other hand, the
strategies of the agents are updated with the shuffle up-
date scheme, because it is more realistic to assume that
agents do not simultaneously update their strategies.
The time scale in updating strategies is assumed to be
much smaller than that of movement. Thus, the crowd is
in a spatial equilibrium at a snapshot of the simulation. Yet
again, it should be reminded that the game the agents play
only models the agents expected outcomes, and it does not
have to correspond to the realization of the CA evacuation
model.
Next, a step-by-step description is given of the spatial
game coupled to the CA evacuation model. In the begin-
ning of the simulation, the agents are located randomly in
the room.
Step 1. At the beginning of each time step, the game pa-
rameters Tij = (Ti+Tj)/2,i6=j,i,jIare calculated
according to Eq. (1).
Step 2. The agents’ strategies are updated with the shuf-
fle update scheme until an equilibrium is reached. The
agents observe the strategies of the other agents in their
Moore neighborhood, and choose a best-response strategy
according to Eq. (4).
Step 3. The agents’ behavior in the CA model is set to
correspond their strategy choice. This is done by altering
kSF for the agents as follows:
(a) For an agent playing Impatient kSF =10.0.
(b) For an agent playing Patient kSF =1.0.
Step 4. Friction parameter
µ
is calculated according to
Eq. (8).
Step 5. The agents’ positions are updated in parallel ac-
cording to Eq. (7). In a case of a conflict, one of the agents
is allowed to move with probability 1
µ
.
Step 6. Go to Step 1. The procedure is repeated until all
agents have evacuated the room.
5. Simulation results
We are interested in how the agent’s type affects the ef-
ficiency of the evacuation on both individual and crowd
level. Thus, in Sec. 5.1, we simulate an evacuation with
both high and low TASET agents, to see how the different
agent types perform against each other. And, in Sec. 5.2,
we simulate an evacuating crowd consisting of a single
type agent, to see how the different types of agents per-
form on a crowd level, and show that the faster-is-slower
effect is a result of agents’ subjective interpretation of the
threat. In the last two experiments, we study the mech-
anism behind the faster-is-slower effect. In Sec. 5.3, we
look at the time evolution of a single simulation, to find
that the agents evacuate in a stepwise manner. Finally, in
Sec. 5.4 we study the distribution of the step lengths, or
time lapses between consecutively evacuated agents.
1334
In the simulations, we are using a crowd of 200 agents.
Thus, smaller values for TH
ASET and TL
ASET are used in com-
parison to the equilibrium simulations in Sec. 3. We set
TH
ASET =120 and TL
ASET =30. The capacity of the exit is
β
=1.25 (1/s), in all simulations. For impatient agents
the coupling parameter is set to kSF =10 and for patient
agents kSF =1. In all simulations, a one cell, i.e., 0.4m
wide exit is used. The friction parameter is of the form
µ
=0.6
ρ
a
ρ
Im p +0.2
ρ
a+0.2
ρ
Im p.
Sensitivity analysis on the effect of
µ
on macroscopic
quantities of an evacuation simulation, e.g., flow at exit,
have been already studied thoroughly in [12, 16]. Thus,
the authors are left with the following conclusion: it ap-
pears that the faster-is-slower effect is not sensitive to
changes in b1, as long as it is not decreased too much from
0.6. The faster-is-slower effect is more sensitive to the
values of b2and b3. They should not be set much higher
than 0.2 for the model to work.
5.1. Performance of the individuals
Next, we study how high TASET agents perform against
low TASET agents. In Fig. 5, a crowd of 200 agents, with
100 high TASET agents, and 100 low TASET agents, have
been set in a room to evacuate through a narrow exit. The
simulation was run 100 times, and the number of evacu-
ated agents as a function of time was monitored. The gray
curve represents the average number of evacuated high
TASET agents, and the black curve the average number of
evacuated low TASET agents.
Figure 5: Evacuation of a crowd consisting of 100 high TASET
agents and 100 low TASET agents. Averaged number of evacu-
ated agents as a function of time for both agent types.
In Fig. 5, it can be seen that the average number of evac-
uated low TASET agents is almost always higher, than the
average number of evacuated high TASET agents. How-
ever, both types of agents have evacuated at about the
same point in time. Thus, our simulations show that the
majority of low TASET agents are able to rush to the exit
before high TASET agents.
Since, at the beginning of each simulation, the agents’
positions were randomized, this shows that on average
low TASET agents are able to overtake high TASET agents
in our simulations. This can be explained by the analyses
from Sec. 3, which show that a low TASET is more prone
to be impatient. Furthermore, in [16] it has been shown
that individual impatient agents are able to overtake pa-
tient agents in the CA evacuation model.
5.2. Performance of the crowd
Next, we study how the agent types perform on a crowd
level. We set all agents in a crowd to be of the same type,
and see which crowd evacuates faster. Otherwise we take
the same simulation setup as in Fig. 5. Two scenarios are
simulated; in the first the crowd consists of high TASET
agents, and in the second, it consists of low TASET agents.
The simulations are run 100 times, and the results are seen
in Fig. 6.
Figure 6: Averaged number of evacuated agents as a function
of time for two different values of TASE T .
The gray curve represents the average number of evac-
uated high TASET agents, and the black curve the average
number of evacuated low TASET agents. Whereas an indi-
vidual low TASET agent was able on average to evacuate
faster than an individual high TASET agent in a twotype
crowd, here the crowd with low TASET agents evacuate
slower than a crowd with high TASET agents. As the crowd
consists of only low TASET agents, there is nobody to over-
take, and the agents just hinder each others’ attempt to es-
cape.
5.3. Time evolution of an evacuation
Because the values in Fig. 6 are averaged, the curves do
not tell us about the development of a single simulation. In
1335
Fig. 7 we take a look at the first 40 seconds of a single sim-
ulation of the evacuation of both a crowd with only high
TASET agents, and a crowd with only low TASET agents.
Figure 7: The development of a single simulation. Number of
evacuated agents as a function of time for both agent types.
It is interesting to note that the curves, for both agent
types, increase in an irregular stepwise manner. Actually,
for a real crowd it is quite typical that people evacuate in
such irregular successions [5]. In real crowds, it is a re-
sult of human arches forming and breaking down [6]. It
is quite fascinating that the simple CA model can simu-
late this phenomenon. This feature of the CA was already
noted in [12].
Though, in the CA, the phenomenon is not a result of
human arches forming and breaking down, because no
actual physical forces in the crowd are modeled. In the
CA, in front of the exit, there are constantly conflict situ-
ations. Whether any of the agents is allowed to move to
the desired cell, is a consequence of the friction parame-
ter
µ
. The stochastic nature of these conflict situations in-
troduces irregularity to the time lapses between consecu-
tively evacuated agents. In [12] an analytical dependence
between
µ
and the number of evacuated agents with re-
spect to time has been derived.
The average amount of evacuated agents is indirectly
proportional to the average time lapse between two con-
secutive agents. The step lengths of the curves show how
long the time lapse is between two consecutive evacuated
agents. Judging from Fig. 7, for the crowd with low TASET
agents, the step lengths are longer, and there is more vari-
ability in their lengths.
5.4. Distribution of time lapses
Next, we calculate the average time lapses between two
consecutive agents for both a crowd of only high TASET
agents, and a crowd of low TASET agents. Since the fric-
tion parameter
µ
gives the probability that none of the
agents in a conflict situation is allowed to move, the length
of the time lapses is dependent on the value of
µ
. Recall
Eq. (8);
µ
depends on the size of the crowd. Thus, the
distribution of the time lapses should depend on the size
of the crowd. As the crowd is evacuating, the size of the
crowd changes with time. Thus, we use data of only the
10 first time lapses from 100 simulations, which results in
1000 data points. For high TASET agents, the mean is 5.40
s and for low TASET agents it is 3.51 s.
In Fig. 8 the complementary cumulative frequency dis-
tribution (complementary CDF) of the time lapses is plot-
ted for a crowd with high TASET agents and for a crowd
with low TASET agents. The complementary CDF equals
1CDF, and tells the probability of the time lapse, say
4x, being larger than some specific time t. Note that both
of the axes are in logarithmic scale.
Figure 8: Complementary cumulative frequency distribution of
the time lapses 4xbetween two consecutive evacuated agents
for both a crowd with high TASET agents and a crowd with low
TASET agents.
Note that the probability for low TASE T agents is always
larger than the probability for high TASET agents. Thus,
our simulations show that longer time lapses are more
probable for a crowd with low TASET agents.
6. Conclusions
The contribution of our article compared to previous arti-
cles [7, 16] is that here we have extended the spatial mul-
tiagent game, originally presented in [7] with only one
agent type, to allow two types of agents, high and low
TASET agents, and given a new explanation to the faster-
is-slower effect with this model.
In Sec. 3, the equilibria of the game were studied. It
was very interesting to notice that an agent, in a crowd
with only high TASET agents, would be impatient in a cer-
tain location, but in a crowd with two types of agents, in
exactly the same location, the agent would be patient. This
can be explained by the low TASET agents’ best-response
1336
strategy allowing a larger proportion of impatient agents
in their neighborhood, compared to high TASET agents’
best-response strategy. Thus, in a twotype crowd, in an
equilibrium, it is the low TASET agents that are going to
fill most of the spaces for impatient agents.
In Sec. 4, the spatial game was coupled to a modified
version of the CA evacuation model from [16]. The dy-
namic floor field, which was included in the original CA
evacuation model, was omitted, as its main purpose was
to model the herding behavior of agents. The friction pa-
rameter
µ
, which describes the probability that none of the
agents in a conflict situation is able to move, was set to de-
pend on the density of the crowd and proportion of impa-
tient agents in it, thus more realistically modeling build-up
of local pressure in the crowd.
In Sec. 5, evacuation of a crowd from a room was sim-
ulated. It was shown that low TASET agents were on av-
erage able to overtake high TASET agents. However, if all
the agents in the crowd were low TASET , they evacuated
slower than a crowd were all the agents were high TASET .
Also, the underlying mechanism of the faster-is-slower ef-
fect was studied, by examining distribution of time lapses
between consecutive agents. It was shown that the aver-
age time lapse between consecutively evacuating agents
is higher for a crowd with low TASE T agents. Moreover,
it seems that longer time lapses are more probable for a
crowd with low TASET agents. Our results coincide nicely
with the experimental results in [5], even though we were
not able to fit a power law to the tail of the complementary
CDF.
In this article, we restricted ourselves to the simple sce-
nario of a crowd evacuating from a room through a sin-
gle exit. When modeling more complex geometries, we
should take into account agents’ exit selection and herd-
ing effects, i.e., that people go where the majority of the
crowd is heading. Also, the static floor field and friction
parameter
µ
should be set suitable for the new geometry.
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