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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.

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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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