## Appendix A
By Equations (21) and (23)  
\[CI\left(H_{g}\right)=\frac{1}{dn^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{s=1}^{d} \left(h_{ij,g}^{\sigma(s)}\times\widetilde{h}_{ji,g}^{\sigma(s)}\right)+\frac{ 2}{nd(n+1)}\sum_{i=1}^{n}\sum_{j\geq i}^{n}\sum_{s=1}^{d}\left|p_{ij,g}^{\sigma( s)}-\widetilde{p}_{ij,g}^{\sigma(s)}\right|. \tag{12}\]Now, first we deal with multiplicative preference degrees  
\[\left(h_{ij,g}^{\sigma(s)}\times\widetilde{h}_{ji,g}^{\sigma(s)}\right) = \left(\prod_{k=1}^{m}\left(\overline{h}_{ij,k}^{\sigma(s)}\right)^ {\omega_{k}}\times\prod_{l=1}^{n}\left(h_{ji,g}^{\sigma(s)}\times h_{li,g}^{ \sigma(s)}\right)^{\frac{1}{n}}\right)\] \[= \left(\prod_{k=1}^{m}\left(\overline{h}_{ij,k}^{\sigma(s)}\right) ^{\omega_{k}}\times\prod_{l=1}^{n}\left(\prod_{k=1}^{m}\left(\overline{h}_{j,k }^{\sigma(s)}\right)^{\omega_{k}}\times\prod_{k=1}^{m}\left(\overline{h}_{li,k }^{\sigma(s)}\right)^{\omega_{k}}\right)^{\frac{1}{n}}\right)\] \[= \left(\prod_{k=1}^{m}\left(\overline{h}_{ij,k}^{\sigma(s)}\times \prod_{l=1}^{n}\left(\overline{h}_{j,l,k}^{\sigma(s)}\times\overline{h}_{li,k }^{\sigma(s)}\right)^{\frac{1}{n}}\right)^{\omega_{k}}\right)\]  
thus, by Lemma 2  
\[\left(h_{ij,g}^{\sigma(s)}\times\widetilde{h}_{ji,g}^{\sigma(s)}\right)\leq \sum_{k=1}^{m}\omega_{k}\left(\overline{h}_{ij,k}^{\sigma(s)}\times\widetilde{ h}_{ji,k}^{\sigma(s)}\right),\]  
therefore Equation (A4) implies  
\[\frac{1}{dn^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{s=1}^{d}\left(h_{ij,g}^{ \sigma(s)}\times\widetilde{h}_{ji,g}^{\sigma(s)}\right)\leq\max_{k}\left( \frac{1}{dn^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{s=1}^{d}\left(\overline{h}_ {ij,k}^{\sigma(s)}\times\widetilde{h}_{ji,k}^{\sigma(s)}\right)\right).\] (A5)  
Consider,  
\[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]0 \[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]1  
For the inductive step, suppose the following  
\[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]2 \[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]3  
then  
\[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]4  
by supposition,  
\[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]5 \[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]6 \[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]7  
Hence, by principle of mathematical induction for all \(t\in\mathbb{N}\)  
\[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]8 (A6) \[\left|\left(p_{ij}^{(t+1)}\right)^{\sigma(s)}-\left(\widetilde{p}_{ij}^{(t+1)} \right)^{\sigma(s)}\right|<\left|\left(p_{ij}^{(t)}\right)^{\sigma(s)}-\left( \widetilde{p}_{ij}^{(t)}\right)^{\sigma(s)}\right|. \tag{11}\]9Therefore,  
\[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]0  
Equations (105) and (107) imply  
\[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]1  
\(\qed\)  
Proof of Theorem 8.: By Equations (4), (5) and (34),  
\[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]2  
Consider,  
\[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]3 \[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]4 \[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]5 \[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]6 \[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]7  
Additionally,  
\[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]8 \[CI\left(H^{(t+1)}\right)<CI\left(H^{(t)}\right),\]9 \[CI\left(H^{\infty}\right)=\lim_{t\rightarrow\infty}CI(H^{(t)})=\inf\{CI(H^{(t )}):t=1,2,3,...\}.\]0  
Now, by Equations (21) and (32)  
\[CI\left(H^{\infty}\right)=\lim_{t\rightarrow\infty}CI(H^{(t)})=\inf\{CI(H^{(t )}):t=1,2,3,...\}.\]1  
virtue of Equations (108) and (109) provide us the conclusion that  
\[CI\left(H^{\infty}\right)=\lim_{t\rightarrow\infty}CI(H^{(t)})=\inf\{CI(H^{(t )}):t=1,2,3,...\}.\]2  
with this, proof is complete.