- Email: [email protected]

S1877-7503(18)30327-2 https://doi.org/doi:10.1016/j.jocs.2018.09.003 JOCS 915

To appear in: Received date: Revised date: Accepted date:

31-3-2018 24-8-2018 2-9-2018

Please cite this article as: Martha Mitsopoulou, Nikolaos I. Dourvas, Georgios Ch. Sirakoulis, Katsuhiro Nishinari, Spatial Games and Memory Effects on Crowd Evacuation Behavior with Cellular Automata, (2018), https://doi.org/10.1016/j.jocs.2018.09.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Journal of Computational Science

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Journal of Computational Science 450 (2018) 1–25

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Spatial Games and Memory Effects on Crowd Evacuation Behavior with Cellular Automata

of Electronics, Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, GR67100, GREECE b Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, 113-8656, JAPAN

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Martha Mitsopouloua , Nikolaos I. Dourvasa , Georgios Ch. Sirakoulisa,∗, Katsuhiro Nishinarib

Abstract

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When it comes to emergency evacuations of a big number of individuals from a certain room or a building, it is crucial that they should be able to exit the under study place in a quick and safe manner. Nevertheless, crowd behavior can be affected from various factors, among which are the distance from the exit, the visibility of the exit, the density of the crowd and how uncomfortable the conditions are inside the room for the crowd. In this study, we propose an evacuation model, in order to achieve a computationally enhanced simulation, based on Cellular Automata modeling, coupled with a Spatial Game to solve conflict situations among the anxious crowd. Additionally, aiming to a more realistic model, we added not only topological, but also behavioral characteristics coupled with the incorporation of different moving velocities and memory features to the evacuees.

1. Introduction

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Keywords: Crowd Evacuation, Cellular Automata, Modeling, Spatial Games, Memory

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When we are part of a large moving crowd inside a building or a room, our safety and comfort depend strongly both from the behavior of the fellow crowd members and the design of the facility we are in. In situations where the crowd needs to evacuate the facility immediately, the process often leads to conflict when seeking a better position, especially around the exits. Former research of crowd dynamics during emergency evacuations has been based both on macroscopic and microscopic models. Macroscopic models face the crowd as a homogeneous mass that has certain characteristics. Such models, just to name a few, are the Regression models [1], the Route Choice models [2], the Queuing models [3], the Gas kinetics [4] and the Flow-based models [5]. On the other hand, the microscopic dynamics models seem more realistic, as the simulated crowd is composed from discrete individuals. These models include both continuous and discrete space models, while the most common categories are the Social Force models [6, 7], the CA models [8, 9], the Rule-based models [10] and the Agents-based models [11]. The Cellular Automata (CA) model offers a computational intelligent technique which approaches the evacuation simulation from the perspective of artificial intelligence. According to the CA models, both time and space are discrete, while the principal of the CA gives us the opportunity to create a parameterized model, as detailed as possible, focusing on specific individual behaviors under certain conditions. In addition, Game Theory is used to solve conflict problems among the crowd, as an acknowledged tool for interpreting human behavior and decision making. Despite ∗ Corresponding

author Email addresses: [email protected] (Martha Mitsopoulou), [email protected] (Nikolaos I. Dourvas), [email protected] (Georgios Ch. Sirakoulis), [email protected] (Katsuhiro Nishinari)

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this fact, very few studies have been conducted to explore crowd conflict behaviors during emergency evacuation with Game Theory [12, 13, 14, 15, 16, 17, 18, 19]. In this paper, we developed a CA evacuation model which investigates conflicts among individuals in the evacuation process by using a spatial game. This model provides us with the ability to predefine almost every evacuation parameter, such as the size of the evacuated area, the emergency and risk scale, the crowd density inside the room, the crowd different behaviors, the individuals’ velocity, etc. Thus, this model incorporates both topological features and features that describe the crowd formation and manners. It should be reminded that every individual constitutes a different entity, which has distinguished characteristics but still can interact with the others. This feature enhances the realism of the model, giving us the capability to study various phenomena under different panic circumstances. Moreover, although previous studies [13] have introduced the scheme of CA crowd behavior combined with game theory, or discrete modeling like Gas Lattice scheme enriched with game theory [15] or snowdrift game theories [14], to simulate crowd behavior, the presented study can be considered enhanced in many different ways. More specifically, taking into account Moore neighborhood instead of von Neumann [13, 14, 16, 17], the ability of the proposed model to better illustrate the movement of people inside the room in more physical way is apparent and state their preferable next position according to a desired direction, considering the position of the exit. Furthermore, the outcome of the proposed game, namely Chicken Game’s, not only sets who of the players will move to the desired position, but also affects the players’ behavior and thus choice of possible strategy in subsequent conflicts. Thus, compared to the aforementioned studies [12, 13, 14, 18, 16, 17, 19], one’s behavior is not an acquired feature but is affected by many parameters, which are both initial and current distance from the exit, along with the level of emergency, and the outcome of previous games; and can change during the evacuation. In other words, the proposed model outmatches previous evacuation models which utilize short term memory characteristics, adopting a fully dynamic behavior for the under study crowd taking advantage of the long memory CA based approach as a prominent feature of the people that instinctively affects their behavior. Finally, compared to the aforementioned confrontations, another novel feature of the proposed model towards realistic modeling of crowd is the incorporation of more than one velocities for the moving agents and how the ratio of different velocities in a heterogeneous crowd affect the evacuation process. This difference in velocities does not affect the introduced mechanism of the considered game played between individuals in a conflict. The slow individuals deal with the drawback that they do not participate in a conflict as often as a faster individual. The advantage is a model that approaches a more realistic situation providing us with much more interesting results. Finally, the efficacy and the robustness of the proposed CA model has been proven in both qualitative and quantitative ways, with the help of fundamental diagrams of flow, speed and density, compared with real life data as reported in literature. In the remainder of this paper, Section 2 describes the contribution of Game Theory in the model when conflicts occur among evacuees. In Section 3, the selected CA modeling approach for crowd evacuation is briefly analyzed in its fundamental characteristics while Section 4 outlines the proposed CA model and its parameters and features, as well as the spatial game developed and how the payoffs affect the individuals involved and consequently the evacuation evolution. The corresponding simulations with various parameter settings, the metrics diagrams and the data results obtained from the simulations are presented in Section 5. Finally, in Section 6 conclusions regarding this model are made and some suggestions for future work are proposed. 2. Game Theory

Game Theory constitutes “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers” [20]. It has been a known scientific field since 1928, when physicist and mathematician John von Neumann proved his minimax theorem. He then extended his research upon games and published the results in his 1944 book, Theory of Games & Economic Behavior along with Oskar Morgenstern [21]. A few years later, another mathematician, John F. Nash, whose contribution to Game Theory is considered also great, introduced a concept solution for non-cooperative games [22], later known as Nash equilibrium. Since 1950’s there has been a huge development in the topic of Game Theory, together with an increase in Game Theory’s applications. Nowadays it is widely used in behavioral relations [23], not only in economics, politics and psychology, but also in diplomacy, biology, computer science as well as everyday life. A normal formal definition of a Game [24] claims that a Game involves three components: 2

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1. A set of players I = {1, . . . , I} , 2. A pure strategy space S i [a set of strategies S i (with s ∈ S i ), i = 1, . . . , n], 3. Payoff functions ui (s1 , . . . , sn ), i = 1, . . . , n, giving von Neumann-Morgenstern utility levels associated with the outcome arising from strategies (s1 , . . . , sI ) ∈ S 1 × . . . × S I .

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As a result, the mathematical definition of such a Normal (N) Game is given by the following normal form representation ΓN : ΓN = {I, {S i } , {ui (·)}} . (1)

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Game theory assumes that each player only cares about maximizing personal payoff. According to the informal definition, in order to recognize which type of Game we have, it is important to determine some of the key Game Theory features. These are the number of players, the number of strategies per player, the number of pure strategy Nash equilibria, the constant sum of the payoffs from the possible outcomes of the game, and the information players can obtain from previous game iterations. Following these common Game features, we can categorize Games as normal or strategic, cooperative or non-cooperative, zero-sum or non zero-sum and simultaneous move or sequential games, which can also has perfect information. 3. Cellular Automata

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Cellular Automata (CA) are computational discrete models of physical systems, where space and time are discrete and interactions are local. A CA consists of a finite number of dimensions, usually two dimensional, lattice of n × n cells. Apart from Pascal’s triangle, which is considered to be the first cellular structure, the original concept was discovered in the 1940s by Stanislaw Ulam and John von Neumann [25]. Despite the studies among the next decades, it was 1970’s that the CA got a spotlight from the academic community, thanks to John Conway’s Game of Life [26] and afterwards due to Stephen Wolfram’s contribution to the development of the CA, as he studied a number of one dimensional CA models and proved CAs’ importance to computing, statistical mechanics and cryptography [27, 28]. CA structure is defined by the following parameters [29]:

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1. The discrete n-dimensional space or lattice of homogenously shaped cells. 2. The discrete states. At each discrete time step, each cell is found in one state, from a finite number of states: C = {C1 , C2 , . . . , Ck }, where k is finite. 3. Local interactions. Each cell’s behavior, and thus current state, depends only on what happens within its local neighborhood of cells (which may or may not include the cell itself), which is described with a set of Boolean variables: C(~r, t + 1) = {C1 (~r, t), C2 (~r, t), . . . , Cm (~r, t)} (2) where ~r is each cell in the CA lattice and t = 1, 2, 3, . . . are the discrete time steps. When it comes to a 2dimensional CA lattice, two are the most common neighborhoods. The von Neumann neighbourhood, which consists of m = 5 cells, i.e. the central cell, whose condition is to be updated, and the four cells located to the north, south, east and west of the central cell, and the Moore neighbourhood, consisted of m = 9 cells, i.e. the same cells with the von Neumann neighbourhood together with the four other adjacent cells of the central cell, i.e. northwester, northeaster, south–east and south west cells. 4. Discrete dynamics. At each time step t, each cell’s current state C(~r, t + 1) his updated according to a local rulei R = {R1 , R2 , . . . , Rm }, as shown in the following equation: Ci (~r, t + 1) = Ri C(~r, t), C(~r + ~δ1 , t), ..., C(~r + ~δm , t) where the ~r + ~δm represents the cells included in the cell ~r neighborhood. It is also assumed that the update is synchronous and Ri takes as input at time step t + 1, the neighborhood states at the immediately previous time step t.

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4. The proposed CA Model

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C{~r + ~δl , t} ≥ 1

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• one or more CA cells in the cell’s neighborhood are currently occupied:

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In this study, we propose an evacuation model, based on CA and coupled with a special game in situations of crowd collisions. To be more specific, the physical space in which the evacuation takes place, is being simulated with a two dimensional CA lattice of n × n cells, while the exit point is being represented by specific number of cells, which corresponds to exit door width equal to the length of the under study exit. Each of the CA cells is considered either empty or occupied by an individual i. Both these situations determine the state of the CA cell, namely C(~r, t). Each cell has the same dimensions, which in the presented study are 40 centimeters length and 40 centimeters width, as this is usually the space occupied from a single average standing person in crowding situations, according to studies [30]. We also use the Moore neighborhood for each cell, which means that in the local rule R = {R1 , R2 , . . . , Rm }, m = 9. Due to the universal implementation of the local rule to all cells, rules interact with each other and an individual can move to one of the unoccupied cells in his Moore neighborhood, as long as this cell is closer to the exit. It is possible for a cell to be occupied during the next time step, if:

(3)

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• and the CA cell is closer to the exit than the occupied neighborhood cells.

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Furthermore, in this study, we insert the concept of different individual velocities. Specifically, given that in a CA model time is discrete, and an individual can only move once in one of the unoccupied cells of his neighborhood during a time step, we came up with the idea of two different velocities v1 , v2 . The maximum one v1 , corresponds to moving or ability of moving towards another cell nearby during one time step, and it is the average walking velocity of an adult person, which is, according to studies, 1.3 meters per second, or else 4.68 kilometers per hour [30, 31]. The minimum velocity v2 , is the one that simulates the movement in a neighbor cell during two time steps. This basically means that the individual stays at the cell which he currently occupies, during the first time step, and attempts to move towards another cell during the second time step. This velocity represents individuals with quite limited moving abilities, such as elders or people with physical disabilities as well as children. In our model, the velocity characteristic is given to each individual before simulation’s inception and cannot be changed during the process of the evacuation. In view of the foregoing, we also use Game Theory, in order for the individuals to have more extended abilities when it comes to decision making. The game helps solving conflict situations among individuals, which wish to occupy the same cell in order to evacuate as fast as possible. The basic concepts of a two players game are the set of strategies that each player can adopt and the payoff that can receive by following each strategy. The symmetrical games give to players the same amount of strategies and payoffs and can be represented onto a n × n matrix, where n is the number of strategies that each player can adopt. The simplest symmetrical games are the twelve 2 × 2 games which are catalogued by [32]. The more interesting ones are the four games where the players do not prefer the same outcome but they confront some dilemmas. The strategies that are possible in such a 2 × 2 game is the cooperation or defection for each player. These strategies are usually noted as D for defection and C for cooperation, i.e. when one player chooses to cooperate with the other one as shown in Table 1 where the corresponding payoffs for each player are also given introducing behaviours such as temptation, coordination, neutrality and punishment. Table 1. Typical strategies and payoffs table for a 2 × 2 strategy game.

C

D

C

C, C

C, D

D

D, C

D, D

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In more details, each cell of Table 1 includes two payoffs, one for each player. The left payoff is for the player that can choose the strategies in the rows and the right payoff is for the player that can choose the strategies of the columns. So, ‘CC’ are the payoffs when the two players cooperate, ‘CD’ are the payoffs when the ‘row’ player cooperates and the ‘column’ player defects, ‘DC’ is the opposite, and ‘DD’ are the payoffs when both player defect. The cooperation seems to be the most sensible solution. Even if we defect the cooperation of the opponent is always preferable. So in a common sense game the sequence of the best payoffs is CC > DC > CD > DD. But in many cases the dilemmas are created because the player has a good payoff if he defects. The four most known games with their conditions are [15]: • The Prisoners Dilemma, DC > CC > DD > CD

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• The Deadlock, DC > DD > CC > CD • The Chicken game, DC > CC > CD > DD

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• The Stag hunt, CC > DC > DD > CD

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Now, in the proposed evacuation game, it is assumed that there is a situation where more than one individuals wish to occupy the same cell as soon as possible, in an attempt to approach the exit sooner, as they have some knowledge about the location of the exit. Nevertheless, according to the prespecified CA model principles, only one of the individuals will finally move to this CA cell. Each individual involved in the evacuation simulation, is considered to be either patient or impatient according to the following formula. if 0 ≤ a < 0.5 patient, (4) b= impatient, if 0.5 ≤ a ≤ 1

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Parameter b is the label that each individual receives according to the a value which is the irritation level of each individual. Note that this behavior can change during the procedure of the evacuation and it affects the individual’s choice of strategy when she/he is implicated to a conflict situation, as the one already described. Therefore, one can choose to “play” either polite or competitive, which means that she/he can claim the cell she/he wishes to move into, or she/he can grant it, respectively. This scheme is described with the 2 × 2 evacuation game, which includes two players and two possible strategies (Table 2). Table 2. A payoff matrix of the proposed evacuation game.

Competitive

Polite

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p3 , p2

Competitive

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If both players claim the same cell by both being competitive, neither of them will occupy the cell and additionally they will have an equal chance of getting injured from the conflict. This risk of injury will lead to a cost, p4 = 0.2, which is an increase in both individuals’ irritation level. At this point, let us mention that the more irritated an individual is, the more likely it is to choose the impatient strategy the next time he will be involved in an evacuation game. On the other hand, if both individuals choose to play patient, then again neither of them will move and there will be an increase of the irritation, p1 = 0.1, not as much as in the previous scenario, which affects both of them. In an impatient versus patient individual contest, the impatient one will overtake the cell (p2 = 0), leading again to the other player’s irritation, p3 = 0.15. It is clear that for each pair of strategies, there is a pair of costs, formed according to non-cooperative, non-perfect information, simultaneous, non-zero-sum games as already described before. These costs have a negative effect for the players, as they increase players’ irritation and thus encourage impatient and aggressive behaviors. When a 5

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player overtakes the cell she/he was claiming, her/his cost is equal to zero, which means that she/he maintains her/his irritation level. We chose not to decrease this value in order to show that by winning the conflict, the player acquires experience and assumes that the choice of his former strategy led the player to victory. So, the payoff matrix of the proposed evacuation game takes the following values for parameters p1, p2, p3, p4, respectively as shown in the 2 × 2 strategy matrix (Table. 3).

C

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0.15, 0

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Table 3. Payoffs in the presented game.

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The desirable is the smallest payoff because those numbers consider irritation level. So, the best sequence for every player is DC > CC > CD > DD which according to the sequences presented before, is clearly a Chicken Game. In other words, the game we propose shows great similarity with the Chicken Game [33] (see also its payoff matrix in Table 4), according to which, two drivers are moving against each other with high speed. The first driver to change direction in order to avoid collision is the “chicken” one, meaning he is the “coward” and she/he loses. If both drivers continue to drive straight ahead they may also die in the crash. Table 4. A payoff matrix of the Chicken Game.

Straight

Swerve

Tie, Tie

Lose, Win

Straight

Win, Lose

Crash, Crash

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At this point, we can logically assume that the individuals choose their strategies based on former iterations of the game, unable to predict what can possibly happen in future iterations. This means that the individuals have memory, not only from former iterations of the game, but from previous games as well. In terms of CA this is translated to a CA cell with states referring to more previous time steps like t − 1, t − 2, etc. [29, 34]. This memory mechanism is described with the form: bt+1 i

T X t = f e, d(i, t0 ), d(i, t), (bi )

(5)

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Parameters that are taken into consideration in order for an individual to choose his strategy are (please consider that in the following, like before, parameter t stands for time, and its maximum duration equals to T ): • How unsafe the conditions are during the evacuation: e ∈ [0, 10] • How far the individual initially (i.e. time step t = 0) was from the exit: d(i, t0 ) = D(x, y). It should be mentioned that in this study, we are using the Euclidean distance, due to the fact that each cell interacts with cells into the Moore neighborhood. The Euclidean distance from x = (x1 , x2 , . . . , xn ) to y = (y1 , y2 , . . . , yn ) in a Euclidean n-space is: D(x, y) = D(y, x) =

p

(x1 − y1 )2 + (x2 − y2 )2 + . . . + (xn − yn )2 =

v t n X

(xi − yi )2 .

(6)

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• How far the individual currently is from the exit: d(i, t) = D(x, y). • And what choices the evacuee made in previous conflict situation in terms of the outcome they had.

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At each iteration all players’ strategies are updated simultaneously and thus every player has no information about the strategies the other players will choose to play. One can only speculate, that the more iterations a game between 2 players has, the more irritated these two players will be at the end of the game. A game between two particular players comes to an end when one of them moves to the cell they were arguing about. The other player, will either remain to her/his current position, or move to a least favorite cell, depending on exit’s direction, avoiding conflicts. With this mechanism we make sure that each individual struggles in order to escape, although some of them will win the conflicts and thus temporarily, only for one time step, accelerate during the evacuation. The irritation a is a characteristic that every individual i in the simulation has: (7)

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where a ∈ [0, 1], p(a, i) the payoff of the games in which the individual participated so far, considering that at time step t = 0 no games have been played and index P stands for the maximum t value of all the previous time steps. The result of irritation a determines the parameter b of an individual (patient or impatient) which finally rules the outcome of the individual’s strategy at the next time step. In this model, we also treat conflicts between more than two individuals, with correlated games. This confrontation is really important when simulating an evacuation process, especially with the cells closest to the exit, where n + 2 individuals wish to move to the n cells of the exit. Of course, as already mentioned, only n of them will eventually move during one time step, as a result of n + 1 games between those individuals, with every game cost affecting the other games. In general, game’s parameters such as under what circumstances one player (individual) becomes impatient due to danger scaling and risk conditions, can be adjusted and hence affect the proportion of patient and impatient pedestrians resulting to different game strategies and accordingly to different evacuation patterns and times. On other words, the game strategies that emerge result to different game outcomes for each individual and the arising effects can be observed during the evacuation, as the behavior and thus the movement of the pedestrians is affected. From the way pedestrians approach the exit, to how persistent they are when claiming their desired position, these phenomena do affect the evacuation process quantitatively and qualitatively and more important, the model parameters influencing them can be changed at the duration of various experiments. In the following, the pseudo-algorithm of the proposed CA model is presented step by step for readability reasons:

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• Step 1: At the beginning, the model parameters are set. Among these parameters are the CA grid dimensions, which correspond to the actual room, the exit position and length, along with the fullness of the room and how unsafe the conditions are inside of it. During this step, the initialization of the CA lattice/grid also takes place. This lattice contains each cell’s condition during the previous evacuation time step and computations of each individual’s distance from the exit are also taken place. • Step 2: The evacuation process starts. When there is crowding right before the exit, which can be depicted by more than one CA cells in the CA matrix, for instance n cells, this means that n + 2 individuals wish to exit the room simultaneously. In this case, n + 1 games will be played among these individuals, in order to decide which ones will exit the room during the current time step. Moreover, the corresponding game parameter b is refreshed according to any game outcome. • Step 3: Before any move is made in the rest of the room, we have to make sure that an empty cell will be occupied by one and only individual. Nevertheless, as it is already mentioned, more than one evacuee wishes to move to the same cell, especially to the cells closer to the exit, found by computing the Euclidean distance from the exit as provided in Eq. 6 for every cell in every individual’s neighborhood. In case of competing interests, the proposed game takes place in order to give a solution. 7

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• Step 4: While the game winners move to their preferred CA cell, the losers will act according to the circumstances. The more unsafe the conditions are perceived, the more they feel their life is in under threat and they will move to another empty CA cell in their neighborhood. Otherwise, they will remain to their current position. It should be also noticed that the corresponding game parameter b is refreshed according to any game outcome.

• Step 6: Steps 2 through 6 are repeated until every individual exits the room.

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• Step 5: At this step, the CA lattices are “reinitialized” since the current cells’ condition becomes previous condition and the new distances from the exit are computed in order to proceed with the evacuation.

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For readability reasons, the flowchart of the proposed CA model is presented in Figure 1. Finally, the computational complexity of a single pass for a N × N 2D CA is O(N 2 ). Now, that we require t time steps of the above procedure, the total complexity will be in the order of O((t)N 2 ). This expression serves as an upper bound to the algorithm’s complexity, since after the first time step, the number of cells that are updated is normally expected to decreasing with the number of iterations depending on the individuals that exit the place, as well as the empty (unoccupied) cells. Consequently, the actual algorithm’s complexity will always be less than O((t)N 2 ), since not all CA cells will be updated after the first time step.

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5. Simulation Results

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We developed a simulation environment using MATLAB, in order to couple the CA evacuation model with the evacuation game. In the following we describe the results of the simulations. In order to better present the underlying mechanism of the proposed overall model, in Fig. 2 (a) a snapshot of a simulation, in which the people evacuating the room have the same velocity, is depicted. In this ground truth example, there are no individuals that are slower than the others. This homogeneity in velocity leads to expected results and the graphs show a high flow of people through the exit (Fig.2 (b)). This is a logical result because there are no slower individuals that will decrease the flow of people as we will check later. However, in order to meet with the realistic conditions of an evacuation process considering people with different physical conditions and acknowledging varying speeds, as proposed earlier in the model description, we incorporate model’s various parameters. More specifically, we used different parameter values to indicate various situations of the evacuation of a large room and we were able to adjust the parameters of the model according to our requirements. These parameters were the portions of individuals moving with low and high velocity, Pv2 and Pv1 = F − Pv2 respectively, where F ∈ (0, 1] the total fullness of the room; and how dangerous or uncomfortable the conditions are inside the room given by the corresponding parameter e, where e ∈ [0, 10], as described in the previous Sections. In order not only to be sure that the results are statistically correct, but also to fully explore the model’s abilities, we have conducted 31 simulations for every set of parameters with different initial conditions each time, which refer to different initial positions of the crowd inside the room and we have calculated the average of the evacuation time steps. In order for the results to be comparable, we maintained the same 31 different conditions for every set of parameters. In these simulations we have investigated the behavior of individuals attempting to evacuate a square room, with only one exit, as tests showed that similar behaviors would have occurred if we have decided to add a second exit point to the room under study. During this analysis we have conducted numerous experiments with a number of different initial conditions for every set of parameters and we have monitored the exit times and the morphology of the crowd as the evacuation proceeds. Please bear in mind that the place is considered homogeneous, with no obstacles found and the people are absolutely free to move inside the room with no other difficulty. In Figure 3, typical stages of the evacuation process are shown for 2 different emergency situations. Low velocity individuals are marked with red color and normal velocity individuals with yellow, while dark blue indicates the empty space inside the room and light blue the walls of the simple under study place with the exit located at the bottom of the room and in the middle. We observed various crowd behaviors for different values of the parameters. For example, crowd seems to be more compact around the exit when high risk conditions exist, e ≥ 5, as the individuals tend to be more impatient about exiting the dangerous area and vying behavior occurs more frequently. On the other hand, when no emergency situation exists, evacuation is processing in proper order as individuals seem to approach the exit with a delay, a behavior that has a great effect on evacuation time steps. 8

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Figure 1. The flowchart of the proposed CA model indicating the possible movements of the under study individuals.

In Figure 4, the progression of the people density around the exit for different values of the initial density is shown. One can observe that it takes less than 5 time steps for the individuals to gather around the exit when the danger is on high levels inside the room. The density remains on high levels for the two thirds of the evacuation evolution and begins to descend afterwards, while the flow of people exiting the room is steady. Figure 5 demonstrates that crowd density and low velocity individuals percentage have as well a great influence on evacuation time steps. The first logical assumption one can make, is that the more the people are, inside the room, the longer the evacuation process will last. Another interesting finding from the simulations was that the increase of 9

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Figure 2. (a) A snapshot of an evacuation process where people are all moving with the same velocity. (b) The corresponding flow/density diagram for this simulation.

Figure 3. Individuals evacuate a one exit room. Displayed three stages of the process for different time steps for low risk conditions (top) and high risk conditions (bottom). Red cells refer to individuals with low velocity, yellow to ones with normal pace speed, empty space is shown with dark blue and walls with light blue.

the total evacuation time steps is not always proportional to the increase of the crowd density. For instance, for danger equal to 10, if we increase the percentage of individuals initially inside the room from 30% to 50%, the evacuation time steps increase by 66%. In both situations we kept the low velocity individuals percentage to 10% of the crowd. If we set the low velocity individuals percentage to 30% and we repeat the simulation for the same crowd density percentages, the evacuation time steps now increase by 69%. And for low velocity individuals’ percentage up to 60% of the crowd, the same percentage becomes 75%. Another interesting finding from the simulations was that the increase of the total evacuation time steps is not always proportional to the increase of the crowd density. For instance, for danger equal to maximum value, i.e. “10”, if we increase the percentage of individuals initially inside the room from 30% to 50%, the evacuation time steps 10

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Figure 4. Evolution of the density in the area of interest which is around the exit point during the evacuation for three different initial densities inside the room, given the same proportion between slow and fast moving evacuees, and the maximum level of emergency.

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increase by 66%. In both situations we kept the low velocity individuals percentage to 10% of the crowd. If we set the low velocity individuals percentage to 30% and we repeat the simulation for the same crowd density percentages, the evacuation time steps now increase by 69%. And for low velocity individuals’ percentage up to 60% of the crowd, the same percentage becomes 75%. From the reference between exit times and the amount of slow moving individuals, one can assume that the more the low velocity individuals are among the escape area the longer the evacuation will last, in a proportional way, despite the fact that there is the same amount of people initially inside the room. This happens because individuals 11

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Figure 6. (a) The under study space of the experiment setup of [35, 36] considered for the bottleneck simulations. http://www.asim. uni-wuppertal.de/en/database/own-experiments/bottleneck/bottleneck-no-2.html#c3211. (b) – (e) Progression of the people evacuation for initial crowd density equal to 80% and low velocity percentage up to 25% for danger parameter equal to 1. (f) – (i) Progression of the people evacuation for the same simulation setup with only one difference that the danger parameter has changed to its bigger value, namely 10. Consequently, the place is evacuated in less time steps.

with low velocity seem to be treated like moving obstacles and they delay the people behind them, especially when they are grouped. It was also observed that these slow moving individuals, as expected, are eventually bypassed from the individuals with regular moving speed and that they tend to be the last to evacuate the room. Finally, for comparison reasons, another experiment setup was designed and considered and corresponding simulation tests were performed by using the setup of bottleneck flow as presented in Figure 6(a) [35, 36]. In specific, in correspondence to the experiments run in [35, 36], the bottleneck width was chosen b = 0.8m, the bottleneck length l = 4cells×0.4m = 1.6m, the room width bc = 30cells×0.4m = 12m, while the initial crowd density inside the room differs as follows: pinitial = 15%, 30%, 45%, 60%, 75% and 90%, respectively and the distance to the entrance equals to dbottleneck = 10cells×0.4m = 4m. In Figure 6 the progression of the people density in the presented bottleneck flow is presented for the lower and maximum values of danger parameter, i.e. equal to 1 and 10, respectively and crowd density of 80% and low velocity individuals percentage up to 25% while the room is evacuated in 256 and 566 time steps (Figs. 6(b)-(i)), respectively. 12

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From the extended simulations, mentioned above, it was clear that certain attributes of the model, such herding phenomena, congestions in exits, motion variation from random to coherent due a common purpose, transition to in–coordination (arching) due to dense gathering in front of escaping points prove in a qualitative way the efficacy of the proposed model in accordance with the real phenomena as recorded and reported in literature [6]. However, to further validate the model’s response in a quantitative way, more measurements are needed. Although overcrowded states in animals have been empirically studied (mice, ants, etc.), similar experiments are quite difficult to take place for human activities. Apparently, there is up to a point a lack of elaborated evidence regarding mass human crowded critical circumstances for experimentation reasons. Moreover, it is very difficult to control “inner” state of pedestrians, i.e. behavioral strategy to others, in experiments. Also it is impossible to “observe” the person’s strategy from the video. It should be noticed, to the knowledge of the authors, there is no any fixed methodology or benchmark data. The most recent studies try to approach answers to fundamental issues of crowd evacuation dynamics, by elaborating past recordings of real, often tragic, events [37] and taking into consideration measurements and simulations regarding flow, density and speed of the under study crowd. As a result, the proposed model is quantitatively evaluated by comparing the flow–dense dependence, i.e. the fundamental diagram, of many different crowd distributions, to the corresponding diagrams of real–life distributions. In particular, the total dense of the crowd is measured, by counting all persons within the area of interest and then dividing with the extent of this area. The total flow is calculated by counting the number of people that pass a vertical intersection in a corridor/time unit and dividing with the length of this intersection. Finally, the proposed model response is further compared to empirical relations that engage the glow of people with the speed of the individuals and the density of the crowd that they form. According to the relation between flow and density [37], the flow/(meter of width), follows the relationship: Q(ρ) = ρV(ρ)

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where Q represents the flow/(meter of width), V velocity and parameter ρ the density of the people. Eq. 8 is frequently used when public venues are designed, regarding matters of safety and study of building evacuation [38]. Experimental measurements are often restricted to density values between 4–6 persons/m2 [29]. In [31], for instance, the maximum density is equal to 5.4 persons/m2 and the corresponding dependency between velocity and density is: !#) ( " 1 1 − (9) V(ρ) = Vo 1 − exp −a ρ ρmax

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where Vo = 1.34 m/sec is the free speed in low densities of people and a = 1.913 persons/m2 is an fit parameter. As already mentioned in the previous Section, each individual is able to cover a cell, i.e. a minimum area equal to 40 cm×40 cm2 =0.16m2 , as this is usually the space occupied from a single average standing person in crowding situations, according to studies [30], resulting to a maximum density equal to 6.25 persons/m2 . The whole area corresponds to a CA grid of Atotal = 30 × 30 cells that is equal to an area that is calculated and measured as Atotal = 900(cells)× [0.4(m)×0.4(m)] = 144m2 (please notice that this area differs slightly when the bottleneck cell effect of the extra space is applying). The area of interest (AoI), i.e. the one that is taken into consideration in the density measurements equals to AoI = 10 × 10(cells) × [0.4(m)×0.4(m)] = 16m2 . For each of the total times steps of the cell simulated experiment (please consider that different times were needed depending on the initially considered crowd density, which as already mentioned differs and scales as presented in Figures 4, 5, 8), the number of people inside the area of interest is measured and the corresponding density is calculated, according to the following relationship: ρt =

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where t corresponds to a particular time step. In the view of the foregoing, the following fundamental diagrams of speed-density and flow-density of the proposed model are presented, using data that emerged from the simulation tests, like the ones shown in Figures 6, for different initial densities and compared with the ones found in literature [36]. More specifically, the diagram of the global flow–global density is depicted in Fig. 8(a). In the same figure, i.e. 8(b) it is depicted the corresponding diagram that is based on the results that they have been provided by Seyfried et al. [36] where experimental data of the flow and the associated density in the bottleneck described before in the simulation test are provided in comparison with experimental data for the fundamental diagram of unidirectional pedestrian streams and the specifications for the 13

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fundamental diagram according to the SFPE Handbook [39] and the guidelines of Weidmann [31] and Predtechenskii and Milinskii [40]. Aiming at further assessment of the model, the diagram of velocity–density is depicted (Fig. 8(c)) for the exact simulation results corresponding to the bottleneck setup as expressed before. The corresponding diagrams are found in very good qualitative and quantitative agreement, thus enhancing the validation of the model and prove both the quantitative and qualitative efficiency of the proposed model. In Fig. 7 the average irritation level in different levels of danger is depicted. Every individual has a long term memory of her/his irritation during the whole evacuation process. The tests were performed for different levels of danger ranging from 0 up to 10 and for different crowd densities. The average irritation of all individuals was measured at every test and then this value was normalized between 0 and 1. According to results, the increment in irritation level is almost linear to the increment of danger level.

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Figure 7. Average irritation of people in different danger levels. Simulations with different levels of danger ranging from 0 up to 10 were considered. The average irritation of all individuals was measured at every test running for different crowd densities and then this value was normalized between 0 and 1. The individuals’ irritation lever at situations with danger levels 8, 9 or 10 are much higher that the individuals’ irritation in conditions with danger levels 0, 1 or 2.

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In this work we introduced a CA based evacuation model, where the evacuees are participants to a spatial game when collisions take place. Thus we have a heterogeneous population inside the room, where every individual can choose either the polite or the competitive strategy, according to his experience and/or other parameters, such as the walking distance from the exit and the hazards of the situation, in order to exit the room as quick as possible, with a memory mechanism that was developed in order for this model to be as realistic as possible. • The adoption of Moore neighborhood instead of a von Neumann one. Others similar studies often use von Neumann neighborhood; however, Moore neighborhood provides the proposed model with the ability to better illustrate the movement of people inside the room in more physical way, while the increment of the complexity is not considered seriously influential. • The Chicken Game, which is used in the proposed model, not only defines the players that will move to the desired position in the next time step, but also affects the players behavior through time and thus choice of possible strategy in subsequent conflict. This property provides the presented model with the advantage that an individual’s behavior is not an acquired feature but is seriously affected by the model’s parameters, such as initial and current distance from the exit, along with the level of emergency/danger, inherent behaviour, the outcome of previous games and more; and, most important, it can be changed during the evacuation. • Moreover, this model incorporates the feature of multiple moving velocities, giving us the opportunity to choose the proportion of fast and slow moving individuals in order to perform several simulations and examine different evacuation scenarios. 14

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Figure 11: Experimental data of the flow and (b) the associated density in the bottleneck in comparison with experimental data for the fundamental diagram of unidirectional pedestrian streams and the specifications for the fundamental diagram according to the SFPE Handbook and the guidelines of Weidmann and Predtechenskii and Milinskii.

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most measurements with densities larger then ρ = 1.8 m−2 are performed with bidirectional streams, see [19, 21–24]. But also data gained by measurements on strictly unidirectional streams has been considered [4,5]. The data of Hanking and Wright gained by measurements in London subway (UK) are in good agreement with the data of Mori and Tsukaguchi who measured in the central business district of Osaka (Japan). Their fundamental diagrams agree very well and differ notable from the fundamental diagram of the SFPE and Weidmann for densities larger ρ = 1.8 m−2 . In accordance with Predtechenskii and Milinskii we assume that the maximum of the fundamental diagram for unidirectional movement is at much higher densities. Thus we conclude from our measurements of densities and velocities that the flow through the bottleneck under normal conditions will not reach the capacity defined through the maximum of the fundamental diagram and that a jam occurs well before the maximum is reached. (c) There are three possible reasons for jamming below the capacity limit. One is clearly flow fluctuations. Local density maxima higher then capacity-density cause obstructions that can bediagram resolved only the incoming flow is less tests, thanlike the which in this Figure 8. (a) The global flow–global people density using dataifthat emerged from the simulation theoutflow, ones shown in Figures 6, case is below for the considered experimental setup of [35, 36] while the initial density inside differsorganization as follows: pinitial 15%, 30%, 45%, 60%, capacity. A crowd second reason is the in room the local of=the pedestrians. At high densities, 75% and 90%, respectively and the rest of the modeling parameters are the same as described in thepositioning text. (b) Global as a function the global the full flow requires a fairly regular of flow persons. Thisofmay be achieved by predensity in experimental data for the fundamental diagram of unidirectional pedestrian streams [36, 41, 42, 39, 31, 40] and the specifications for the arranging them as in the experiments in [17, 18] or possibly by funneling the flow into the fundamental diagram according to the SFPE Handbook [39] and the guidelines of Weidmann [31] and Predtechenskii and Milinskii [40]. (c) The bottleneck. Neither the random arrival at a narrow passage out of the almost free motion global speed–global people density diagram, as derived from the simulation results of Figs. 6 for the considered experimental setup of [35, 36] and a wide also random - starting into the passage from waiting in a jam can for different initial crowd densities, namely pinitial in = 15%, 30%,room 45%, nor 60%,the 75%-and 90%. achieve this. Therefor the actual bottleneck is the entrance into the passage, while in the passage itself the flow corresponds to the lower density associated with the observed flux • Finally, the proposed CA model meets successfully in both and ways withusually experimental (compare PM [25]). The qualitative third reason is quantitative motivational, people prefer larger distances

data as previously 1reported in literature. As earlier studies propose, we used a combination of CA and Game theory in an evacuation process where conflicts 15 arise between individuals and the goal is to enhance each of the combined computational methods advantages and reduce their disadvantages. In this perspective, CA make the model ideally suited for large-scale computer simulations because they are discrete in space and time with appropriate state variables while they also have strong expressive 15

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power to represent many collective behaviors. In the presented studies, CA have also proven to reproduce quite good simulation results, very close to real experiments. Nevertheless, due to their aforementioned discrete nature, most of the presented CA models do not take always into consideration contact forces in crowd panic that may generate high pressures, which can asphyxiate people in the crowd and even push down a brick wall, or may ignore situations like stumbling, falling, ache, injury, walking over fallen people that can occur during a fast evacuation process. In other words they are not always dealing successfully with the psychological parameters of the people involved in the evacuation and as such they may not simulate individuals’ behavior efficiently when they are in stress and try to act/respond/behave during an evacuation process. This is why in this paper we coupled CA with Game Theory. We have successfully presented that in the Game Theory, the behavior of individuals and the psychological aspects can be mimicked. Moreover, game theoretical approach can also rationalize the interaction of the evacuees with the environment. Nevertheless, the drawback of such an approach is that is characterized by a payoff matrix which many times is difficult to define when many players participate in the game. Further characteristics and parameters can be added and in our future research, we intend to study thoroughly the mathematical perspectives of the Chicken Game varying payoffs with the proposed evacuation game parameters and the resulting consequences. Since our model is computationally inexpensive and after the aforementioned comparisons, it can be used for real-time safety analyses when combined with real data for accurate initialization of its parameters.

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[1] I. Joseph Milazzo, N. Rouphail, J. Hummer, D. Allen, Effect of pedestrians on capacity of signalized intersections, Transportation Research Record: Journal of the Transportation Research Board 1646 (1998) 37–46. doi:10.3141/1646-05. [2] S. Hoogendoorn, P. Bovy, Pedestrian travel behavior modeling, Networks and Spatial Economics 5 (2) (2005) 193–216. doi:10.1007/ s11067-005-2629-y. [3] G. G. Løvås, Modeling and simulation of pedestrian traffic flow, Transportation Research Part B: Methodological 28 (6) (1994) 429–443. [4] L. F. Henderson, The statistics of crowd fluids, Nature 229 (5284) (1971) 381–383. doi:10.1038/229381a0. [5] G. Santos, B. Aguirre, A Critical Review of Emergency Evacuation Simulation Models, DRC preliminary paper, Disaster Research Center, University of Delaware, 2004. [6] D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic, Nature 407 (6803) (2000) 487–490. doi:10.1038/ 35035023. [7] A. Kłusek, P. Topa, J. Was, R. Lubas, An implementation of the social distances model using multi-GPU systems, The International Journal of High Performance Computing Applications 0 (0) (0) 1094342016679492. doi:10.1177/1094342016679492. [8] K. Nishinari, A. Kirchner, A. Namazi, A. Schadschneider, Simulations of evacuation by an extended floor field CA model, in: Traffic and Granular Flow 2003, 2005, pp. 405–410. [9] M. Kanai, S. Isojima, K. Nishinari, T. Tokihiro, Ultradiscrete optimal velocity model: A cellular-automaton model for traffic flow and linear instability of high-flux traffic, Phys. Rev. E 79 (2009) 056108. doi:10.1103/PhysRevE.79.056108. [10] W. Reynolds, Craig, Flocks, herds, and schools: A distributed behavioral model, Computer Graphics (ACM) 21 (4) (1987) 25–34. [11] S. Musse, D. Thalmann, Hierarchical model for real time simulation of virtual human crowds, IEEE Transactions on Visualization and Computer Graphics 7 (2) (2001) 152–164. doi:10.1109/2945.928167. [12] S. Lo, H. Huang, P. Wang, K. Yuen, A game theory based exit selection model for evacuation, Fire Safety Journal 41 (5) (2006) 364–369. doi:10.1016/j.firesaf.2006.02.003. [13] X. Zheng, T. Zhong, M. Liu, Modeling crowd evacuation of a building based on seven methodological approaches, Building and Environment 44 (3) (2009) 437–445. doi:10.1016/j.buildenv.2008.04.002. [14] D.-M. Shi, B.-H. Wang, Evacuation of pedestrians from a single room by using snowdrift game theories, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 87 (2). doi:10.1103/PhysRevE.87.022802. [15] S. Bouzat, M. Kuperman, Game theory in models of pedestrian room evacuation, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 89 (3). doi:10.1103/PhysRevE.89.032806. [16] X. Zheng, Y. Cheng, Conflict game in evacuation process: A study combining cellular automata model, Physica A: Statistical Mechanics and its Applications 390 (6) (2011) 1042–1050. doi:10.1016/j.physa.2010.12.007. [17] X. Zheng, Y. Cheng, Modeling cooperative and competitive behaviors in emergency evacuation: A game-theoretical approach, Computers and Mathematics with Applications 62 (12) (2011) 4627–4634. doi:10.1016/j.camwa.2011.10.048. [18] A. von Schantz, H. Ehtamo, Spatial game in cellular automaton evacuation model, Phys. Rev. E 92 (2015) 052805. doi:10.1103/ PhysRevE.92.052805. [19] H.-N. Wang, D. Chen, W. Pan, Y. Xue, H.-D. He, Evacuation of pedestrians from a hall by game strategy update, Chinese Physics B 23 (8). doi:10.1088/1674-1056/23/8/080505. [20] R. Myerson, Game Theory, Harvard University Press, 1997. [21] J. von Neumann, O. Morgenstern, Theory of games and economic behavior, 3rd Edition, Princeton Univ. Press, Princeton, NJ, 1953. [22] J. Nash, Non-cooperative games, Annals of Mathematics 54 (2) (1951) 286–295. [23] L. Dugatkin, Dynamics of the tit for tat strategy during predator inspection in the guppy (poecilia reticulata), Behavioral Ecology and Sociobiology 29 (2) (1991) 127–132. doi:10.1007/BF00166487.

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Martha M. Mitsopoulou received her Diploma in Electrical and Computer Engineering in April, 2014 from Democritus University of Thrace (DUTh), Greece, followed by her MSc from the same University on June, 2017. During her studies she was an active member of IEEE Student Branch of Thrace and a member of the Organizing Committee of the 5th Panhellenic Electrical and Computer Engineering Students Conference. Her research interests lie in the areas of Cellular Automata, Real-time Models and Algorithmic Game Theory.

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Nikolaos I. Dourvas received his Diploma and his M.Sc degree in Electrical and Computer Engineering from the Democritus University of Thrace (DUTh) in Xanthi, Greece in 2013 and 2015 respectively. He is now working as a PhD candidate at the same department. His main interests are cellular automata, modeling of large scale systems, Design and Integration of Novel and Emergent Micro-Nano-Bio devices and systems, unconventional and parallel computing, modeling and simulation and bio-inspired algorithms. He is member of IEEE.

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Georgios Ch. Sirakoulis received his Ph.D degree in Electrical and Computer Engineering from Department of Electrical and Computer Engineering (ECE) in the Democritus University of Thrace (DUTh), Greece, in 2001, where he is serving as Full Professor while he is also Visiting Research Professor in University of West England. published more than a hundred papers in leading international journals, more than 140 papers in international conference proceedings and 14 guest-editorials. He is co-editor of 7 books as well as coauthor of 20 book chapters. He is Associate Editor in “Journal of Cellular Automata”, “International Journal of Parallel, Emergent and Distributed Systems”, “International Journal of Unconventional Computing”, “IEEE Trans. on Nanotechnology”, “Microelectronics Journal”, “Integration, the VLSI Journal”, “Parallel Processing Letters”, etc. His research emphasis is on Complex and Smart Electronic Systems, Cellular Automata Theory & Applications, Future and Emergent NanoElectronic devices, circuits, models and architectures, beyond CMOS computing devices and circuits, Memristors, Bioelectronics and Bioengineering, Unconventional computing, High performance Computing, FPGAs, Modelling and Simulation. Katsuhiro Nishinari received his Ph.D. degree in aerospace engineering from The University of Tokyo, Japan, in 1995. He is currently a professor at the Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, Japan, and the director of the crowd management research society. He is a member of the Japan Society for Industrial and Applied Mathematics, and an editor, Journal of Cellular Automata. He has published more than a hundred research papers in leading international journals, and wrote several books concerning traffic jam and applied mathematics. He has won awards for his work including “The 23th Scientific Publication Award in Japan by the book “Jamology” (in Japanese)” (2007), and NISTEP Award 2013 from National Institute of Science and Technology Policy. His research interests are “Jamology”, i.e. interdisciplinary research on transportation and jamming phenomena (vehicular traffic, pedestrian motion, queue network and supply chain, etc.), Cellular Automata, Soliton theory and its Applications and Network system, Non-linear Dynamics.

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Spatial Games and Memory Effects on Crowd Evacuation Behavior with Cellular Automata Martha Mitsopouloua, Nikolaos I. Dourvasa, Georgios Ch. Sirakoulisa,*, Katsuhiro Nishinarib

Laboratory of Electronics, Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, GREECE b

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Research Center for Advanced Science and Technology, The University of Tokyo,

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Tokyo, JAPAN

Highlights

> Cellular Automata (CA) modeling of crowd evacuation behavior with spatial game.

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> Novel long memory CA based approach models in a fully dynamic way crowd behavior. > Realistic crowd modeling by incorporating more than one velocity for

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moving agents. > Multi-parameter modeling including level of emergency, and previous games outcome. > Study of people behavior and choice of possible strategy

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in subsequent conflicts.

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