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Spatial game in cellular automaton evacuation model

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For numerical simulations of crowd dynamics in an evacuation we need a computationally light environment, such as the cellular automaton model (CA). By choosing the right model parameters, different types of crowd behavior and collective effects can be produced. But the CA does not answer why, when, and how these different behaviors and collective effects occur. In this article, we present a model, where we couple a spatial evacuation game to the CA. In the game, an agent chooses its strategy by observing its neighbors' strategies. The game matrix changes with the distance to the exit as the evacuation conditions develop. In the resulting model, an agent's strategy choice alters the parameters that govern its behavior in the CA. Thus, with our model, we are able to simulate how evacuation conditions affect the behavior of the crowd. Also, we show that some of the collective effects observed in evacuations are a result of the simple game the agents play.
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Spatial game in cellular automaton evacuation model
Anton von Schantzand Harri Ehtamo
Systems Analysis Laboratory, Aalto University, P.O. Box 111000, FI-00076 Aalto, Finland
For numerical simulations of crowd dynamics in an evacuation we need a computationally light
environment, such as cellular automaton model (CA). By choosing the right model parameters,
different types of crowd behavior and collective effects can be produced. But the CA does not
answer why, when and how these different behaviors and collective effects occur. In this article, we
present a model, where we couple a spatial evacuation game to the CA. In the game, an agent chooses
its strategy by observing its neighbors’ strategies. The game matrix changes with the distance to
the exit as the evacuation conditions develop. In the resulting model, an agent’s strategy choice
alters the parameters that govern its behavior in the CA. Thus, with our model, we are able to
simulate how evacuation conditions affect the behavior of the crowd. Also, we show that some of
the collective effects observed in evacuations are a result of the simple game the agents play.
I. INTRODUCTION
When a large crowd evacuates from a public place or
building, it gives birth to many collective effects. These
collective effects include clogging at bottlenecks and the
faster-is-slower effect. In the faster-is-slower effect, the
attempt of individuals to move faster results in a slower
evacuation for the whole crowd.
There are experiments studying microscopic pedes-
trian behavior in bottlenecks. In [1] it was shown that
a clogging can occur without the flow exceeding the ca-
pacity of the bottleneck. It has also been shown that
humans form lanes in bottlenecks. These lanes overlap,
which is why the phenomenon is called the zipper effect.
Because of the zipper effect, the flow in a bottleneck is a
stepwise function of the bottleneck width [2, 3].
In [4] experimental proof for the faster-is-slower effect
is given. In the experiment, a group of people was set to
evacuate a room through an exit. Time lapses between
consecutive individuals were monitored. An interesting
result was that the higher the desired velocity of the evac-
uees was, more likely longer time lapses were.
The time lapses are a result of the formation and break
down of arch-like blocks of humans at exits. These arches
are able to hinder the flow at exit. An irregular succes-
sion of arch-like blocks forming and breaking also causes
the underlying phenomena for clogging at bottlenecks [5].
Note, that a higher probability for longer time lapses, not
only means a slower evacuation, but also a higher prob-
ability for people to get suffocated and squeezed.
In contrast, there is also experimental evidence of phe-
nomena, which can improve the evacuation efficiency. In
[6] it was shown that placing an obstacle in front of an
exit makes the evacuation more efficient. Also, maybe we
could learn something from how ants behave when facing
danger. In [7] it was shown that the faster-is-slower effect
is not witnessed in ants, because ants are always evenly
distributed in the space, unlike humans which form dense
areas in front of exits.
anton.von.schantz@aalto.fi
What is evident, is that for a safe evacuation, it is
crucial to understand the mechanism behind dangerous
collective effects. The agent-based social force model
describes the actual physical forces arising in a rush-
ing crowd [5]. It has been implemented to the soft-
ware FDS+Evac, which allows simultaneous simulation
of both fire spreading and evacuation process [8]. Collec-
tive effects in evacuations can be realistically simulated
with FDS+Evac by setting specific model parameters [8–
11]. However, it does not answer why, when and how
these effects occur.
There lies a consensus among social theorists that peo-
ple behave rationally in evacuation situations [12–14]. In
his famous experiment, Mintz showed that escape panic
is due to the reward structure innate to the evacuation
situation [15]. Moving orderly towards the exit is reward-
ing, as long as everybody else moves orderly. When one
agent starts to rush, rushing behavior becomes rewarding
also for the other agents.
Brown attempted to model Mintz’ results with game
theory. He suggested the agents to play a one-shot Pris-
oner’s Dilemma (PD) game [16]. In the game, the agents
choose simultaneously, not knowing the other agents
choice, whether to rush or evacuate orderly. The outcome
of the game is for all agents to rush. However, this is not
exactly in line with Mintz’ results, and with how people
behave in evacuation situations. In reality, people can
observe the behavior of other agents. Thus, it has been
suggested that the agents rather play a repeated version
of PD, where the agents make their decisions based on
what the other agents have played in the previous rounds
of the game [17].
Heli¨ovaara et al. [18] have pointed out that an evac-
uation situation is spatial in nature, and, since time is
a limited resource, the reward structure should change
according to how far from the exit the agent is. Based
on these remarks, Heli¨ovaara et al. proposed a spatial
game, where the agents play against their nearest neigh-
bors. The agents can choose between two strategies: Pa-
tient and Impatient. The game will be either Hawk-Dove
(HD) or PD, depending on how far they are from the
exit. The strategy Patient corresponds to the strategy of
evacuating orderly, and Impatient to rushing in Mintz’
2
experiment. The agents are myopic in the sense that
they choose their strategies based on the previous pe-
riod play of their opponents, not considering the play at
farther history, or of future periods.
The spatial game has been implemented to FDS+Evac
in [18]. There, the physical behavior of the agents is
obtained by altering the parameters of the social force
model in the following way: impatient agents do not
avoid contacts with other agents as much; they accelerate
faster to their target velocity, and move more nervously.
On the other hand, patient agents, avoid contact with
other agents.
Simulations with this version of FDS+Evac gives an
explanation to the faster-is-slower effect [18]. Due to
the game agents play, the proportion of impatient agents
grow farther back in the crowd. Impatient agents push
the crowd in front of them. As a result of the driving
force of the agents back in the crowd and frictional in-
teraction between agents, a clogging is formed [5]. The
clogging slows down the evacuation, and becomes more
severe as the proportion of impatient agents grow.
However, FDS+Evac is computationally rather heavy.
This is because the social force model is continuous and
interaction forces between all agents are modeled. In
this paper, our goal is to couple the spatial game [18] to
a computationally lighter evacuation model. Such model
could be used by the incident commander, e.g., through
the internet, to give timely and accurate instructions in
evacuation situations.
For our purposes, the computationally very light cel-
lular automaton evacuation model (CA) is appropriate.
With it, it will be possible to produce many collective ef-
fects observed in evacuation situations, for example, lane
formation, clogging and faster-is-slower effect [19–22]. In
CA the agents move in a discrete square grid according to
some transition probabilities. These transition probabil-
ities depend on the so called coupling parameters, which
describe the degree to which the agents move straight to-
wards the exit, or move to regions of higher crowd flow.
There is some literature on agents’ decision-making in
CA evacuation models [23–27]. In these articles, game
theory has mainly been used to solve conflict situations,
i.e., situations where several agents try to occupy the
same space simultaneously. The agents play a game when
in a conflict, to determine who gets to move to the cell.
Thus, the decision-making is limited to these conflicts
only. However, it is reasonable to assume that the agents
interact with their immediate neighborhood, and make
decisions, also when not in a conflict. Furthermore, in
large scale simulations, the games the agents play at dif-
ferent distances from the exit should differ.
In this paper, we propose a model, where the spatial
evacuation game presented in [18] is coupled with the CA
presented in [21]. The agents observe their neighbors’
strategies in the discrete square grid, and choose their
strategy accordingly. The game the agents play depends
on the distance to the exit and the available time to safely
evacuate. The choice of strategy alters the agents’ behav-
ior in the CA: if the agent plays Impatient, it will rush
straight towards the exit, and, if it chooses Patient, it
will move towards the exit using regions of higher crowd
flow. These behaviors are obtained by choosing the cou-
pling parameters for the agents accordingly. As far as we
know, there are no earlier studies where agents in a CA
can have different values for the coupling parameters, or
the agents’ coupling parameters would change systemat-
ically during the simulation.
Our model takes into account that there are differently
behaving individuals in the crowd and that their behavior
can change during the evacuation. Simulation results
with our model show that some of the collective effects
observed in evacuations are a result of the simple game
the agents play.
The paper is structured as follows: in Sec. II we present
how crowd movement is modeled with a CA; in Sec. III we
present the spatial evacuation game; in Sec. IV we couple
the CA evacuation model with the game; in Sec. V we
present simulation results with our model; Sec. VI is for
conclusions.
II. CELLULAR AUTOMATON MODEL
A two-dimensional cellular automaton (CA) is a grid
of cells, where each cell has a finite number of states.
The state of every cell can be updated in discrete time
steps, so that it simulates the movement of a crowd in a
two-dimensional space. The update rules can be defined
in many ways. Next, we present a well-known CA evacu-
ation model [21] by Schadschneider et al. In Sec. IV, the
spatial evacuation game will be coupled to this model.
A. Movement in the CA
In the CA, the agents are located in a discrete square
grid, so that a single agent occupies a single cell. The
agents move towards the exit according to some transi-
tion probabilities. A cell is assumed to be 40 cm ×40 cm.
The maximal possible moving velocity for an agent is one
cell per time step, i.e., 40 cm per time step. Empirically
the average velocity of a pedestrian is about 1.3 m/s [28].
Thus, a time step in the model corresponds to 0.3 s.
At each time step of the simulation, the agent can move
to one of the unoccupied cells in his von Neumann neigh-
borhood, i.e., the nearest cells in the orthogonal direc-
tions. Diagonal movement is not allowed, since moving
diagonally in a square grid would account for a higher
velocity than moving orthogonally.
The transition probabilities depend on the values of
the static and dynamic floor field in the cells. The static
floor field SF is based on the geometry of the room. The
values associated with the cells of SF increase as we move
closer to the exit, and decrease as we move closer to the
walls. The dynamic floor field D F , in turn, represents a
virtual trace left by the agents. An agent leaving a cell,
3
causes the value of DF in that cell to increase by one
unit. Over time, the virtual trace decays and diffuses to
surrounding cells. The values of the fields DF and SF
are weighted with two coupling parameters kDF [0,)
and kSF [0,).
Now, for each agent, the transition probabilities pmn,
for a move to a neighbor cell (m, n) are calculated as
follows
pmn =1
ZekDF DFmn ekSF S Fmn (1 ξmn),(1)
where
ξmn ={1 for forbidden cells (walls and occupied cells)
0 else
and the normalization
Z=
(m,n)
ekDF DFmn ekSF S Fmn (1 ξmn).
The agents’ desired movement directions are updated
with a parallel update scheme, i.e., the directions are up-
dated simultaneously for all agents. The parallel update
of movement directions can result in conflict situations,
i.e., situations where several agents try to occupy the
same cell simultaneously. In these situations, all the
agents are assigned equal probabilities to move, and with
probability 1 µone of the agents is allowed to move to
the desired cell. Here, µ[0,1] is a so called friction
parameter. In [21], a conflict situation was interpreted to
represent the effect of pressure in the crowd. The impact
of the friction parameter is depicted in Fig. 1.
FIG. 1. The impact of friction parameter on the agents’ movement.
With probability µneither of the agents gets to move, and with
probability 1 µone of the agents moves.
B. Different crowd behaviors
It has been shown that by altering the coupling pa-
rameters kSF and kDF different crowd behaviors can be
observed [20, 21]. The different crowd behaviors were
called ordered, disordered and cooperative in [20, 21]. In
Fig. 2, the coupling parameter combinations responsible
for different behavioral regimes are plotted in a schematic
phase diagram.
FIG. 2. Altering the coupling parameters kSF and kDF , in the
CA model, produces different crowd behaviors.
Before analyzing the behavior of agents in the differ-
ent regimes, let us consider the dynamic floor field DF .
Because the value of DF in a cell at a specific time is
proportional to how often agents have been able to leave
that cell recently, it can be thought to describe temporal
crowd flow in different parts of the grid. For cells where
agents get stuck due to conflict situations, the value of
the DF is small.
Now we can give an interpretation to the different be-
havioral regimes. In the ordered regime, an agent’s tran-
sition probabilities mainly depend on S F , i.e., the agent
moves straight towards the exit. In the disordered regime,
an agent’s transition probabilities mainly depend on DF ,
i.e., the agent moves to cells of locally higher crowd flow.
Between the ordered and disordered regime is the coop-
erative regime around the values kDF =kSF = 1. In the
cooperative regime, an agent’s transition probabilities de-
pend as much on SF as DF . That is, the agent moves
towards the exit, using cells of locally higher crowd flow.
Typical movement patterns of a single agent of different
behavioral regimes are illustrated in Fig. 3. In the figure,
the agent is part of crowd, i.e., it is not moving indepen-
dently in the space. For the sake of clarity, the movement
patterns of the rest of the crowd is not depicted.
4
FIG. 3. Typical movement patterns of an agent in the different
behavioral regimes.
In Fig. 3, we can see that if the agent is in the ordered
regime, it moves straight towards the exit. In the co-
operative regime, the agent moves towards the exit, but
using a longer route with a higher average crowd flow.
In the disordered regime, the agent just blindly moves in
the grid to regions of locally higher crowd flow.
C. Collective effects in evacuations
With the right parameter choices the CA model is able
to generate many of the collective effects in evacuations,
e.g., the faster-is-slower effect and clogging at bottlenecks
[21].
The faster-is-slower effect, where the attempt of the in-
dividuals to evacuate faster results in a slower evacuation
for the whole crowd, is a typical phenomenon observed
in evacuation situations. In the CA, if an agent would
be able to move towards the exit without getting into
conflicts, ordered behavior would make the agent evac-
uate fastest. However, in [21], it was shown that if µis
sufficiently large, a crowd of ordered agents to evacuate
slowest, and a crowd of cooperative agents fastest.
The reason is that ordered agents often try to move si-
multaneously to the same cells, which causes conflicts
that slow down the evacuation. In the cooperative
regime, all the agents move to exit using cells of higher
crowd flow. Still, there will not be as much conflicts as
in the ordered regime. If too many agents get into con-
flicts in a cell of higher crowd flow, it ceases to be a cell
of higher crowd flow and is a less desirable movement
direction. Thus, conflict situations are not an unwanted
result of the parallel update scheme, but an important
feature to be able to describe the dynamics of an evacu-
ation properly.
A dense crowd trying to move through a bottleneck can
cause a clogging. A typical situation where a clogging is
formed is when many people try to leave a room at the
same time. The crowd will self-organize in a half-circle
like formation in front of the bottleneck, this slows down
the movement of the crowd or completely hinders it [5].
In CA, a half-circle like formation in front of a bottle-
neck builds up quite fast, without it actually having to
slow down the evacuation. Thus, we can’t speak of a clog-
ging until the density of the crowd is such that more than
about every third cell in the half-circle like formation is
occupied. Then, the number of conflicts drastically in-
creases. An example of a clogging at an exit in the CA
model is shown in Fig. 4. In the figure, the gray squares
represent agents moving towards the exit.
FIG. 4. A clogging formed at an exit. The gray squares represent
agents moving towards the exit.
III. SPATIAL EVACUATION GAME
Next, we present the spatial evacuation game defined
in [18]. In the game, naagents, indexed by i, i I=
{1, ..., na}, are in an evacuation situation. Each agent
has an estimated evacuation time Ti, which depends on
the number λiof agents closer to the exit than him, and
on the capacity of exit β.Tiis defined as
Ti=λi
β.(2)
Each agent has a cost function that describes the risk of
not being able to evacuate before the conditions become
intolerable. The cost function depends on Ti.
5
The game is played for the remaining time to evacuate.
The agents interact with their nearest neighbors. Each
agent has 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 impa-
tient behaviors of the agents, respectively. Let us denote
the average of the evacuation times of agents iand j,
Tij = (Ti+Tj)/2. When agents iand jinteract with
each other, we assume that the agents expect the follow-
ing outcomes:
1. In an impatient vs. patient agent contest, an impa-
tient agent ican overtake his patient neighbor j. This
reduces agent i’s evacuation time by Tand increases
j’s evacuation time by the same amount. The cost of i
is reduced by u(Tij) and increased for jby the same
amount. Here
u(Tij ) = u(Tij )u(Tij − △T)u(Tij )T. (3)
2. In a patient vs. patient agent contest, the patient
agents do not compete with each other, they keep their
positions 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.
The above mentioned conflict, should not be confused
with a conflict situation in the CA, which is a mere con-
sequence of the parallel update scheme [21].
From the aforementioned assumptions, a 2 ×2 game
matrix is constructed in Table I.
TABLE I. The game matrix for the spatial evacuation game.
Impatient Patient
Impatient C, C −△u(Tij ),u(Tij )
Patient u(Tij),−△u(Tij ) 0,0 .
In the game matrix in Table I, 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. Now, each
number in a game matrix can be multiplied by a number
>0 without affecting the equilibria of the game. So,
divide every number in our matrix by u(Tij ).
Then the game matrix only depends on the parame-
ter C/u(Tij ). When 0 < C/u(Tij )1, the game
played is PD, and the only Nash equilibrium is (Impa-
tient, Impatient). If C/u(Tij )>1, the game played
is HD, and there are two pure strategy Nash equilibria
(Impatient, Patient) and (Patient, Impatient). There is
also a mixed strategy equilibrium, where the strategy
Impatient is played with probability u(Tij)/C , and the
strategy Patient with probability 1 − △u(Tij )/C. For
general comments on 2 player 2 ×2 games of this form
see the appendix in [18].
It should be emphasized that the outcomes of the
above game are what the agents think will happen in dif-
ferent encounters. The agents’ decision making is based
on these expectations. In Sec. IV, we couple the sim-
ple decision making model with the CA model presented
in Sec. II. As the crowd is a large complex system, we
are not able to capture the full complexity of the system
with the simplified assumptions behind the game and CA
model. Hence, when coupling the game strategies with
the CA model, the actual outcome of an interaction is
not necessarily exactly what the agents expected when
selecting their strategies.
A. The cost function
The cost function depends on the parameter TASE T ,
available safe egress time, which is a term extensively
used in the literature [29]. It describes the time, in which
the conditions in a building in fire become intolerable.
Additionally, the cost function depends on a parameter
T0. It describes how short the time difference between
TASET and Tij has to be before agents iand jstart
playing the game with each other. If TAS ET Tij > T0,
or, equivalently Tij < TASET T0, agents iand jdo not
play the game with each other. In order to make cal-
culations with the model, some additional assumptions
about the cost function are made.
1. For Tij < TASET T0the game is not played. Then
u(Tij )0.
2. For Tij TASET T0the game is played. The cost
function starts to increase quadratically.
3. u(TASET ) = C.
A quadratic cost function that fulfills these conditions
is
u(Tij ) =
0,if Tij < TASET T0,
C
2T0
(Tij TASET +T0)2,if Tij TASET T0.
(4)
The derivative of the cost function is
u(Tij ) =
0,if Tij < TASET T0,
C
T0
(Tij TASET +T0),if Tij TASET T0.
(5)
6
Two different cost functions meeting these assumptions
are depicted in Fig. 5. The function illustrated by the
gray dashed curve starts to grow immediately at Tij = 0,
because for it TASE T T0= 0. The function illustrated
by the black curve, on the other hand, starts to grow
when Tij = 40, because for it TASET T0= 40.
FIG. 5. Two different cost functions. The gray dashed curve has
the parameter values: TASET = 60, T0= 60, C = 3. The black
curve has the parameter values: TASE T = 60, T0= 20, C = 3.
The cost of overtaking can be expressed as
u(Tij )u(Tij )T=C
T0
(Tij TASET +T0)T. (6)
We will suppose that T= 1s. Then the parameter
C/(Tij ) appearing in the game matrix is
C
u(Tij )T0
Tij TASET +T0
.(7)
Note that whether the game played is PD or HD, depends
only on the value of T0/(Tij TASET +T0). Thus the
game only depends on the estimated evacuation time Tij,
since T0and TASET are constants. When Tij increases,
the game turns from HD to PD.
B. Spatial setting and update of strategies
We assume that the agents are located in a discrete
square grid. At this point, we do not assume the agents
to move. Let Nibe the set of agents in the Moore neigh-
borhood of agent iI, i.e., the agents surrounding agent
iimmediately in the orthogonal and diagonal directions.
Note that when we couple the game model to the CA, the
agents in the Moore neighborhood of agent iwill change
as agent imoves in the square grid.
We assume that in period t, t 1, agent iobserves the
strategies of its neighbors, and updates its strategy based
on its best-response function BRi. The best-response
strategy s(t)
iof agent iin period tis defined by
s(t)
i=BRi(s(t1)
i;Ti, Ti)
= arg min
s
iS
jNi
vi(s
i, s(t1)
j;Tij ).(8)
Here, the function vi(s
i, s(t1)
j;Tij ) gives the cost de-
fined by the game matrix to agent i, when he plays strat-
egy s
i, and agent jhas played strategy s(t1)
jon period
(t1). The strategy set S={Patient, Impatient}. The
notation s(t1)
iis used to denote the strategies of all other
agents than agent iat period t1, and Tiincludes the
estimated evacuation times of these agents. The initial
configuration of strategies is s(0)
i=s0
i,iI.
The best-response strategy (8), is a strategy that min-
imizes the total cost of an agent against the strategies
played by the neighboring agents in the previous period.
Thus, the agents are myopic in the sense that they choose
their strategies based on the previous period play of their
opponents, not considering the play at farther history, or
of future periods.
For numerical simulations it is convenient to assume
that in each period only one agent updates its strategy.
To obtain convergence we further assume that each agent
in the crowd Iupdates its strategy several times. Techni-
cally, a suitable iteration scheme is shuffle update scheme,
where after every naperiods each agent has updated its
strategy exactly once, in random order [30].
C. Equilibrium configuration in a square grid
Numerical simulations show that regardless of the
number of agents, the spatial equilibrium is found, when
all the agents have updated their strategies less than ten
times, or so, with the shuffle update scheme.
As stated earlier, the game turns from a HD to PD as
we move farther away from the exit. It can be shown,
that in the region, where the agents play HD, we observe
subregions with different proportions of impatient agents.
Let the set of impatient agents in the Moore neighbor-
hood of agent ibe NI mp
i. Moreover, let the number of
agents in the Moore neighborhood of agent ibe |Ni|and
the number of impatient agents |NImp
i|, then the number
of patient agents is |Ni|−|NImp
i|.
Recall the costs associated with interactions of differ-
ent strategists. Now, agent ishould play Impatient, if
the cost of playing Impatient against his neighbors is less
than or equal to the cost of playing Patient:
jNImp
i
T0
Tij TASET +T0
+(|Ni|−|NImp
i|)(1) ≤ |NImp
i|.
(9)
To make the analysis simpler, let us approximate Ti=
Tj, j NI mp
i. Then the above inequality can be written
7
|NImp
i|
|Ni|TiTASET +T0
T0
.(10)
Here, |NImp
i|/|Ni|is the proportion of impatient agents
in the Moore neighborhood of agent i. Thus, Eq. (10)
gives an upper limit for the proportion of impatient
agents in the Moore neighborhood of an impatient agent
iat a given distance from the exit. If this limit is ex-
ceeded, agent ishould switch strategy to Patient. We
can see that this limit increases as we move farther away
from the exit.
In an equilibrium configuration, all agents choose their
strategy based on Eq. (10). Thus, the proportion of im-
patient agents grows as we move farther away from the
exit. We will actually notice eight subregions, with dif-
ferent proportions of impatient agents. This is due to the
game being played in the Moore neighborhood, which im-
plies exactly eight possible proportions. This can be seen
in Fig. 6, which illustrates an equilibrium configuration
of the game. Along the semicircles A, B, C , the number
of impatient agents in the Moore neighborhood of agents
are at most 1,2,4, respectively.
FIG. 6. An equilibrium configuration for 3180 agents with param-
eter values TASET = 2200 and T0= 2200. Black squares represent
impatient agents and gray squares patient agents.
More such simulations, with different patient and im-
patient agent proportions, can be found in [18]. Similar
results were obtained in [31] for a square lattice of peri-
odic boundary conditions. There, contrary to our model,
the parameters in the HD game did not depend on the
agents’ locations.
IV. CELLULAR AUTOMATON EVACUATION
MODEL COUPLED WITH A SPATIAL GAME
Recall Sec. II; if kS F is high, and kDF low, an indi-
vidual agent is able to evacuate fast, but the evacuation
time of the whole crowd using these coupling parameters
is slow. On the other hand, if kDF kS F 1, an in-
dividual agent evacuates slowly, but the evacuation time
for the whole crowd is fast [21].
Next, we are going to couple the CA with the spatial
game from Sec. III. In the resulting model, an agent’s
coupling parameters kSF and kDF are a realization of
the agent’s strategy choice in the spatial game. As far as
we know, there are no earlier studies where agents can
have different values for kSF and kDF , or that they would
change during the simulation.
As discussed in Sec. II, in the CA, the agents move in a
discrete grid in discrete time steps. A time step in the CA
is equal to 0.3 s. At the beginning of every time step, in
the CA, before the agents move, the averaged estimated
evacuation times Tij for the neighboring agents i, j I
are calculated. After that, before moving in the grid, the
agents observe the strategies of their neighboring agents
and react by updating to their best-response strategy.
The agents’ best-response strategies are updated until
an equilibrium is reached. Thus, the crowd is in an equi-
librium configuration always when moving. Note that
the equilibrium configuration may slowly change in form
over time, since the game parameter Tij changes for the
agents.
We let the agent’s strategy choice alter its behavior
in the CA: playing Impatient results in faster movement
for the agent, than playing Patient. Thus, for an agent
playing Impatient, the coupling parameters are set to
kSF = 10, kDF = 1, and for an agent playing Patient
kSF = 1, kDF = 1.
So, in our model, kSF and kDF change for agent i
based on the game parameter Tij and neighboring agents’
strategies. Again, it should be emphasized that Tij de-
scribes what agent iexpects its averaged estimated evac-
uation time with neighbor jto be, and has nothing to do
with the elapsed time in the CA.
After kSF and kDF have been updated for the agents,
the agents’ positions are updated in parallel according to
the transition probabilities determined by Eq. (1). In a
case of a conflict, one of the agents is allowed to move
with probability 1 µ,
For technical reasons, the agents’ positions are updated
with parallel update scheme. If the positions were not
updated in parallel, no conflicts would occur. These con-
flicts are a crucial feature to achieve clogging and faster-
is-slower effect with our model. On the other hand, the
shuffle update used for updating agents’ strategies is not
only technically suitable, but is a fair description of how
agents really update their strategies. Since, in many real
social systems, agents do not update their strategies si-
multaneously. Alternatively, the agents could be set to
update their strategies according to a Poisson process.
A. Model description
Next, a recap of the model is given in a form of a step-
by-step description. In the beginning of the simulation,
the agents are located randomly in the room, and are all
considered patient.
Before the simulation, we set a specific value for the
8
available safe egress time TASET , however, TASET can
change with time. The linear change in the value of
TASET in 1sof the simulation is described by the param-
eter TASET . So, TASE T describes the rate at which
the evacuation conditions get worse or improve.
Step 1.: At the beginning of each time step, the game
parameters Tij = (Ti+Tj)/2, i ̸=j, i, j Iare
calculated according to Eq. (2). If Tij < TASET
T0,jNi, agent idoes not play the game and
behaves patiently.
Step 2.: The agents’ strategies are updated with the
shuffle 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. (8).
Step 3.: The agents’ behavior in the CA model is set to
correspond their strategy choice. This is done by
altering the agents’ coupling parameters as follows:
(a) For an agent playing Impatient, the coupling
parameters are set to kDF = 1.0 and kSF =
10.0.
(b) For an agent playing Patient, the coupling pa-
rameters are set to kDF = 1.0 and kSF = 1.0.
Step 4.: The agents’ positions are updated in parallel
according to Eq. (1). In a case of a conflict, one
of the agents is allowed to move with probability
1µ.
Step 5.: The available safe egress time is updated:
TASET =TASET +TAS ET ·0.3.
Step 6.: Go to Step 1. The procedure is repeated until
all agents have evacuated the room.
V. EVACUATION SIMULATIONS
We have presented an evacuation model, where the
agents’ behavior in the CA is a result of the spatial game
they play. Preliminary results with the model were ob-
tained in [32]. In the following, we will extend these
simulations for a few typical evacuation situations.
Additionally, the performance of the strategies in an
evacuation is investigated. We show that playing the
strategy Impatient results in fastest evacuation for an in-
dividual, but if the whole crowd adopts it, it will result
in slowest possible evacuation for the crowd. This is the
well-known faster-is-slower effect, which was also found
in the original formulation of the spatial game [18]. In
the CA model discussed in Sec. II, the phenomenon is
achieved by pre-simulation setting specific model param-
eters. It is remarkable that here we are able to obtain
the phenomenon as a result of the simple game the agents
play.
A sufficiently large friction parameter µ= 0.6 is used
in all the simulations. Later in this section, we analyze
the influence of µand kSF and kDF on the faster-is-slower
effect.
A. Simulation of typical situations
When the remaining time to evacuate is scarce, people
start to behave impatiently. This especially holds for
people located farther away from the exit. It is striking
that our model describes this feature of human beings.
It is clearly seen in Fig. 7; recall also the explanation in
Sec. II I C.
In Fig. 7, there are three snapshots from different
stages of an evacuation from a large room. The black
squares represent impatient agents and grey squares pa-
tient. As can be seen, the agents organize in a half-circle
like formation in front of the exit already in the early
stages of the evacuation. Even though the agents are
moving, the whole crowd is in a spatial equilibrium in
every snapshot. This is a consequence of fast updating
of agents’ best-response strategies compared to the time
step of their move. This can also be seen in Fig. 8 and
9.
(a) (b)
(c)
FIG. 7. Different stages of an evacuation, with 1000 agents, from
a large room. Here T0=TASET = 100 and TASET = 0. Black
squares represent impatient agents and gray squares patient agents.
(a) Early stages. (b) Middle stages. (c) Late stages.
In Fig. 8 the agents evacuate from the same room as
9
in Fig. 7. In comparison to Fig. 7, here TASE T =2,
which means that the situation gets worse over time.
This is typical for a fire evacuation. Due to the wors-
ening conditions, more agents feel pressured to evacuate,
resulting in a higher proportion of impatient agents in
the crowd.
Recall that impatient agents move straight towards the
exit, and thus try to occupy same cells simultaneously,
which causes conflicts in the CA. These conflicts slow
down the evacuation. Patient agents, on the other hand,
move to regions of higher local crowd flow, and thus have
a tendency to avoid conflicts. Thus, the evacuation is
slower in Fig. 8 than in Fig. 7. This can be seen by
comparing the amount of agents in the late stages of the
evacuation in both figures. Later in this section, a more
in depth analysis of the subject is given.
(a) (b)
(c)
FIG. 8. Different stages of an evacuation, with 1000 agents, from
a large room. Here T0=TASE T = 100 and TASE T =2. Black
squares represent impatient agents and gray squares patient agents.
(a) Early stages. (b) Middle stages. (c) Late stages.
In Fig. 9, there is a crowd moving from a room to
another through a narrow hallway. The exit is located
in the second room. In the beginning, the agents are
located in the first room. Already in the early stages
of the evacuation, some of the agents have reached the
exit, and the hallway is filled with agents. Because, the
hallway is narrow conflicts are likely to happen. After
the early stages, the amount of conflicts in the hallway
starts to increase. Thus, the flow in the hallway will start
to decrease. This is because the proportion of impatient
agents in the hallway is increasing.
(a)
(b)
(c)
FIG. 9. Different stages of an evacuation, with 800 agents, through
a hallway. Here T0=TASET = 150 and TAS ET = 0. Black
squares represent impatient agents and gray squares patient agents.
(a) Beginning of evacuation. (b) Early stages. (c) Middle stages.
B. Simulations with fixed strategies
To gain knowledge about the performance of differ-
ent strategists, we will simulate an evacuation, where the
strategies of the agents are held fixed, i.e., the agents do
not update their strategies.
In the simulation, 50 impatient and 50 patient agents
are randomly distributed in a 7.2 m ×7.2m -room, with a
0.8 m -wide exit. The agents have 60 seconds to evacuate,
and the simulation is run 40 times with different random
initial locations. The results can be seen in Fig. 10. It
shows the average number of patient (gray curve) and
impatient (black curve) agents inside the room over time,
and the difference in these numbers (dashed curve).
Point B and C in Fig. 10 represent the points in time
when on the average all the impatient and patient agents,
respectively, have evacuated the room. Because B occurs
earlier than C, impatient agents are on the average able
to evacuate faster than patient agents. Thus, the as-
sumption behind the spatial game that impatient agents
10
can overtake patient agents seems to approximately hold
also for our model.
FIG. 10. The average number of patient and impatient agents
inside the room over time, and the difference in these numbers.
Impatient agents are able to evacuate faster, because
the majority of them are able to rush to the front of the
exit before a clogging is formed. Naturally, the agents in
front of the exit are also the first ones to evacuate. After
they have evacuated, we are left with a crowd in front
of the exit, where the proportion of impatient agents is
small. The remaining few impatient agents are able to
overtake patient agents, but they now have a smaller ad-
vantage than before, because the crowd is so dense that
there is not much room to move. In Fig. 10, this can be
seen by looking at the difference in the average numbers
of patient and impatient agents in the room. The dif-
ference increases until point A, after which it starts to
decrease. Point A represents the time when the group of
impatient agents that has first arrived in front of the exit
has managed to evacuate.
So, on average an individual impatient agent is able to
evacuate faster than a patient agent. Then what effect
has the proportion of impatient agents on the evacuation
efficiency of the whole crowd? Next, we study how the
proportion of impatient agents affects the average egress
flow of the crowd.
In the simulation, 200 agents are set in a half-circle in
front of the exit, and update their strategies until equi-
librium is reached. Immediately afterwards, their strate-
gies are fixed, the exit is opened, and the agents start
to evacuate. The simulation is performed 100 times for
11 different values of TASET , to get results for 11 differ-
ent proportions of impatient agents. The results of the
simulations can be seen in Fig. 11.
In Fig. 11 it is clearly seen that the more agents be-
have impatiently, the smaller the egress flow is. This is
because, the more there are impatient agents, the more
agents move straight towards to the exit, resulting in an
increased amount of conflict situations in the CA. These
conflicts decrease the egress flow.
We have shown that even though an impatient agent
FIG. 11. Average egress flow of a crowd, with 200 agents, for
different proportions of impatient agents. In the simulations, 11
different values of TASET were used. TAS ET = 0 in each simu-
lation.
evacuates faster than a patient agent, the whole crowd
evacuates the slower the larger the proportion of impa-
tient agents in the crowd is. In other words, the at-
tempt of an individual agent to evacuate faster results in
a slower evacuation for the whole crowd. This astounding
result shows that in our model the faster-is-slower effect
is a result of the 2-strategy game the agents play in their
nearest surrounding.
C. Simulations with adaptive strategies
Next, we let the agents update their strategies through-
out the simulation. We study how the number of evac-
uated agents and proportion of impatient agents vary in
different scenarios. The scenarios are listed in Table II.
TABLE II. The initial values of TASET ,T0and the linear change
of TASET in 1 s of the simulation, i.e., TAS ET , for Scenarios 1-5.
Scenario TASET T0TAS ET
Scenario 1 200 100 -2
Scenario 2 100 100 -2
Scenario 3 100 100 -0.5
Scenario 4 10 100 +4
Scenario 5 10 100 +1
Each scenario in Table II is simulated 50 times. During
the simulations, 200 agents are set to evacuate from a
room. In Fig. 12 the proportion of impatient agents over
time for Scenarios 1-5 are shown.
In Scenarios 1-3 of Table II, the value of TASET de-
creases over time. It can be seen that this results in
more agents behaving impatiently over time. On the
other hand, in Scenarios 4-5 of Table II, the value of
TASET increases over time, which improves the evacua-
tion conditions for the agents. Thus, over time less agents
behave impatiently. At a point in time, they even stop
11
playing the game, and behave all patiently, because their
estimated evacuation times are larger than TASET T0.
FIG. 12. Proportion of impatient agents over time for Scenarios
1-5.
Also, the cumulative exit counts were monitored in the
simulations, the results are shown in Fig. 13.
FIG. 13. Cumulative exit count over time for Scenarios 1-5.
It can be seen that the better the conditions are, the
more agents are able to evacuate. The cumulative exit
count curve grows almost linearly over time in all scenar-
ios. Actually, for Scenarios 1-3, there is a slight decrease
in the slope of the curve over time, because as the propor-
tion of impatient agents grow, the egress flow decreases.
And, in Scenarios 4-5, there is a slight increase in the
slope of the curve over time. This is because, as the pro-
portion of patient agents grow, the egress flow increases.
The same simulations have been run with FDS+Evac
in [18]. There, the cumulative exit count curves differ
quantitatively. In [18] the slopes of the curves increase
or decrease, depending on TASE T , more over time.
As has been mentioned before, in [18], impatient agents
actually push the crowd in front of them. The driving
force of the impatient agents in the back of the crowd,
and the frictional interactions between the agents, result
in arch-like blocks in front of the exit, which form and
break down [5].
In our model, an increase in the proportion of impa-
tient agents increases the amount of conflicts. These con-
flicts slow down the evacuation, but not as drastically as
the arch-like blocks in FDS+Evac. The reason is that an
increase in impatient agents only increases conflicts in the
locations where the impatient agents are, not necessary
in front of the exit. Whereas in [18], an increase in im-
patient agents adds up to the driving force of the back of
the crowd. Thus, an increase in impatient agents in [18]
has a more severe effect on the egress flow in FDS+Evac,
and because of that the slope of the cumulative exit count
curve will decrease more.
Additionally, in FDS+Evac, the agents can make a
use of their body ellipse, and thus two persons can move
through an exit simultaneously [8]. This makes an evacu-
ation for a crowd with a high proportion of patient agents
more efficient, and thus the slope of the cumulative exit
count curve will increase more in FDS+Evac.
D. Influence of µand different strategists’ kSF and
kDF on the faster-is-slower effect
So far, we have only run simulations with the param-
eter values: µ= 0.6, and kSF = 10, kDF = 1 for impa-
tient agents and kSF = 1, kDF = 1 for patient agents.
In Sec. V B, it was seen that using these parameter val-
ues the faster-is-slower effect is a result of the game the
agents play. Next, we investigate whether there are other
suitable parameter values.
First, we investigate the influence of µon the faster-
is-slower effect. We have run the same simulations as in
Fig. 10 with different values of µ. For each value of µ, we
have calculated the average evacuation time for a patient
and an impatient agent, ¯
tP at and ¯
tImp , respectively. The
difference ¯
tP at ¯
tImp for given µis shown in Table III.
TABLE III. Difference in average evacuation time for a patient
and and impatient agent, i.e., ¯
tP at ¯
tImp for given µ.
µImpatient (kSF /kDF ) Patient (kSF /kDF )¯
tP at ¯
tImp
0 10/1 1/1 11.8918
0.3 10/1 1/1 12.1972
0.6 10/1 1/1 12.5511
0.9 10/1 1/1 13.7004
For all values of µin Table III, ¯
tP at ¯
tImp is positive,
i.e., impatient agents are on average able to overtake pa-
12
tient agents and evacuate faster.
It seems that the advantage impatient agents have over
patient agents grows as µincreases, because ¯
tP at ¯
tImp
is larger for larger values of µ. However, in Table III, it
is not shown that also ¯
tImp is larger for larger values of
µ. So, the advantage impatient agents have over patient
agents, in relation to their own evacuation time, in fact,
decreases as µincreases.
In addition to impatient agents being able to overtake
patient agents, for the faster-is-slower effect to take place,
the egress flow needs to decrease, as the proportion of
impatient agents increases. To study how the egress flow
depends on the proportion of impatient agents, we have
run the same simulations as in Fig. 11 for different values
of µ. The results are shown in Fig. 14. Notice that the
vertical axis scale is larger in Fig. 14 than in Fig. 11.
FIG. 14. Average egress flow of a crowd for different proportions
of impatient agents. The parameter µis altered. In all cases kSF =
10, kDF = 1 for impatient agents and kSF = 1, kDF = 1 for patient
agents.
For µ= 0.6 and µ= 0.9, the faster-is-slower effect
takes place, because the average egress flow decreases as
the proportion of impatient agents increases. Whereas,
for µ= 0 and µ= 0.3, the attempt of agents to move
faster results in a faster evacuation for the whole crowd.
Recall Fig. 2; kSF = 10, kDF = 1, used here for impa-
tient agents, is in the ordered regime, and kSF =kDF =
1, used here for patient agents, in the cooperative regime.
If µis sufficiently large, a crowd of ordered agents evac-
uates slower than a crowd of cooperative agents [21]. In-
versely, if µis low enough, a crowd of ordered agents will
evacuate faster than a crowd of cooperative agents.
Thus, there should exist a limit value for µ, at which
the faster-is-faster effect turns into faster-is-slower effect.
Based on Fig. 14, it is between 0.3 and 0.6. This value
should vary with crowd density, and kSF and kDF for
impatient and patient agents.
In Fig. 14, it can also be seen that for larger values of
µ, the faster-is-slower effect becomes more pronounced.
However, if µ1, the egress flow should approach 0, for
all proportions of impatient agents, because the proba-
bility for an agent to move in a conflict would approach
0.
Next, we investigate the effect impatient agents’ kSF
and kDF have on the faster-is-slower effect. We have run
the same simulations as in Fig. 10 with different values
of impatient agents’ kSF and kDF . The difference ¯
tP at
¯
tImp for given parameter values is shown in Table IV.
TABLE IV. Difference in average evacuation time for a patient and
an impatient agent, i.e., ¯
tP at ¯
tImp , for given parameter values.
µImpatient (kSF /kDF ) Patient (kSF /kDF )¯
tP at ¯
tImp
0.6 10/4 1/1 11.8356
0.6 10/0 1/1 13.5004
0.6 3/1 1/1 11.0358
Again, because ¯
tP at ¯
tImp is positive in all the cases,
impatient agents are able on average to overtake patient
agents and evacuate faster.
Next, we investigate how the egress flow depends on
the proportion of impatient agents for different values of
impatient agents’ kSF and kDF . Same simulations were
run as in Fig. 11 for different values of impatient agents’
kSF and kDF . The results are shown in Fig. 15.
FIG. 15. Average egress flow of a crowd for different proportions
of impatient agents. kSF and kDF are altered for impatient agents.
In all cases, µ= 0.6, and kSF = 1, kDF = 1 for patient agents.
For all the parameter combinations in Fig. 15, an in-
crease in the proportion of impatient agents decreases
the egress flow, i.e., the faster-is-slower effect takes place.
From Fig. 15, we see that kSF can be set as low as 3 for
impatient agents. On the other hand, there is no upper
limit for kSF for impatient agents, because as kSF → ∞,
the agent walks as straight forward to the exit as possi-
ble, which makes the agent evacuate as fast as possible
and cause conflicts as much as possible [20].
If kSF is fixed for an agent, increasing kDF makes the
agent evacuate slower, and cause less conflicts [20, 21].
Thus, there should be an upper limit for the impatient
agents’ kDF for a fixed kSF . In Fig. 15, it can be seen
that if kSF = 10, kDF can be set at least up to 4.
If kSF is fixed and large enough for an agent, decreas-
ing kDF makes the agent evacuate faster, and cause more
13
conflicts [20]. Thus, the lower limit for kDF for an impa-
tient agent is 0. In Fig. 15, it can be seen that kSF = 10,
kDF = 0 is suitable for an impatient agent.
Finally, we investigate the influence of patient agents’
kSF and kDF on the faster-is-slower effect. We have run
the same simulations as in Fig. 10 with different values of
patient agents’ kSF and kDF . The difference ¯
tP at ¯
tImp
for given parameter values is shown in Table V.
TABLE V. Difference in average evacuation time for a patient and
an impatient agent, i.e., ¯
tP at ¯
tImp , for given parameter values.
µImpatient (kSF /kDF ) Patient (kSF /kDF )¯
tP at ¯
tImp
0.6 10/1 1/0.5 11.1327
0.6 10/1 2/2 9.3429
0.6 10/1 10/10 6.9855
Again, because ¯
tP at ¯
tImp is positive for all param-
eter combinations, impatient agents are on average able
to overtake patient agents and evacuate faster. We still
need to investigate if the egress flow decreases, as the pro-
portion of impatient agents increases, for different values
of patient agents’ kSF and kDF . In Fig. 16, there are
shown the results of the same simulations as in Fig. 11
for the parameter combinations in Table V.
FIG. 16. Average egress flow of a crowd for different proportions
of impatient agents. kS F and kDF are altered for patient agents.
In all cases, µ= 0.6 and kSF = 10, kDF = 1 for impatient agents.
For all the parameter combinations, an increase in the
proportion of impatient agents decreases the egress flow,
i.e., the faster-is-slower effect takes place. Naturally,
kSF = 1, kDF = 0.5 is suitable for patient agents, since it
is in the cooperative regime of Fig. 2. However, it is sur-
prising to see that kSF =kDF = 2 and kS F =kDF = 10
are suitable for patient agents.
Also, we have run simulations with kDF > kSF for
patient agents, which results are not presented here,
since kDF > kSF resulted in patient agents performing a
random-walk in the room. Thus, for patient agents, kDF
should not be so high that the dynamic floor field DF
dominates the agents movement. It seems that there is an
upper limit for patient agent’s kDF which is kDF kSF ,
at least when kSF 10.
We omitted the investigation of large and really small
values of kSF and kDF for both impatient and patient
agents. The larger the coupling parameters are, the more
amplified the differences in the values of the static SF
and dynamic floor field DF become. Thus, the agent
will either follow DF or SF , which means that the be-
havior either resembles a random-walk or is suitable for
impatient agents. On the other hand, if kSF , kDF 0
the agents movement resembles a random-walk [20].
We can conclude that our model is not defined merely
for a narrow range of parameters, but there is room to
alter both µ, and kS F and kDF for patient and impa-
tient agents. For a patient agent, kSF and kDF needs to
be chosen in such a way that a patient agent evacuates
slower than an impatient agent, but causes less conflicts
than an impatient agent. Thus, the coupling parameters
for patient and impatient agents can not be chosen in-
dependently. It seems to hold approximately, that the
coupling to DF in relation to the coupling to S F , i.e.,
kDF /kSF , needs to be larger for patient than impatient
agents. Coincidentally, in [20] the ratio kDF /kSF was
compared to the panic parameter introduced by Helbing
et al. in [5].
VI. CONCLUSIONS
The aim in this paper was to develop a computationally
light evacuation simulation model, which quantitatively
answers the questions posed in [18], and in earlier studies,
namely, why, when and how collective effects of crowd
dynamics occur.
For simulation of the movement of the crowd, we
choose the computationally very light CA [21]. With the
CA, different crowd behaviors can be produced by alter-
ing the coupling parameters kSF and kDF for the agents.
In [21], the coupling parameters are adjusted before sim-
ulations. Thus, there all the agents behave in the same
way, at all times and everywhere.
To equip the agents with decision-making abilities, we
coupled the spatial evacuation game [18] to the CA. In
the game, an agent observes the strategies of its neigh-
boring agents and reacts by choosing its best-response
strategy. The agents have two strategies to choose from:
Patient and Impatient. An agent’s best-response strategy
depends, in addition to the neighboring agents’ strate-
gies, on the agent’s estimated evacuation time Tij and
available safe egress time TASET .
The choice of strategy determines the agent’s coupling
parameters kSF and kDF . To our knowledge, there are
no previous studies where agents in the CA have differ-
ent values for the coupling parameters, or the coupling
parameters would change for the agents during the sim-
ulation.
The agents update their best-response strategies so
that they are in a one shot spatial equilibrium configura-
14
tion always when moving. Because the game parameter
Tij changes for the agents when they move, the equilib-
rium also changes as the crowd moves. In equilibrium,
the proportion of impatient agents grows farther away
from the exit. This reflects the fact that human beings
farther away from the exit feel more pressured to evac-
uate and tend to behave more impatiently. Since the
assumptions behind the model are fairly simple, and the
best-response dynamics was found out to converge quite
fast, it is plausible that similar equilibrium patterns could
exist in real-life situations.
In Sec. V, we showed that impatient agents are able
to reach the exit faster than patient agents. However,
the more agents were impatient, the slower the evacua-
tion was for the whole crowd. This is because, the more
there are impatient agents, the more agents move straight
towards the exit, resulting in increased amount of con-
flict situations in the CA. These conflicts slow down the
evacuation. This remarkable result shows that when we
couple the spatial game to the CA evacuation model, the
faster-is-slower effect is found as a result of the Hawk-
Dove game the agents play in their nearest surrounding.
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... Another global pushing mechanism is proposed in [20]. Furthermore, game theory is combined with CA models in some studies to better reproduce the movement of pedestrians [21,22]. Prolonged clogs and stable clogs can also be observed in the social force models for pedestrian dynamics by increasing the desired velocity of the agents [23]. ...
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This article studies clogging phenomena using a velocity-based model for pedestrian dynamics. First, a method to identify prolonged clogs in simulations was introduced. Then bottleneck simulations were implemented with different initial and boundary conditions. The number of prolonged clogs was analyzed to investigate the decisive factors causing this phenomenon. Moreover, the time lapse between two consecutive agents passing the exit, and the trajectories of agents were analyzed. The influence of three types of factors was studied: parameters of the spatial boundaries, algorithmic factors related to the implementation of the model, and the movement model. Parameters of the spatial boundaries include the width and position of the bottleneck exit. Algorithmic factors are the update methods and the size of the time step. Model parameters cover several parameters describing the level of motivation, the strength and range of impact among agents, and the shape of agents. The results show that the occurrence of prolonged clogs is closely linked to parameters of the spatial boundaries and the movement model but has virtually no correlation with algorithmic factors.
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The game-theoretic approach is an essential tool in the research of conflicts of human behaviors. The aim of this study is to research crowd dynamic conflicts during evacuation processes. By combining a conflict game with a Cellular Automata model, the following factors such as rationality, herding effect and conflict cost are taken into the research on frequency of each strategy of evacuees, and evacuation time. Results from Monte Carlo simulations show that (i) in an emergency condition, rationality leads to “vying” behaviors and inhibited “polite” behavior; (ii) high herding causes a crowd of high rationality (especially in normal circumstances) to become more “vying” in behavior; (iii) the high-rationality crowd is shown to spend more evacuation time than a low-rationality crowd in emergency situations. This study provides a new perspective to understand conflicts in evacuation processes as well as the rationality of evacuees.
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Game theory is introduced to simulate the complicated interaction relations among the conflicting pedestrians in a pedestrian flow system, which is defined on a square lattice with the parallel update rule. Modified on the traditional lattice gas model, each pedestrian can move to not only an empty site, but also an occupied site. It is found that each individual chooses its neighbor randomly and occupies the site with the probability W(x→y)=1/1+exp[-(P_{x}-U_{x})/κ], where P_{x} is the x's payoff representing his personal energy, and U_{x} is the average payoff of its neighborhood indicating the potential well energy if he stays. Two types of pedestrians are considered, and they interact with their neighbors following the payoff matrix of snowdrift game theory. The cost-to-benefit ratio r=c/(2b-c) (where b is the perfect payoff and c is the labor cost) represents the fear index of the pedestrians in this model. It is found that there exists a moderate value of r leading to the shortest escape time, and the situation for large values of r is better than that for small ones in general. In addition, the pedestrian flow system always arrives at a consistent state in which the two types of walkers have the same number and evolve by the same law irrespectively of the parameters, which can be interpreted as the self-organization effect of pedestrian flow. It is also proven that the time point of the onset of the steady state is unrelated to the scale of the pedestrians and the square lattice. Meanwhile, the system exhibits different dynamics before reaching the consistent state: the number of the two types of walkers oscillates when P_{C}>0.5 (i.e., probability to change the present strategy), while no oscillation happens for P_{C}≤0.5. Finally, it is shown that a smaller density of pedestrians ρ induces a shorter average escape time.