c. Mathematical Model of Size of Power and Cohesion Force

c. Mathematical Model of Size of Power and Cohesion Force

I consider the content I explained based on [Diag.2.A.6] to be only a means of providing the intuition necessary to easily understand the size of power and cohesion force, not that all values (the area of the circle and the area of the rectangle) depicted geometrically here accurately represent the precise values to be used in the mathematical model. (It is only an approximation.) This is because the basic values in the mathematical model can be defined more simply and efficiently.

To form a simpler mathematical model, I will consider the relative size of political power between a and b to be M':N'=M:N. In reality, if line segment ab is long, the ratio of the area of power quantity of a to that of b converges to M:N as shown in [Diag.2.A.6], but if line segment ab becomes shorter, the error increases. That is, the ratio of the area of triangle acd to triangle bcd in the diagram is not exactly M:N.

 

Relative Size of Power

 

Let's call the political influence that political actors a and b can exert on each other "(relative) political power," when the political capacity of a and b is different, the size of power each has on each other will also be different^[Ch.2.9a]. When a and b's political capacity are \( \vec{L_a} \) and \( \vec{L_b} \), respectively, the size of political power \( \vec{P_{WR_{a/b}}} \)  that a holds over b is calculated as follows.

 

       [Fmla.2.2]           \( \vec{P_{WR_{a/b}}}  =_{def}     \frac{ \vec{L_a} }{ \vec{L_b} }\)

 

Here, the right-hand side of the division operation of the trivector in [Fmla.2.2] is not defined in general vector calculus. Therefore, in this book, it will be defined as the division operation of each component.

 

        [Fmla.2.3]           \( \frac{ \vec{L_a} }{ \vec{L_b} }\)   \(=_{def} \)   \( \begin{bmatrix} \frac{A_a}{A_b} \\ \\ \frac{E_a}{E_b} \\ \\ \frac{\vec{I_a}}{\vec{I_b}}  \end{bmatrix} \)

 

Here is an example of what this mathematical model means. Let's say a is a morally respected military officer, and b is a corrupt military officer who has made money through unethical means. However, a and b are both military officers of the same rank. In this case, a is a military officer, so their armed capacity is large, and they are also a respected military officer, so their ideological capacity is also large. Let's say that value is 10. However, their money is limited, so their economic capacity is only about 2. On the other hand, b is also a military officer, so their armed capacity is the same, and they also have money, so their economic capacity is also large. However, they have become corrupt, so their ideological capacity is very limited and only amounts to 1. Therefore, the survival capacity of a and b is as follows.
 
                    \( \vec{L_a} \)   \(= \)   \( \begin{bmatrix} 20 \\ \\ 2 \\ \\ 10  \end{bmatrix} \)  and    \( \vec{L_b} \)   \(= \)   \( \begin{bmatrix} 20 \\ \\ 20 \\ \\ 1  \end{bmatrix} \) 

Then the political power that a has over b () and the political power that b has over a () are as follows.

                     \( \vec{P_{WR_{a/b}}} \)   \(= \)   \( \begin{bmatrix} 1 \\ \\ 0.1 \\ \\ 10  \end{bmatrix} \)  and      \( \vec{P_{WR_{b/a}}} \)   \(= \)   \( \begin{bmatrix} 1 \\ \\ 10 \\ \\ 0.1  \end{bmatrix} \) 

The political power () that a has over b and the political power that b has over a are as follows:
In terms of military power, a has a score of 1 and a score of 0.1 in terms of economic power, but a score of 10 in terms of ideological power. This can be condensed, considering the spatiotemporal effects of the Samjae capacities mentioned earlier. That is, in the short term, a has equal political influence over b, but in the medium term, a is politically suppressed and under pressure, and in the long term, a will exert superior political influence. On the other hand, in the short term, b has equal political influence over a, and in the medium term, b will exert superior political influence, but in the long term, b will be politically dominated.

 

Cohesion force of power relation

On the other hand, the power relations among political actors become stronger as the necessity for cooperation and profit increases [Ch.2.9b]. Let the cooperation profit between a and b be  and the measurement of a's power regarding b be . Then  is calculated as follows.

 

        [Fmla.2.4]             \( \vec{S_{a/b}}  =_{def}     \frac{ \vec{H_{a,b}} }{ \vec{L_a} }\)

 

The measurement of b's power regarding a,  is calculated in the same way. Now let the cohesion force between a and b be  and  be the amount of their cooperation activities. This cohesion force can be modeled as follows.

 

       [Fmla.2.5]              \( \vec{S_{a,b}}  =_{def}     \frac{ \vec{H_{a,b}} }{ \vec{L_a} + \vec{L_b} }\)

 

Now, let's consider the mathematical model for the cohesion force of the entire group. This refers to the measurement of power not only in the relationship between two individuals, a and b, but in relationships involving more than three people. In everyday language, it can be understood as the attachment to a group or organization. The higher the attachment, the stronger the tendency to follow the rules of the organization and its power holders. This is what is meant by the "cohesion force."

 

If there is a group { \(m_{1}\) , \( m_{2}\) , ... , \( m_{n}\) } composed of n members, and the capacity of each \( m_{i} \)  is \( \vec{L_{i}} \) , the size of the cohesion force of group K is as follows.

 

        [Fmla.2.5]               \( \vec{S_{a,b}}  =_{def}     \frac{ \vec{H_{a,b}} }{ \sum \vec{L_i} }\)
                              ( \( \vec{H_{K}} \) is the quantity of total cooperation in group \(K\)  )

 

\( \vec{H_{K}} \) represents the amount of cooperation activity, which refers to the profit obtained by the members of group K through their cooperation. \( \vec{H} \)   is also a survival capability, just like \( \vec{L} \)  , and is a three-dimensional vector. This capability belongs indirectly to each member of the group, but directly to the group as a whole. Therefore, the strength of the power that each member  feels depends on how much \( \vec{H_{K}} \) is distributed.

In this aspect, the explanation of the intuitive model [Diag.2.A.6] and the more efficient mathematical model differ. In the intuitive model, square acbd represents the amount of cooperation activity, not the result of cooperation activity.

It may seem that the model only considers the size of profit from cooperation, not the necessity of it, when looking at the model superficially. However, the necessity of the profit from cooperation can be somewhat modeled as a three-vector (Samjae capacities) that composes survival capacities. This will be discussed later.