# Detection of Decision-Making Manipulation in the Pairwise Comparison Method
## 3 Preliminaries
### Inconsistency  
It is commonly assumed that in the optimal scenario, experts comparing each pair of alternatives assign \(c_{ij}\), a value corresponding to the actual priority ratio between the two alternatives \(a_{i}\) and \(a_{j}\). It is, therefore, natural to expect that  
\[c_{ij}=\frac{w(a_{i})}{w(a_{j})}. \tag{7}\]  
If \(c_{ij}=c_{ik}c_{kj}\) holds for all \(i,j,k\in\{1,\ldots,n\}\), the PC matrix \(\mathbf{C}=\left[c_{ij}\right]\) is said to be multiplicatively consistent. Otherwise, if this condition is not satisfied, \(\mathbf{C}\) is considered inconsistent.  
Each priority vector \(w\) induces exactly one consistent PC matrix \(\mathbf{C}\) and vice versa: a consistent PC matrix \(\mathbf{C}\) induces exactly one vector \(w\). Thus, for a consistent pairwise comparison matrix, both methods mentioned, the EVM and GMM, result in the same vector \(w\).  
Like the PC matrices themselves, their inconsistency is quantitative. Thus, one may be tempted to quantify to what extent the PC matrix (decision data in the form of a set of pairwise comparisons of alternatives) is inconsistent.  
One of the most recognizable and frequently used indexes is [PERSON]'s consistency index \(CI\)[19], defined as follows:  
\[CI(\mathbf{C})=\frac{\lambda_{max}-n}{n-1}. \tag{8}\]  
Additionally, [PERSON] also introduced the consistency ratio \(CR\)[19], defined as follows:  
\[CR(\mathbf{C})=\frac{CI(\mathbf{C})}{RI(\mathbf{C})}, \tag{9}\]  
where \(RI(\mathbf{C})\) is the average inconsistency for the entirely random \(PC\) matrix with the exact dimensions as \(\mathbf{C}\). As proposed by [PERSON], every \(PC\) matrix \(\mathbf{C}\) for which \(CR(\mathbf{C})>0.1\) is deemed too inconsistent to form the basis of a ranking.  
Another consistency index used to determine inconsistency is the Geometric Consistency Index [56], defined as follows:\[GCI(C)=\frac{2}{(n-2)(n-1)}\sum_{i<j}ln^{2}\left(c_{ij}\frac{w(a_{j})}{w(a_{i})} \right), \tag{10}\]  
where \(w\) is derived using the GMM method (6). Both the \(CI\) and \(GCI\) depend on the method of calculating the vector of weights. There are also indexes based on the pairwise comparison matrix itself. An example of such a solution is the determinant inconsistency index \(CI^{*}\) proposed by _[PERSON]_ and _[PERSON]_[57], defined as follows:  
\[CI^{*}(C)=\begin{cases}0&n<3\\ \det C&n=3\\ \frac{1}{(3)}\sum_{\begin{subarray}{c}i\
eq j,i\
eq k-1\\ i\
eq j,i\
eq k,j\
eq k\end{subarray}}^{n}\det C_{ijk}&n>3\end{cases}, \tag{11}\]  
where \(\det C_{ijk}\) is the minor of a \(3\times 3\) sub-matrix \(C\), defined as  
\[\det C_{ijk}=\frac{c_{ik}}{c_{ij}c_{jk}}+\frac{c_{ij}c_{jk}}{c_{ik}}-2. \tag{12}\]  
This approach first calculates the local inconsistency values determined by the \(\det C_{ijk}\). Then, the global inconsistency value for matrix \(C\) is calculated as an arithmetic mean of individual local inconsistency indicators.  
The threshold value below which the matrix \(C\) is considered sufficiently consistent can be calculated empirically following the approach proposed by [19]. There are many different inconsistency indexes for PC matrices. There are at least a dozen different inconsistency indexes for pairwise comparison matrices, including [PERSON]'s index [58], [PERSON]'s index [59], [PERSON]'s index [60], [PERSON]'s ordinal consistency indicator [61], and others [22; 62]. For a \(3\times 3\) matrix, monotonic increasing functional relationships exist between multiple indexes, including [PERSON]'s \(CI\) and the \(GCI\)[63]. The properties of inconsistency indexes are the subject of numerous studies [64; 65; 66].