A Game-Theoretic Model for Bystanders’ Behaviour in Classes with Bullying ()
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:
(1)
(2)
(3)
(4)
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.
(5)
(6)
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
(7)
It follows that
(8)
where
.
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.
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Table 1. Changes to player gain by the number of reporters when selecting either behaviour R or S.
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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-
shion.
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.
(9)
2) For i such as
we have the following in equations.
,
. (10)
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Figure 3. A graph of
when increasing n and t while maintaining
.
Since
(11)
where
(12)
using the relation
(13)
it is left for us to prove
(14)
can be written as
(15)
From relations
(16)
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.
Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers 25350468.