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Cellular Automaton Evacuation Model Coupled
with a Spatial Game
Anton von Schantz and Harri Ehtamo
formal publication DOI: 10.1007/978-3-319-09912-5_31
c
2014. This manuscript version is made available under the CC-BY-NC-ND 4.0
license http://creativecommons.org/licenses/by-nc-nd/4.0/
Abstract
For web-based real-time safety analyses, we need computationally
light simulation models. In this study, we develop an evacuation model,
where the agents are equipped with simple decision-making abilities.
As a starting point, a well-known cellular automaton (CA) evacuation
model is used. In a CA, the agents move in a discrete square grid ac-
cording to some transition probabilities. A recently introduced spatial
game model is coupled to this CA. In the resulting model, the strat-
egy choice of the agent determines his physical behavior in the CA.
Thus, our model offers a game-theoretical interpretation to the agents’
movement in the CA.
1 Introduction
To avoid losses, e.g., in evacuation situations, the rescuing authorities should
make timely and accurate decisions. A successful operation requires real-
time safety analysis to forecast various disasters and accidents that may take
place in events involving human crowds. Thus, safety simulations should
be computationally light enough to run in real-time, e.g., in the internet.
Recent research sites aiming at these goals are [17,18].
Our ultimate goal is to create a computationally light evacuation simula-
tion model suited for web-based real-time analyses. Our focus in this paper
is on two computational evacuation models: the cellular automaton (CA)
model [7–9] and the social-force model [10]. FDS+Evac is a validated evacu-
ation simulation software based on the social-force model [6]. In FDS+Evac,
the agents’ exit selection is modeled using optimization and game theory [2].
Computationally very light CA model is especially suitable to simulate
moving agents in traffic jams and evacuation situations. Hence, it could be
1
used to develop web-based tools to simulate these matters as well. Although,
agent movement in the CA model is rather realistic resembling granular flow,
it lacks agents’ explicit decision-making abilities. In CA the agents move
according to some transition probabilities defined by the so called static
and dynamic floor fields. The influence of the floor fields on the transition
probabilities depend on two parameters, or coupling constants, resulting in
different behaviors of the crowd.
So far, in the CA literature [11–15], game theory has been used to solve
a conflict situation, i.e., a situation where several agents try to move simul-
taneously to the same cell.
In this paper, we couple the spatial game defined in [5] to CA. In our
approach, each agent plays the Hawk-Dove game in his neighborhood leading
to two types of strategies for each agent described by two possible values of
coupling constants. In our model, the agent does not just choose his strategy
when in conflict, but optimizes it constantly to minimize his evacuation
time.
2 Cellular Automaton Model
The agents’ movement is simulated with a CA introduced by Schadschneider
et al. [9]. Next, we give brief overview of the CA model. In the model,
the agents are located in a room divided into cells, so that a single agent
occupies a single cell. At each time step of the simulation, the agent can
move to one of the unoccupied cells orthogonally next to him, i.e., in the
Moore neighborhood, where the transition probabilities associated with the
diagonal cells are set to zero.1
2.1 Movement in the CA
The transition probabilities depend on the values of the static and dynamic
floor field in the cells. The static floor field Sis based on the geometry of
the room. The values associated with the cells of Sincrease as we move
closer to the exit, and decrease as we move closer to the walls. On the other
hand, the dynamic floor field Drepresents virtual paths left by the agents.
An agent leaving a cell, causes the value of Din that cell to increase by one
unit. Over time, the virtual path decays and diffuses to surrounding cells.
The values of the fields Dand Sare weighted with two coupling constants
kD∈[0,∞) and kS∈[0,∞).
1Also called von Neumann neighborhood.
2
Now, for each agent, the transition probabilities pij , for a move to a
neighbor cell (i, j) are calculated as follows
pij =NekDDij ekSSij (1 −ξij ),(1)
where
ξij =1 for forbidden cells (walls and occupied cells)
0 else
and the normalization
N=
X
(i,j)
ekDDij ekSSij (1 −ξij )
−1
.
The agents’ desired movement directions are updated with a parallel
update scheme, i.e., the directions are updated simultaneously for all agents.
In a conflict situation, i.e., a situation where several agents try to occupy the
same cell, 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 friction parameter, illustrating the internal pressure
caused by conflicts. The impact of the friction parameter is depicted in
Figure 1.
Figure 1: The impact of friction parameter on the agents movement. With probability
µneither of the agents get to move, and with probability 1 −µthe other agent moves.
Here, E refers to the exit cell.
A cell is assumed to be 40 cm ×40 cm. The maximal possible moving
velocity for an agent, who does not end up in conflict situations, is one cell
3
per time step, i.e., 40 cm per time step. Empirically the average velocity of
a pedestrian is about 1.3 m/s. Thus, a time step in the model corresponds
to 0.3 s.
2.2 Different Crowd Behaviors
In [8], Schadschneider showed that by altering the coupling constants kS
and kDdifferent crowd behaviors can be observed. He named the different
crowd behaviors ordered, disordered and cooperative. In Figure 2, the cou-
pling constant combinations responsible for different regimes are plotted in
a schematic phase diagram.
Figure 2: Altering the coupling constants kSand kD, in the CA model, produces
different crowd behaviors.
In the ordered regime, the agents move towards the exit using the shortest
path. The regime is called ordered, because the movement of the agents is
in a sense deterministic. In the disordered regime, the agents just blindly
follow other agents’ paths, whether the path they are following is leading
to the exit or not. In this study, we are only focusing on ordered and
cooperative behavior, as disordered behavior is thought to occur mainly
in smoky conditions. Between the ordered and disordered regime is the
cooperative regime around the values kD=kS= 1. There, the agents move
towards the exit using paths of higher flow, i.e., paths where the amount of
conflict situations is small.
Consequently, for a freely moving agent, ordered behavior makes the
agent evacuate fastest. However, a sufficiently large µcauses a faster-is-
4
slower phenomenon, where a crowd of ordered agents will evacuate slowest.
The reason is that ordered agents cross paths often, which causes conflicts
that slow down the evacuation. In the cooperative regime, even though the
whole crowd moves to the paths of higher flow, there will not be as much
conflicts as in the ordered regime. If too many agents get into conflicts in
a path of higher flow, the path ceases to be a path of higher flow and the
agents change path.
3 Spatial Evacuation Game
Next, we present the spatial game defined by Heli¨ovaara et al. in [5]. It
should be noted that the spatial game and CA are two separate models. In
the game, naagents, indexed by i, i ∈I={1, ..., na}, are in an evacuation
situation, and located in a discrete square grid. Each agent has an estimated
evacuation time Ti, which depends on the number λiof agents between him
and the exit, 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 u(Ti) is
a function of Ti. The shape of the cost function depends on the parameter
TASET ,available safe egress time, which describes the time, in which the
conditions in the building become intolerable. Additionally, a parameter T0
describes the time difference between TASE T and when the agents start to
play the game.
The agents interact with other agents in their Moore neighborhood. Each
agent can choose to play either Patient or Impatient. Let us denote the
average evacuation time of agent iand j,Tij = (Ti+Tj)/2. In an impatient
vs. patient agent contest, an impatient agent ican overtake his patient
neighbor j. This reduces agent i’s evacuation time by 4Tand increases j’s
evacuation time by the same amount. The cost of iis reduced by 4u(Tij )
and increased for jby the same amount. Here
4u(Tij ) = u(Tij )−u(Tij − 4T)'u0(Tij )4T. (3)
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.
In an impatient vs. impatient agent contest, neither agent can overtake
the other, but they will face a conflict and have an equal chance of getting
5
injured. The risk of injury is described by a cost C > 0, which affects
both agents. The constant Cis called the cost of conflict. We assume
that u0(TASET ) = C. Also, we assume that u0(Tij )>0. Thus, based on
Equation 3, we have 4u(Tij )>0. Now, an illustration of a quadratic cost
function can be drawn (see Figure 3).
Figure 3: Illustration of the parameters of the cost function. The function in the figure
has the parameter values: TAS ET = 90, T0= 45, C = 3.
From the aforementioned assumptions, a 2 ×2 game matrix can be con-
structed:
Agent 2
Impatient Patient
Agent 1 Impatient C/4u(Tij ), C/4u(Tij )−1,1
Patient 1,−1 0,0 .
Here, all the elements of the more intuitive form of the game matrix have
been divided by 4u(Tij ). When a particular pair of strategies is chosen, the
costs for the two agents are given in the appropriate cell of the matrix. The
cost to agent 1 is the first cost in a cell, followed by the cost to agent 2.
Because this is a cost matrix, the agents want to minimize their out-
come in the game. Depending on the number C/4u(Tij ), the matrix game,
considered as a one-shot game, is a Prisoner’s Dilemma game or a Hawk-
6
Dove game. In addition to pure Nash equilibria (NE) the latter has mixed
strategy NE. These equilibria are analyzed in detail in [5].
3.1 Update of Strategies
During a simulation round, all naagents update their strategies once, so
that a simulation round consists of naiteration periods. Hence, on an it-
eration period t, there is only one agent updating its strategy once. The
strategies are updated with a shuffle update scheme, i.e., the order in which
the strategies are updated is randomized. At this point, we do not assume
the agents to move. In the next section, it is explained how the game is
coupled to the CA model presented in the previous section. Thus, do not
confuse a simulation round or iteration period of the game with a time step
in the CA.
The total cost for an agent is the sum of the costs against all of his
neighbors, and the agent’s best-response strategy is a strategy that minimizes
his total cost. The agents are myopic in the sense that they choose their
strategies based on the previous iteration period of the game, not considering
the play of future iteration periods. The best-response strategy s(t)
iof agent
ion iteration period tis given by his best-response function BRi, defined
by
s(t)
i=BRi(s(t−1)
−i;Ti, T−i) = arg min
s0
i∈SX
j∈Ni
vi(s0
i, s(t−1)
j;Tij ).(4)
Here, Niis the set of agents in agent i’s Moore neighborhood. Note that
when we couple the game model to the CA, the Niwill change as agent i
moves in the square grid. The function vi(s0
i, s(t−1)
j;Tij ) gives the loss defined
by the evacuation game to agent i, when he plays strategy s0
i, and agent jhas
played strategy s(t−1)
jon iteration period (t−1). That is, vi(s0
i, s(t−1)
j;Tij )
is equal to the corresponding matrix element. Here, s(t−1)
−iis used to denote
the strategies of all other agents than agent ion iteration period t−1, and
T−iincludes the estimated evacuation times of these agents.
Simulations in [5] have been done with an experimental (undocumented)
version of FDS+Evac software [6]. There, playing the game actually changes
the physical behavior of the agents. Impatient agents do not avoid contacts
with other agents as much; they accelerate faster to their target velocity,
and move more nervously. Whereas, patient agents avoid contact with other
agents.
7
4 Cellular Automaton Evacuation Model Coupled
with a Spatial Game
There are similarities between the presented spatial game and CA model. As
noted above, impatient agents end up in conflicts by competing with other
agents, whereas patient agents avoid conflicts. The description of impatient
agents resembles the movement of agents in the ordered regime of CA; recall
Section 2.2. Agents in the ordered regime are set to move towards the exit
using the shortest path, and thereby have a tendency to get into conflicts.
On the other hand, the description of patient agents resembles the movement
of agents in the cooperative regime. Agents in the cooperative regime move
towards the exit using paths of higher flow, i.e., paths where the amount of
conflict situations is small, and thereby have a tendency to avoid conflicts.
From the aforementioned observations, we propose a model, where we
couple the CA model with the spatial evacuation game. In our model, we let
the strategy choice of playing Impatient result in ordered behavior, i.e., the
agent to move towards the exit using the shortest path, and playing Patient
in cooperative behavior, i.e., the agent to move towards the exit using paths
of higher flow. For an agent playing Impatient, the coupling constants are
set to kS= 10, kD= 1, and for an agent playing Patient kS= 1, kD= 1.
The coupling constant values chosen to represent ordered and cooperative
behavior are chosen to be such that they are clearly inside the appropriate
regimes in Figure 2. The effect of strategy choice on the agent’s behavior is
depicted in Figure 4.
(a) If the agent plays Im-
patient, he moves towards
the exit using the short-
est path, regardless of the
awaiting conflict situation.
(b) If the agent plays Pa-
tient, he moves towards
the exit using the path of
higher flow, avoiding the
awaiting conflict situation.
Figure 4: Effect of strategy choice on the agent’s behavior.
8
It should be noted, that the strategy choice the agent makes, does not
reflect an optimal path to the exit, i.e., it is not an optimal strategy for
the whole evacuation over time. Rather, the strategy choice is optimal in
a snapshot of the evacuation against his immediate neighbors (actually the
whole crowd is in an NE in a snapshot [5]).
4.1 Model Description
Next, a step-by-step description of our model is given. In the beginning of
the simulation, the agents are located randomly in the room. None of the
agents play the game, and all agents are considered patient.
Step 1. At the beginning of each time step, Tiis calculated for i= 1, ..., na.
If Ti> TASET −T0, the agent iplays the game.
Step 2. The agents’ strategies are updated with the shuffle update scheme.
The agents observe the strategies of the other agents in their Moore
neighborhood, and choose a best-response strategy according to Equa-
tion 4.
Step 3. The agents’ behavior is updated in the CA model, to correspond
to their strategy choice. This is done by altering the agents’ coupling
constants as follows:
(a) Playing Impatient results in ordered behavior. The agents cou-
pling constants are set to kD= 1.0 and kS= 10.0.
(b) Playing Patient results in cooperative behavior. The agents cou-
pling constants are set to kD= 1.0 and kS= 1.0.
Step 4. The agents move in the CA.
Step 5. Go to Step 1. This procedure is repeated until all agents have
evacuated the room.
Remark 1 : Here, a time step refers to a time step in the CA, i.e., the
agents are able to move once.
Remark 2 : In Step 2, the shuffle update scheme is repeated multiple
times, to ensure that the agents are in an equilibrium configuration all the
time. Figure 5 illustrates a snapshot of the evacuation in such a config-
uration. Note that because the estimated evacuation times of the agents
9
increase farther from the exit, the proportion of impatient agents do so; this
is explicitly shown in [5]. More such simulations, with different patient and
impatient agent densities, can be found in [1], [5]. The convergence of the
best-response dynamics in the spatial Hawk-Dove game has previously been
studied in [16].
Figure 5: An equilibrium configuration for 378 agents with parameter values TASET =
450 and T0= 400. Black cells represent impatient agents and white patient.
5 Evacuation Simulations
We have presented an evacuation model, where the agents’ coupling con-
stants appear as a result from the game the agents play. In the following,
we illustrate how the agents behave in a typical evacuation simulation. Addi-
tionally, we show that the faster-is-slower effect, already found in the original
formulation [9], now appears as a result of the game the agents play. The
result is compared to a similar analysis made by Heli¨ovaara et al. with an
experimental (undocumented) version of FDS+Evac [5].
5.1 Evacuation of a Large Room
Here, we simulate a typical evacuation situation, i.e., the evacuation of a
large room. In Figure 6 there are three snapshots from different stages of
this evacuation simulation. The black squares represent impatient agents
and the white patient.
As can be seen, the agents form a half-circle rather quickly in front of
the exit. Notice, that the agents play their equilibrium strategies at each
snapshot of the simulation. At these snapshots, the impatient agents move
towards the exit using the shortest path, whereas the patient agents use a
path of higher flow.
10
(a) Early stages (b) Middle stages
(c) Late stages
Figure 6: Snapshots of the simulation in different stages of the evacuation process.
The black squares represent impatient agents and the white patient.
5.2 Faster-is-Slower Effect
Some people experience the evacuation situation more threatening than oth-
ers, and thus start to behave more impatiently in relation to the other people.
It is striking that our model describes this feature of human beings. It is
clearly seen in Figure 5; see also the explanation in Remark 2.
In [5] the dependence of the proportion of impatient agents on egress flow
was studied with an experimental (undocumented) version of FDS+Evac.
The agents were set in a half-circle in front of the exit, and they updated their
strategies until equilibrium was reached. Afterwards, the agents’ strategies
were fixed, the exit was opened and the agents start to evacuate. The same
simulations were run with our model. Here, we want to demonstrate that
both models describe qualitatively the faster-is-slower effect. The results of
the simulations with these two models can be seen in Figure 7.
It is clearly seen, from both Figures 7 (a) and (b), that the more agents
behave impatiently, the smaller the egress flow is. Since the effective velocity
of an impatient agent is larger than that of a patient, a faster-is-slower effect
can be distinguished. In the experimental version of FDS+Evac, this is
11
(a) Simulations with the experimental version of
FDS+Evac [5] (a 0.8 m wide exit).
(b) Simulations with our model (a 0.4 m wide exit).
Figure 7: Average egress flow for 200 agents with different proportion of impatient
agents in the population. In the simulations, 11 different values of TASET were used. Note
that the vertical scales in the figures differ.
caused by impatient agents pushing harder towards the exit, which results
in jams and reduced flows [5]. In our model, it is caused by impatient agents
moving straight towards the exit, resulting in more conflict situations and
slowing down the evacuation. The quantitative differences can be explained
by the different geometries of both the agents and the exits. Also, the
velocities of the agents are different in the two models.
12
6 Discussion and Conclusions
We introduced a CA evacuation model, where the agents are equipped with
simple decision-making abilities. For the simulation of the agents’ move-
ment, we used the simulation platform by Schadschneider et al. [9]. In it,
ordered and cooperative crowd behaviors can be obtained by altering the
coupling constants kDand kS. To provide decision-making abilities, we
coupled it with a spatial game introduced by Heli¨ovaara et al. [5].
In our model, the choice of strategy actually changes the physical be-
havior of the agent in the CA. Patient agents move towards the exit using
paths of higher flow, i.e., have a tendency to get avoid conflicts, whereas
impatient agents move towards the exit using the shortest path, i.e., have a
tendency to get into conflicts.
In the original model by Schadschneider et al., the values of the coupling
constants should be fixed before simulation starts. In our formulation, the
agents’ coupling constants depend on their strategy choice in the spatial
game. Moreover, the agents’ parameters change dynamically according to
their perception of the surrounding conditions, i.e., the risk of not being able
to evacuate in time, and the behavior of neighboring agents.
In the end of the numerical section, we noticed that our model in some
aspects give qualitatively similar results as in [5]. To map the full potential of
our model, further comparisons with evacuation simulation software should
be done. Since our model is computationally light, it could be used for
web-based real-time safety analyses.
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