A Game-Theoretic Model for Bystanders’ Behaviour in Classes with Bullying


In this paper, the behaviour of bystanders in a classroom in which bullying is occurring is analyzed using Game theory. We focus on bystander’s behaviour and formulate a threshold model. Our analysis shows that as class sizes become smaller, the probability of bullying being stopped increases.

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Isada, Y. , Igaki, N. and Shibata, A. (2015) A Game-Theoretic Model for Bystanders’ Behaviour in Classes with Bullying. Open Journal of Social Sciences, 3, 97-102. doi: 10.4236/jss.2015.39015.

1. Introduction

According to a survey on problematic behaviours by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) in Japan, the number of recognized incidents of bullying at all grade levels nationwide in 2012 was 198,108, 2.82 times than that of the previous year. This represents the largest number of recorded incidents since the survey began [1]. Additionally, there were 196 student suicides in 2012, of which 3.1% were understood to have resulted from bullying; this made the issue of bullying a serious social problem that must be solved.

Morita [2] emphasised the importance of bystanders’ behaviour. Shibata et al. [3] conducted an economic analysis on the behaviour of bystanders. Glass and Smith [4] established that a reduced class size can be expected to produce academic achievement. Additionally, Smith and Glass [5] have shown that small class size is effective in improving student attitudes and behaviour.

This paper comprises several sections in which different aspects of bullying analyses are discussed. Section 2 explains our model in detail. Section 3 analyses the Nash equilibrium within the model. Section 4 discusses numerical experiments with changes to class size and the impact of these changes on behaviour. Section 5 summarizes our results.

2. A Game-Theoretic Model for Bullying

There are three kinds of people in this situation: the bully, the bullied child and bystanders. In this paper we only focus on bystanders’ behaviour in a class where there is bullying. Suppose that there are bystanders in the class each bystander can take behaviour R, where a student reports bullying to a teacher, or behaviour S, where a student does not report the bullying. Bullying is resolved when more than students report the bullying. All players are initially granted a utility level. When bullying occurs, players incur a negative externality (disutility). Cost is constantly incurred for student who selects behaviour R, regardless of whether bullying is stopped or not.

Then, a non-cooperative n-person game model [6] [7] is formulated, shown in Table 1 . Each value shows the player’s gain in each case, where X denotes the number of reporters other than himself of herself.

3. Nash Equilibrium in the Bullying Model

Each bystander play this game according to Table 1 . Suppose each bystander has the same probability of reporting, q. When players other than oneself select behaviour R with a probability q, the probability of case 1, 2 and 3 are respectively as shown below:





The Expected utility of when a player selecting behaviour R, and the Expected utility of when a player selecting behaviour S are expressed with the following equations.



is the state when all players select behaviour R and is the state when all players select behaviour S. When, the result is as.

From Equation (2), we have, and


It follows that



Figure 1 shows that there are two values of which hold when. Let them denote,.

Figure 2 shows the relationship between Expected utilities and and for player behaviours R and S where.

As we saw in Figure 2, and have two intersections for the range, occurs. This indicates that a free rider phenomenon occurs where many other players report bullying, but the player in question decides it is better not to report. Based on the above, we can make the following proposition.

Table 1. Changes to player gain by the number of reporters when selecting either behaviour R or S.

Figure 1. Two intersections of and.

Figure 2. Expected utility when there are two intersections of and.

Proposition 1

1) A pure strategy Nash equilibrium always exists in which no player reports bullying. Only when is true, two mixed strategies Nash equilibrium exists.

2) When, segment exists for q where. Conversely, when

it is always true.

When examining Figure 1, we see that we can expand the range of by reducing, since is decreased and is increased. Based on the above, we can make the following propositions.

Proposition 2

An increase in b or a decrease in due to a decrease in e causes a decrease in and an increase in.

By differentiating both sides of results in. Because satisfies

and satisfies, , , and are solved in similar fa-


4. Behavior Resulting from Changing the Number of Bystanders in the Class

Let us examine changes in and that occur at the two intersections with and when changing only, the number of bystanders in the class, while the ratio of threshold to the number of bystanders is kept constant at. Figure 3 is a graph of where the number of is changing. As the value of becomes smaller, becomes smaller and becomes larger. Thus, the range of expands.

Figure 4 is a graph that shows the value of and for four cases, (n, t) = (20, 10), (40, 20)…, where the ratio of keeps a constant. As becomes smaller, becomes smaller and becomes larger. Again, the range of expands. On the other hand, as class size becomes larger, becomes larger and becomes smaller. The values of the upper limit decrease, and the lower limit increase, and the range of shrinks.

Proposition 3.

1) For, the following relations hold.


2) For i such as we have the following in equations.


. (10)

Figure 3. A graph of when increasing n and t while maintaining.

Figure 4. A graph of, when increasing and t while maintaining.



where (12)

using the relation (13)

it is left for us to prove (14)

can be written as


From relations


we obtain, then the proof is completed.

Proposition 3 (2) shows that adopting smaller class sizes is effective for reducing bullying. It gives the effectiveness of small-group education.

5. Conclusion

In this paper we modelled the behaviour of bystanders of students in a non-cooperative n-player game. We showed by making, the ratio of a threshold number of reporters to the number of bystanders, constant and decreasing becomes possible to decrease the lower limit and to increase the upper limit of the probability of reporting bullying. If class sizes are smaller, the number of bystanders should be fewer. This shows the possibility of eliminating bullying by using the smaller number of bystanders. Note that the reason we insist that smaller classes are better, not because it is easier for a teacher to manage smaller classes. More intuitively, Proposition 3 shows that bystanders can report bullying more easily if they are in a smaller class.

Furthermore, to expand the range of, it is useful to raise the disutility associated with continued bullying and to reduce the cost e of reporting on bullies.

Finally, this study has demonstrated the existence of the “free riding” phenomenon: if the majority of other people report, it is advantageous for any given person not to do so.


This work was supported by JSPS KAKENHI Grant Numbers 25350468.

Conflicts of Interest

The authors declare no conflicts of interest.


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[2] Morita, Y. (2010) Ijime to ha nani ka [What Is Bullying]. Chuko Shinsho, Tokyo.
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[5] Smith, M.L. and Glass, G.V. (1980) Meta-Analysis of Research on Class Size and Its Relationship to Attitudes and Instruction. American Educational Research Journal, 17, 419-433. http://dx.doi.org/10.3102/00028312017004419
[6] Nash, J. (1950) Equilibrium Points in n-Person Games. Proceed-ings of the National Academy of Sciences of the United States of America, 36, 48-49. http://dx.doi.org/10.1073/pnas.36.1.48
[7] Nash, J. (1951) Non-Cooperative Games. Annals of Mathematics Second Series, 54, 286-295. http://dx.doi.org/10.2307/1969529

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