# Validation of Value-Driven Token Economy: Focus on Blockchain Content Platform
## 3 Research Methodology
### AHP Analysis
The response consistency is critical to the validity of the AHP analysis. The significance of each response is determined by combining the results from the individual raters and focusing on responses with a consistency ratio (derived from the pairwise comparison matrix and geometric mean calculation) of less than 0.1. Responses with consistency ratios above 0.1 are excluded to ensure the validity of the analysis. The pairwise comparison matrix (\(A\)), generated by the AHP process, reflects the relative importance of the 'n' factors being compared within each stratum. For elements (\(A_{1}\), \(A_{2}\), \(\ldots\), \(A_{n}\)), the results of the pairwise comparisons between \(a_{\text{i}}\) and \(a_{\text{i}}\) are labeled as \(a_{\text{ij}}\) to form the matrix \(A=(a_{\text{ij}})\). If each factor is important in \(w_{\text{i}}\) (\(I=1\),\(2\), \(\ldots\), \(n\)), matrix \(A=(a_{\text{ij}})\) is used to calculate these weights to ensure the accuracy and validity of the derived values. \[A= \begin{pmatrix}a_{\text{ij}}\end{pmatrix} = \begin{bmatrix}w_{1}/w_{1}&w_{1}/w_{ 2}&\cdots&w_{1}/w_{n}\\ w_{2}/w_{1}&w_{2}/w_{2}&\cdots&w_{2}/w_{n}\\ \vdots&\vdots&\ldots&\vdots\\ w_{n}/w_{1}&w_{n}/w_{2}&\cdots&w_{n}/w_{n}\end{bmatrix} = \begin{bmatrix}1&w_{ 1}/w_{2}&\cdots&w_{1}/w_{n}\\ w_{2}/w_{1}&1&\cdots&w_{2}/w_{n}\\ \vdots&\vdots&\ldots&\vdots\\ w_{n}/w_{1}&w_{n}/w_{2}&\cdots&1\end{bmatrix}  \tag{3}\]  
In the AHP analysis, the relative importance of each factor is described by a ratio. For example, \(w_{1}/w_{1}\) compares itself to \(A_{1}\), resulting in a logical value of 1. Similarly, \(w_{1}/w_{2}\) describes the relative importance of \(A_{1}\) compared to \(A_{2}\), and \(w_{1}/w_{n}\) describes the relative importance of a compared to \(A_{1}\). Multiplying Equation (3) by the column vector \(w=[w_{1},w_{2},\ldots,w_{n}]\), which, again, describes the approximate value of the relative importance between the evaluated factors, results in the following equation:  
\[ \begin{bmatrix}w_{1}/w_{1}&w_{1}/w_{2}&\cdots&w_{1}/w_{n}\\ w_{2}/w_{1}&w_{2}/w_{2}&\cdots&w_{2}/w_{n}\\ \vdots&\vdots&\ldots&\vdots\\ w_{n}/w_{1}&w_{n}/w_{2}&\cdots&w_{n}/w_{n}\end{bmatrix} \cdot \begin{bmatrix}w_{ 1}\\ w_{2}\\ w_{2}\\ w_{n}\end{bmatrix} = \begin{bmatrix}nw_{1}\\ nw_{2}\\ \vdots\\ nw_{n}\end{bmatrix}  \tag{4}\]  
Using the eigenvalue calculation method, this can be shown as  
\[A\cdot w=\lambda_{\text{max}}\cdot w \tag{5}\]  
The maximum eigenvalue of the pairwise comparison matrix (\(A\)), labeled as \(\lambda_{\text{max}}\), is important for deciding these weights. This can be shown by solving the characteristic equation for a nonzero solution in a system of \(n\) simultaneous equations to find the value of \(\lambda_{\text{max}}\) that satisfies the equation below:  
\[|A-\lambda I|=0 \tag{6}\]  
The value of \(\lambda_{\text{max}}\) is always greater than or equal to n, the number of factors, and the closer the value of \(\lambda_{\text{max}}\) is to \(n\), the more consistent the pairwise comparison matrix (\(A\)) is seen to be. The consistency of these comparisons is quantified using \(CI\) and \(CR\), which are calculated as follows:  
\[CI=\frac{\lambda_{\text{max}}-n}{(n-1)} \tag{7}\]  
\[CR=\frac{CI}{RI} \tag{8}\]where n stands for the matrix dimension, and \(RI\) stands for the Saaty random index [62], which depends on the matrix dimension. \(RI\) is the average consistency index obtained from randomly generated reciprocal matrices and serves as a standard benchmark for acceptable levels of consistency. \(RI\) values are shown in Table 5. A \(CR\) of 0.1 or less indicates allowable consistency in the survey returns, which validates the reliability of the AHP analysis. For each survey item, a weight (\(w\)) was decided using the eigenvalue method, and then the overall weight was calculated using the geometric mean method. The \(CI\), \(CR\), and \(\lambda\)max values were eliminated by taking the arithmetic mean of the values obtained for each survey item to yield the overall value.