Bullying is one of many serious problems in the field of education. The number of recognized incidents of bullying in 2013 was 185,860 according to the results of survey by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) in Japan . In 2013, compared to 2012, the total number of recognized incidents of bullying remained almost unchanged, however, bullying incidents increased at elementary schools. Furthermore, there were 240 student suicides in 2013 compared to 196 students in 2012, 3.8% of those had been caused by bullying compared 3.1% in 2012. These facts suggest that the damage caused by bullying had been aggravated and bullying was recognized as crime recently. Magata  indicated that most incidents of bullying are criminal acts and illustrated the types of bullying that are criminal activities. Morita  emphasised the importance of by standers’ behaviour.
In this paper, we consider both the retaliation cost and the reporting cost and focus on the bystander’s behaviour. he retaliation cost was dealt in Shibata et al.  and the reporting cost was dealt in Isada et al. . Actually, our model in this paper is a generalised model of both models  .
This paper comprises several sections in which different aspects of the bystander’s behaviour around bullying are discussed. Section 2 describes our model in detail. The Nash equilibrium within the model is analysed in section 3. Section 4 discusses numerical experiments with changes to class size and the impact of these changes on bystander’s behaviour. Our conclusions are presented in Section 5.
2. An Economic Model of by Standers’ Behaviour in Bullying
There are three kinds of students in this situation; the bully, the bullied child and by standers. In this paper we only focus on bystanders’ behaviour in a classroom where there is bullying. Suppose that that is n by standers 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 t students report the bullying.All students are initially granted a utility level w. When bullying occurs, students incur a negative externality (disutility) b. The retaliation cost c is incurred for students who select behaviour R, only when the bullying could not be stopped despite reporting it. The reporting cost e 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   is formulated shown in Table 1 . Each value shows the student’s gain in each case, where X denotes the number of reporters other than himself of herself.
3. Nash Equilibrium in the Economic Model
Each bystander play this game according to Table 1. Suppose each bystander has the same probability of reporting, q. When bystanders 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 bystander selecting behaviour R, and the Expected utility of when a bystander selecting behaviour S are expressed with the following equations.
is the state when all bystanders select behaviour R and is the state when all bystanders select behaviour S. When, the result is as. Let’s denote that .
As we saw in Figure 1, from and, there are two values of q which hold when and are satisfied, where, q0 is vertex of Conversely, if and, and have only one intersection. When, it is shown in Shibata et al. . If, either of b and e must be negative value. This indicates that when bullying occurs students obtain utility or they get some benefit from reporting the bully. Consequently, we assumed that both of band e are positive values because such a situation is not demonstrated appropriately by the bystander’s behaviour.
Table 1. Changes to student’s gain by the number of reporters when selecting either behaviour R or S.
Figure 1. Two intersections of and.
Let them denote,.
Figure 2 shows the relationship between expected utilities and and for by stander behaviour R and S where and.
As we saw in Figure 2, and always have two intersections for the range, occurs. This indicates that a free rider phenomenon occurs where many other students report bullying, but the student in question decides it is better not to report. When, and have only one intersection. This case shows a situation where a free rider phenomenon is disappeared. Based on the above, we can make the following proposition.
1) A pure strategy Nash equilibrium always exists in which no student reports bullying.
2) If is satisfied, there are two intersection and, then, we have two mixed strategies Nash equilibrium exists.
3) When, a range exists for q where. Conversely, when, is always true.
Figure 3 is a graph of and where the number of c is changing and the number of b and e are kept constant. Figure 4 is a graph of and where the number of e is changing and the number of band care kept constant.
When examining Figure 1 and Figure 4, we see that we can expand the range  of q by reducing, since is decreased and is increased. As we saw in Figure 3 the range  of q is expanded by reducing c. Based on the above, we can make the following propositions.
1) A decrease in due to an increase in b or a decrease in e causes a decrease in and an increase in.
2) A decrease in c causes a decrease in and an increase in.
4. The Impact of Changing the Class Size
Let us examine changes in and that occur at the two intersections with and when changing only n, the number of bystanders in the class, while the ratio of threshold to the number of bystanders is kept constant at. Figure 5 is a graph of where the number of n is changing. As the value of n becomes smaller, becomes smaller and becomes larger. Thus, the range  of q expands.
Figure 6 is a graph that shows the value of and for six 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 q expands. On the other hand, as class size n becomes larger, becomes larger and becomes smaller. The values of the upper limit decrease, and the lower limit increase, and the range
 of q shrinks. It is shown in Shibata et al.  that when, that is, is monotonically decreasing for n.
Figure 2. Expected utility when there are two intersections of and.
Figure 3. A graph of when increasing c while b and eare kept constant.
Figure 4. A graph of when increasing e while b and c are kept constant.
Figure 5. A graph of when increasing n and t while maintaining.
Figure 6. A graph of, when increasing n and t while maintaining.
In this paper we modelled the bystanders’ behaviour with bullying in a non-cooperative n-player game. The three costs are the cost of negative externality, the retaliation cost and the reporting cost, were considered in this model. If the probability of reporting q is larger than the lower limit, it may increase the chance that bullying is resolved. If the probability of reporting q is larger than the upper limit, there is a possibility that the free rider phenomenon may arise. This shows that in order to expand the range of q, it is useful to raise the negative externality b associated with continued bullying and to reduce the retaliation cost c or the reporting cost e of reporting on bullies. We showed by making, the ratio of a threshold number of reporters to the number of bystanders, constant and decreasing nit becomes possible to decrease the lower limit and to increase the upper limit of the probability of reporting bullying. This shows that if class sizes are smaller, it may increase the chance that bullying is stopped because the range of q is expanded.
This work was supported by JSPS KAKENHI Grant Numbers 25350468.
 MEXT (2014) Results of the 2013 Survey of Problem Behaviour of Students and Various Issues with Instructing Students. http://www.mext.go.jp/b_menu/shingi/chukyo/chukyo3/gijiroku/__icsFiles/afieldfile/2014/12/18/1354117_5.pdf
 Shibata, A., Mori, T., Okamura, M. and Soyama, N. (2008) An Economic Analysis of Apathetic Behavior: Theory and Experiment. The Journal of Socio-Economics, 37, 90-107. http://dx.doi.org/10.1016/j.socec.2006.12.026
 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. http://dx.doi.org/10.4236/jss.2015.39015