Abstract
A computer simulation based on both game theory and the cellular automaton model was conducted to investigate the effects of individuals’ interactions and certain conditions on the spread of rule-breaking behavior in public places. Four decision matrices representing obedient, deviant, conforming, and contrary principles were prepared experimentally. The ratio of these principles ranged from 1:2:3:4 to 4:3:2:1 (obedient:deviant:conforming:contrary). There were a total of 24 data sets with each data set composed of 10 decision matrices. In addition to these 24 data sets, the ratio 1:1:1:1 was prepared. This data set was composed of four decision matrices. The results indicated that rule-breaking behavior spreads under the following conditions: (A) when people act according to both individual factors (e.g., their attitudes) and situational factors (e.g., their neighbors’ behavior); (B) when there are more deviant people than obedient people; (C) when the intensity of rule-breaking behavior is higher than that of rule-obeying behavior. Condition A is an important factor. If conditions B or C are satisfied, the spread of rule-breaking behavior does not occur necessarily. At a certain probability threshold, rule-breaking behavior spreads sharply when people act based on individual factors (e.g., their attitudes); the spread of such behavior cannot be attributed solely to situational factors (e.g., neighbors' behavior). There is only a fine line between rule breaking and obeying near the probability threshold.
Introduction
The purpose of this study was to investigate the effects of individuals’ interactions on rule-breaking behavior in public places and to determine the conditions associated with the spread of such behavior. Various rule-breaking behaviors in public places, for example, littering (Cialdini, Reno, & Kallgren, 1990) and violating traffic regulations (Osman, 1982; Verkuyten, Rood-Pijpers, Elffers, & Hessing, 1983), have been investigated in previous studies. Rule-breaking behaviors such as talking in class have been frequently reported in school and college classrooms (e.g., Durmuscelebi, 2010; Özben, 2010; Sacks, 1996; Urabe & Sasaki, 1999). Why do people engage in them?
Interactions Between Individuals
Cialdini, Reno, and Kallgren (1990) found that people broke a rule according to their perceptions of other people’s behavior. They indicated that people’s behavior could be affected by others’ behavior as a descriptive norm. The descriptive norm indicates “what is typical or normal” (Cialdini, Kallgren, & Reno, 1991, p. 1015); “it is what most people do, and it motivates by providing evidence as to what will likely be effective and adaptive action” (p. 1015). Urabe and Sasaki (1999) found that students talked privately even though they thought that it should not be done and that each student also perceived their classmates to be more tolerant toward talking during classes than they themselves do. The authors proposed that students would talk according to their perceptions of group norms in the class rather than their own norms or attitudes on talking. These studies imply that not only the individuals’ own attitudes but also others’ attitudes or behaviors can influence rule-breaking behavior and there are many interactions between the people.
The cellular automaton model has been used to study interactions between individuals (e.g., Latané, Nowak, & Liu, 1994; Nowak, Szamrej, & Latané, 1990). In this model, each cell is considered as an individual, and the grid and the inhabitants of the cells are regarded as social. The cells are arranged in a lattice and change their states according to the same set of rules (Kosugi, Fujisawa, Mizutani, & Ishimori, 2001). The cellular automaton model thus allows the analysis of relationships between microlevel behavior as interactions between individuals and macrolevel phenomena as public opinion (Nowak et al., 1990).
Latané, Nowak, and Liu (1994) conducted computer simulation studies using dynamic social impact theory (DSIT). This theory was derived from social impact theory (Latané, 1981; Latané & Wolf, 1981), where interpersonal influence is expressed by the equation I = f(SIN), where “I” denotes the impact on a target. According to Latané and Wolf, “S” stands for “the strength or intensity (status, power, ability, etc.) of the source persons” (pp. 440–441), “I” for “their [source persons’] immediacy or proximity in space or time to the target” (p. 441), and “N” is “the number of source persons present” (p. 441). DSIT could analyze the effects of more than one impact and the processes of interpersonal interactions (Latané et al., 1994; Nowak et al., 1990). DSIT studies have been conducted using cellular automata (Nowak et al., 1990). In these simulations, hundreds of cells are placed in a matrix with each cell having one of several states. These states represent the cells’ opinions. These cells interact with their neighbor cells and change their states in accordance with a rule. One model that uses DSIT is expressed (Latané et al., 1994) by the equations Ip = [Σ(Si/di2)2](1/2) and Is = [Σ(Si/di2)2](1/2).
This model is called the accumulative-influence model. In these equations, “Si” is the strength of a cell’s neighbor, and “di” is the distance between the cell and its neighbor. There are two types of impact: persuasive (Ip) and supportive (Is). Persuasive impact is calculated using the Si and di of neighbor cells with opposing opinions, and Is is computed using cells with the same opinion as the target. If Ip is higher than Is, then the cell changes its state. In DSIT, the phenomena of consolidation, clustering, and continuing diversity can occur (Latané & L'Herrou, 1996). According to Latané and L'Herrou, consolidation is “the tendency for diversity within the group to be reduced as the proportion of people who hold the minority position decreases” (p. 1219), and clustering is “the propensity for group members to end up more similar to their neighbors than to those at a greater distance” (p. 1219). Continuing diversity means that minorities can survive as a result of clustering. DSIT is used to analyze the effects of individual factors, such as individual opinions, on group level consequences such as public opinions (Latané et al., 1994; Nowak et al., 1990).
Individual Differences
People can have various attitudes or norms toward rule-breaking behavior (e.g., Deguchi, 2013; Koike & Yoshida, 2011). One person may think that rule breaking should not be done while another may think that it should be tolerated. In a situation consisting of many people with different attitudes, when and by what means is rule-breaking behavior transmitted?
Game theory was incorporated into the cellular automaton model to investigate the effects of individual attitude distribution on the spread of rule-breaking behavior. The rules of the simulation model are presented in Table 1. In the simulation, a decision matrix was applied (Table 2). Each cell was in one of two states representing cell behavior: “obeying a rule” and “breaking a rule.” This decision matrix resembles the payoff matrix in game theory (e.g., Kreps, 1990; Rapoport & Guyer, 1966; Scodel, Minas, Ratoosh, & Lipetz, 1959) and the interdependence theory (e.g., Kelly et al., 2003; Thibaut & Kelley, 1959). The payoff matrix has payoffs of both “you” and “the other” (i.e., neighbors); however, the decision matrix in Rule 1 has a payoff of only “you.” In Rule 1, the decision matrix represents the relationships between a cell (you) and its (your) neighbors, and each cell has its own decision matrix. You and your neighbors have two choices: to obey a rule or to break it. The decision matrix includes the following four attitudes (or degrees of satisfaction), namely, the attitude where both you and your neighbors obey the rule (M11), where you obey the rule and your neighbors break it (M12), where you break it and your neighbors obey it (M21), and where you and your neighbors break the rule (M22).
Rules of Simulation.
Decision Matrix.
The first and second formulas in Rule 1 mean that the cells change their states according to their neighbors’ states. In other words, each cell is affected by only some of the cells in the matrix. For example, a person (cell) in a large area (matrix) such as a town or classroom cannot simultaneously perceive the actions of all the other people. Similarly, it is difficult for a person at the edge of a town or classroom to perceive the exact behavior of the people at the opposite edge because his or her observations would be limited by distance and capacity. Therefore, the two formulas in Rule 1 imply that the perceptions are not unlimited but have boundaries.
Two types of neighborhoods, Moore and von Neumann, are frequently used in simulations of cellular automata (e.g., Axelrod, 1984; Kosugi et al., 2001). The Moore neighborhood consists of eight neighbor cells that join in a straight line or diagonally while the von Neumann neighborhood consists of only four neighbor cells connected in a straight line. The Moore neighborhood is applied in Rule 1 because it is not plausible that diagonally adjoining people have no effect.
“Intensity” denotes the relative power of rule-breaking behavior. An intensity higher than 1 implies that rule-breaking cells are more powerful than obeying cells. For example, an intensity of 1.5 means that rule-breaking cells have 1.5 times as much power as obeying cells do. This parameter resembles “strength” in the social impact theory or DSIT. Strength is determined by “the source’s status, age, socio-economic status, and prior relationship with, or future power over, the target” (Latané, 1981, p. 344), and each cell can have its own value. However, in Rule 1, intensity depends on the cells’ status, that is, whether they are obeying or breaking a rule. Cells of the same status have the same intensity.
The scores of O and B in the third and fourth formulas include parameters of individual attitudes. These scores are calculated from four attitudes (“M11,” “M12,” “M21,” and “M22”) in a decision matrix. The scores of O and B are impacts that are weighted based on these four attitudes. In other words, they are impacts that reflect individual attitudes or individual differences. In Axelrod’s (1984) setting, all the cells had the same payoff matrix. In this study, however, the cells had different decision matrices.
Namatame (2004) showed that principles indicating how humans behave are explained with payoff matrices. He termed obeying in Table 2 as “Site 1” (S1) and rule breaking as “Site 2” (S2). In the payoff (decision) matrix whose M12 and M21 are 0 and the sum of M11 and M22 is more than 0, if M22/(M11 + M22) is 0 or less, the payoff matrix is a “hard core of S1.” If M22/(M11 + M22) is 1 or more, it is a “hard core of S2,” and if M22/(M11 + M22) is more than 0 and less than 1, it is “dependent on the situation.” In the payoff matrix whose M11 and M22 are 0 and the sum of M12 and M21 is more than 0, if M12/(M12 + M21) is 1 or more, it is a hard core of S1. If M12/(M12 + M21) is 0 or less, it is a hard core of S2, and if M12/(M12 + M21) is more than 0 and less than 1, it is “dependent on the situation.”
If the difference between M11 and M21 in a cell’s decision matrix is a positive value, then obeying neighbors change the cell’s state to obeying; but if the difference is negative, then they change to rule breaking. Similarly, if the difference between M12 and M22 is positive, rule-breaking neighbors change the cell’s state to obeying; if it is negative, they change to rule breaking. Therefore, decision matrices can be classified into the following four principles according to the comparisons of M11 with M21 and of M12 with M22; when M11 > M21 and M12 > M22, the cells always obey the rules (obedient). For example, a ratio of 1:1:0:0 (M11:M12:M21:M22) represents the obedient principle (1 > 0 and 1 > 0). When M11 < M21 and M12 < M22, the cells always break the rules (deviant). When M11 > M21 and M12 < M22, the cells’ behaviors correspond with their neighbors’ (conforming). When M11 < M21 and M12 > M22, the cells’ behaviors contrast with their neighbors’ (contrary). Therefore, in comparison with Namatame’s (2004) classification, the obedient principle corresponds to a hard core of S1, deviant to a hard core of S2, and conforming and contrary correspond to “dependent on the situation.”
Ambiguity in Human Behavior
In a study on a prisoner’s dilemma game (e.g., Axelrod, 1984, 1997), agents in the simulation can choose to “cooperate” or “defect” (cooperation enables both players to receive high payoffs, whereas defection enables one player to receive the highest payoff and the other receives the lowest). However, the agents behave contrary to their decision at a certain probability called “noise” (Wu & Axelrod, 1995). This behavior is thought to be human error, that is, noise represents the vagueness or uncertainty in human behavior. If there is no noise, the “tit-for-tat” (TFT) strategy (to cooperate first then follow the opponent’s previous behavior) will receive the highest payoff (e.g., Axelrod, 1980a, 1980b). This strategy is believed to be adaptive. However, if there is noise, the TFT strategy will not be the most adaptive (Godfray, 1992; Nowak & Sigmund, 1993). These studies suggest the importance of considering human ambiguity or uncertainty.
Asch’s (1951) experiment on conformity revealed that individuals’ own perceptions do not always reflect their behavior: one third of the participants conformed in half or more of the trials when they were asked to reveal their judgments in the presence of others (confederates). The results of this study imply that individual factors such as perceptions, attitudes, and decisions do not always reflect behavior. Similarly, environmental factors, such as the behavior of the majority, do not always affect human behavior directly. In other words, humans sometimes behave in accordance with individual factors and sometimes according to situational factors; they exhibit ambiguity or uncertainty.
The M-prob in Rule 2 represents the ambiguity or uncertainty of human behavior. When M-prob is 0, the cells always conform to their neighbors’ behaviors. At M-prob = 1, they act only according to their attitudes. Cells with M-prob values of more than 0 and less than 1 are affected by both neighbors’ behaviors and their own attitudes. Thus, the cells change their states in accordance with individual factors at the probability of M-prob and change it according to situational factors at the probability of 1 − M-prob.
Rule 3-1 represents the effect of a situational factor on the cell’s behavior, such as the behavior of the majority (Asch, 1951) or the descriptive norm (e.g., Cialdini et al., 1991; Reno, Cialdini, & Kallgren, 1993). Rule 3-2 exemplifies the effect of an individual factor on the cell’s behavior.
Method
A computer simulation based on both game theory and the cellular automaton model was conducted to examine the effects of interactions between various individuals on rule-breaking behavior in public places. A 21 × 21 square matrix was prepared. Four hundred and forty-one cells were entered into the matrix, and each cell represented an individual or a person. All the cells’ states were set to obeying at the beginning. One decision matrix was selected by random sampling with a replacement from the data set and entered into one cell, that is, a decision matrix selected from the data set was returned to the data set. This process was repeated until all cells had one decision matrix.
The change in the cells’ states was repeated from the second step to the 200th step (repeated 199 times), and all the cells changed their states at the same time (step). The simulation was executed 100 times per condition. The matrix was not a torus and the Moore neighborhood was used. It is difficult for a person at the periphery of a classroom or city to perceive exactly what the people at the opposite side are doing. Therefore, the matrix was not a torus. In the DSIT simulations (Latané et al, 1994; Nowak et al., 1990), cells are affected from cells that are not adjacent to themselves, but their impacts declined according to distance. However, in Rule 1, cells are affected only from up to eight neighbors and there is no parameter such as “distance.” Kosugi, Fujisawa, Mizutani, and Ishimori (2001) indicated that a spatial clustering in the DSIT can be indicated in the simulation using Moore neighborhood. To simplify the model of rule-breaking behavior, the Moore neighborhood was applied in the present study. There were differences in the number of neighbor cells in the center (eight), at the edges (five), and at the apexes (three). The cells interacted with the neighbor cells and changed their states in accordance with rules 1, 2, 3-1, and 3-2.
In Axelrod’s setting (Axelrod, 1984), the cells could decide their behaviors based on the neighbors’ in the same step. For example, they could choose to cooperate (obeying) with some of their neighbors and defect (rule breaking) from others. However, it is impossible for one student to both obey and break a rule at the same time. Therefore, in this article, the cells can decide on only one behavior in the same step and each cell has one of two states (obeying or rule breaking).
The M-prob was set to vary by 0.01 between the values of 0.00 and 1.00. The intensity was set at 1.0 or 1.5. At an intensity of 1.5, an obeying cell would change its state to rule breaking if it uses Rule 3-1 and there are four or more rule-breaking cells in its neighborhood. The simulation program was written in Microsoft Visual Basic.NET.
Four decision matrices representing obedient, deviant, conforming, and contrary principles were prepared experimentally. Each of the four decision matrices was composed of the four attitudes M11, M12, M21, and M22. The values of these four attitudes were set at either 0 or 1. Thus, each decision matrix was 1:1:0:0 (M11:M12:M21:M22; obedient), 0:0:1:1 (deviant), 1:0:0:1 (conforming), or 0:1:1:0 (contrary). The ratio of these principles ranged from 1:2:3:4 to 4:3:2:1 (obedient:deviant:conforming:contrary). There were a total of 24 data sets, with each data set composed of 10 decision matrices. For example, a data set of 1:2:3:4 would include 1 obedient, 2 deviant, 3 conforming, and 4 contrary decision matrices.
In addition to these 24 data sets, the ratio 1:1:1:1 was prepared. This data set was composed of four decision matrices.
Results
The rule-breaking rate, which is the mean rate of all the rule-breaking cells in 200 steps, was used as the output index and was calculated for each execution of the simulation. When analyzing the spreading processes of rule-breaking behavior, however, this index represented the rate of rule-breaking cells at each step.
There were linear and nonlinear relationships between M-prob and rule-breaking rate (Figures 1-1, 1-2, 1-3). The following three patterns were created according to scatterplots of the relationship between the M-prob and rule-breaking rate. “Pattern A” is the linear relationship between M-prob and the rule-breaking rate. As M-prob increases, the rule-breaking rate increases. Because “pattern B” is nonlinear, as M-prob increases, the rule-breaking rate rises but this gradually slows. This seems to be an exponential saturation. In “pattern C,” the relationship between M-prob and the rule-breaking rate follows an inverted J curve; that is, as M-prob increases, the rule-breaking rate reaches a maximum and then decreases. However, the rule-breaking rate at M-prob = 1.00 is much higher than at 0.00.
Relationship between rule-breaking rate and M-prob (ratio: 1:1:1:1). “B” and “C” represent patterns.
Relationship between rule-breaking rate and M-prob (ratio from 1:2:3:4 to 4:3:2:1). Letters in upper right of scatterplots represent patterns.
Relationship between rule-breaking rate and M-prob (ratio from 1:2:3:4 to 4:3:2:1). Letters in upper right of scatterplots represent patterns; 3142, 4123, and 4231 could not be classified into any patterns.
The author, together with two psychology undergraduate students, classified each of the relationships between M-prob and rule-breaking rate into the three patterns. The two undergraduate students first classified them separately; the differences and appropriateness of the classifications were then discussed.
At the intensity of 1.0, the ratio of patterns A, B, and C was 9:11:4. At 1.5, it was 0:1:20, and there were three data sets (ratios of 3:1:4:2, 4:1:2:3, 4:1:3:2) that could not be classified. Except for these three data sets, all the patterns were categorized as C when deviant cells outnumbered obedient cells. The rule-breaking rates of pattern C were generally higher than those of pattern A. The analysis of variance (ANOVA) showed the significant main effects of the patterns at intensity 1.0, partial η2 = .80, F(2, 21) = 41.84, p = .000. Further, the results of multiple comparisons (i.e., Tukey’s method) ranked patterns A (mean [M] = 21.18, standard deviation [SD] = 4.84), B (M = 39.23, SD = 7.87), and C (M = 54.38, SD = 4.11) in order of low to high (ps < .010).
Table 3 shows correlations between the proportions of the principles and rule-breaking rates. The proportions of obedient and deviant cells correlated strongly with the rule-breaking rates. However, the differences between the respective proportions of obedient and deviant cells showed the strongest correlations (intensity 1.0: r = −.95; intensity 1.5: r = −.96). For ratios of 1:3:4:2 and 2:3:4:1 at the intensity of 1.0 (i.e., when the proportion of deviant cells was 30% and the proportion of obedient cells was lower), the rule-breaking rates were sometimes over 60%. Conversely, for the ratios of 4:3:1:2 and 4:3:2:1 (when there was a higher proportion of obedient cells), the rule-breaking rates did not go over 60% although the percentage for the deviant cell remained the same. Therefore, it is important for the proportion of the deviant cell to be higher than that of the obedient cell for the spread of rule-breaking behavior. There were no significant correlations for conforming and contrary cells.
Correlations Between Proportions of Principles and Rule-Breaking Rates.
Note. a Difference between obedient and deviant cells. (N = 24)
Although the principle ratios were the same, there was a difference in rule-breaking rates between the intensities of 1.0 and 1.5 (Figures 1-1, 1-2, 1-3). Overall, the rule-breaking rates were higher at 1.5 than at 1.0. This indicates that not only the principle ratios but also the intensity affects the rule-breaking behavior. Even if the number of deviant people was the same as (ratio of 1:1:1:1) or somewhat fewer (e.g., ratio of 4:3:2:1) than that of obedient people, rule-breaking behavior spreads when its intensity was higher than that of the rule-obeying behavior. However, when the number of deviant people was much less (e.g., ratio of 4:1:3:2) than that of obedient people, rule-breaking behavior did not spread, even at higher intensity.
Additional analyses were conducted to investigate the spreading processes of rule-breaking behavior. The spreading processes in 1:1:1:1 at intensities of 1.0 and 1.5 are shown in Figure 2, and the relationships between the rule-breaking rates for each principle and the steps for the ratio 1:1:1:1 are shown in Figure 3. The deviant cell obtained higher rule-breaking rates than the other cells. At 1.0, the total rule-breaking rate was about .20 and almost constant. At 1.5, the total rule-breaking rate increased immediately to over .80.
Spread of rule-breaking cells. White dots are rule-breaking cells. Values below matrices indicate steps (ratio: 1:1:1:1, M-prob: .30).
Rule-breaking rates in each step (ratio: 1:1:1:1, M-prob: .30).
Discussion
Critical Factors in the Spread of Rule-Breaking Behavior
In all the simulations, the states of all the cells were set to obeying in the first step. In the second step, only deviant and contrary cells using Rule 3-2 broke a rule and caused the spread of rule-breaking behavior. Obedient cells changed to rule breaking only when using Rule 3-1 and when their neighbors broke the rule. Therefore, if deviant cells outnumber obedient cells, then the obedient and conforming cells would gradually break the rule. However, if obedient cells outnumber deviant cells, then the obedient and conforming cells would not break the rule.
At an intensity of 1.5, the rule-breaking rate was sometimes over 50%, even if there were not many deviant cells in the matrix (e.g., ratios of 3:2:4:1, 4:2:3:1). Therefore, if the intensity of rule-breaking behavior is higher than that of rule-obeying behavior, then a few people who do not intend to obey the rule can spread rule-breaking behavior.
In pattern C, the rule-breaking rates were low when M-prob was around 0, implying that a situational factor had forced the cells to be obeying or prevented the cells from changing to rule breaking. When the M-prob was more than 0.20, however, the rule-breaking rates increased sharply, suggesting that a situational factor had forced the cells to be rule breaking. In other words, the directivity of the neighborhoods’ pressure in the matrix had changed from obeying to rule breaking; the descriptive norm oriented toward rule breaking was formed. This means that rule-breaking behavior can be expanded by obscure behavior that reflects both individual factors such as their principles and situational factors such as their neighbor’s behavior. In pattern C, there was a clear threshold (M-prob = .20–.40) that the directivity of the majority’s pressure changed from obeying to rule breaking. In pattern B, however, the threshold was unclear, and there was no threshold in pattern A.
The results indicated that rule-breaking behavior spreads under the following conditions: (A) when people act according to not only situational factors (e.g., their neighbors’ behavior) but also individual factors (e.g., their attitudes); (B) when there are more deviant people than obedient people; (C) when the intensity of rule-breaking behavior is higher than that of rule-obeying behavior.
Condition A is important. If conditions B or C are fulfilled, rule-breaking behavior does not always spread. Whether rule-breaking behavior spreads or not depends on the probability that people act according to individual factors. At a certain probability threshold, rule-breaking behavior spreads drastically when people act based on individual factors; the spread of such behavior cannot be attributed solely to situational factors. There is only a fine line between rule breaking and obeying near the probability threshold.
Elaborating Models
Various Decision Matrices
The differences between M11 and M21 and between M12 and M22 represent the magnitudes of the neighbors’ effects on their holders. At an intensity of 1.0, the magnitude of the obeying neighbors is equal to the absolute value of the difference between M11 and M21 and the magnitude of the rule-breaking neighbor is the absolute value of the difference between M12 and M22. For example, in the ratio 2:3:1:5 (M11:M12:M21:M22), the difference between M11 and M21 is 1 and between M12 and M22 is 2. Therefore, the magnitude of the rule-breaking neighbors is twice as much as that of the obeying neighbors. In the present study, the values of M11, M12, M21, and M22 were either 0 or 1. Therefore, all magnitude values were identified as 1. Deguchi (2013) investigated college students’ decision matrices with a questionnaire; the range of values for the matrices was 1–7. Many payoff matrices would contain the same values (e.g., M11:M12:M21:M22 = 6:3:3:3) in the questionnaire data. Therefore, Deguchi expanded the definitions of “obedient” and “deviant” as follows: Obedient is determined when M11 > M21 and M12 > M22, M11 > M21 and M12 = M22, or M11 = M21 and M12 > M22; deviant is determined when M11 < M21 and M12 < M22, M11 < M21 and M12 = M22, or M11 = M21 and M12 < M22. People with M11 < M21 and M12 < M22 or M11 < M21 and M12 = M22 can break a rule even without rule-breaking neighbors. In addition, the definition “neutral” (M11 = M21 and M12 = M22) was added, in which the cell does not change its state according to its neighbors. This cell always maintains its state as long as it uses Rule 3-2 and changes its state in accordance with only Rule 3-1. There were some obedient (26%) or deviate (11%) students for “talking about subjects not concerned with the class” (Deguchi, 2013). Conducting simulation using their decision matrices is useful for determining the applicability of the simulation model.
Awareness of Neighbors’ Attitudes and Individuals’ Adaptations
In Rule 3-2, cells are unaware of their neighbors’ decision matrices (or principles). They decide their state based on only their own decision matrix. From the perspective of game theory, the model comprising rules 1, 2, 3-1, and 3-2 is classified as a game with incomplete information (Kreps, 1990). A model in which cells can be aware of their neighbors’ decision matrices should be examined because such a model would enable the analysis of many types of games. For example, if both a cell’s and its neighbors’ M11:M12:M21:M22s are 3:1:4:2, the decision matrix becomes a prisoner’s dilemma game (e.g., Barash, 2003; Ishihara & Kanai, 2002). If their M11:M12:M21:M22s are 3:2:4:1, the decision matrix becomes a chicken game (e.g., Barash, 2003; Ishihara & Kanai, 2002). Strategies of changing cells’ states should be developed to examine this model.
In a series of studies on game theory, each cell’s adaptation was calculated using a payoff matrix (e.g., Axelrod, 1984, 1997). The cells played a prisoner’s dilemma game with their neighbors and earned points based on the payoff matrix. The points were regarded as an index of the cells’ adaptations. In the present study, each cell had a decision matrix; therefore, the adaptation of the cell could be calculated with the decision matrix. The relationships among decision matrices, strategies of changing cells’ states, frequencies of rule-breaking behavior, and cells’ adaptations should be investigated to determine not only the factors that cause the spread of rule-breaking behavior but also the principles or strategies that are adaptive for people.
Subgroups
A large group such as a classroom usually contains some subgroups (e.g., Hymel, Wagner, & Butler, 1990; Ladd, 1983). French and Raven (1959) classified social powers into five types, for example, coercive and referent. These social powers imply that people in a subgroup have more influence on their members than out-group people. To examine the effects of these subgroups, a person’s strength should be added into Rule 1 such that “impact of a person = a person’s strength × intensity” and the strength should vary according to whether neighbor cells belong to a subgroup (Deguchi, 2008).
Generalizability and Limitations
The model used in the present study might be applicable to other types of social behavior such as the spread of fashion or public opinion (Nowak et al., 1990). The matrix in the present study was not a torus; however, using a torus matrix would be required for investigating these behaviors because these phenomena may spread globally. This model could provide a framework for analyzing the processes of interactions between different individuals from both microperspectives and macroperspectives. However, the model did not include all factors that could explain rule breaking or other behaviors, such as the influence of a teacher or group leader. It is important to consider the effect of these factors and to investigate what factors determine people’s attitudes, or “decision matrices.”
Footnotes
Author’s Note
I express appreciation to Hiroshi Toyota, Kohsuke Yamamoto, Haruka Imanishi, Saki Sunada, Chika Komatsu, Megumi Yoshinaga, and my colleagues.
Declaration of Conflicting Interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers 18730414, 22730508.