## 2 Literature Review
### [PERSON] et al.'s Conflict Analysis Model
### [PERSON] et al. 's Conflict Analysis Model  
[PERSON] et al. in Reference [4] tried to answer [PERSON]'s questions and focussed on answering the first and second questions raised by [PERSON] in Reference [2]. Let \(U=\left\{u_{1},...,u_{m}\right\}\) be the universe of agents and \(V=\left\{a_{1},...,a_{n}\right\}\) be the universe of issues in the conflict situation. For any subset \(A_{2}\) of \(V\), we call \(A_{2}\) the feasible strategy for a conflict situation. Let the set \(f=\left\{f^{+},f^{-}\right\}\) be the collection of set valued mappings defined as follows:  
\[f^{+}:U\to P\left(V\right),\,f^{+}\left(u\right)=\left\{a\in V\mid f \left(u,a\right)=+\right\},\text{ for all }u\in U\]  
and:  
\[f^{-}:U\to P\left(V\right),\,f^{-}\left(u\right)=\left\{a\in V\mid f \left(u,a\right)=-\right\},\text{ for all }u\in U.\]  
The mapping \(f^{+}\) gives the subset of issues with which agent \(u\) agrees while the mapping \(f^{-}\) gives the subset of issues with which the agent \(u\) disagrees. For any feasible strategy \(A_{2}\in\mathcal{P}\left(V\right)\) on an issue, the approximations are:  
\[\underline{apr_{f}^{+}\left(A_{2}\right)}=\left\{u\in U\mid f^{+} \left(u\right)\subseteq A_{2}\right\},\,\,\,\,\underline{apr_{f}^{-}\left(A_ {2}\right)}=\left\{u\in U\mid f^{-}\left(u\right)\subseteq A_{2}\right\},\] \[\overline{apr_{f}^{+}}\left(A_{2}\right)=\left\{u\in U\mid f^{+} \left(u\right)\cap A_{2}\
eq\emptyset\right\},\,\,\,\overline{apr_{f}}\left(A_ {2}\right)=\left\{u\in U\mid f^{-}\left(u\right)\cap A_{2}\
eq\emptyset\right\}.\]  
**Example 1**. : _Consider the Middle East conflict as in Table 1._  
_Let \(A_{2}=\left\{a_{2},a_{3},a_{5}\right\}\subseteq V.\) Then \(\underline{apr_{f}^{+}\left(A_{2}\right)}=\left\{u_{6}\right\}\), \(\overline{apr_{f}^{+}}\left(A_{2}\right)=\left\{u_{1},u_{6}\right\}\), \(\underline{apr_{f}^{-}}\left(A_{2}\right)=\left\{u_{4},u_{6}\right\}\), and \(\overline{apr_{f}^{-}}\left(A_{2}\right)=\left\{u_{2},u_{3},u_{4},u_{5},u_{6}\right\}.\)_  
_According to [PERSON] et al. [4]:_  
_Agreement subset \(=R_{f}^{+}\left(A_{2}\right)=\underline{apr_{f}^{+}\left(A_{2}\right)}- \underline{apr_{f}^{-}}\left(A_{2}\right)=\emptyset\),_  
_Disagreement subset \(=R_{f}^{-}\left(A_{2}\right)=\underline{apr_{f}^{-}}\left(A_{2}\right)- \underline{apr_{f}^{+}}\left(A_{2}\right)=\left\{u_{4}\right\},\)_  
_Neutral subset \(=U-\left(R_{f}^{+}\left(A_{2}\right)\cup R_{f}^{-}\left(A_{2}\right)\right)= \left\{u_{1},u_{2},u_{3},u_{5},u_{6}\right\}.\)_  
\begin{table}
\begin{tabular}{l c c c c c} \hline \(\boldsymbol{U}\backslash\boldsymbol{V}\) & \(\boldsymbol{a_{1}}\) & \(\boldsymbol{a_{2}}\) & \(\boldsymbol{a_{3}}\) & \(\boldsymbol{a_{4}}\) & \(\boldsymbol{a_{5}}\) \\ \hline  
It is observed that:  
1. For the feasible strategy \(A_{2}=\left\{a_{2},a_{3},a_{5}\right\}\in P\left(V\right)\) using [PERSON] et al. 's technique, the agreement subset \(=\emptyset\), the disagreement subset \(=\left\{u_{4}\right\}\), and the neutral subset \(=\left\{u_{1},u_{2},u_{3},u_{5},u_{6}\right\}.\) This is because, in the information system (Table 1) the values for \(u_{4}\) in the columns of \(a_{2},a_{3}\) and \(a_{5}\) are in disagreement, that is \"\(-\)\". Additionally, the case of \(u_{5}\) is not different from \(u_{4}\), but in Reference [4] it is treated as neutral, so the agents of the same behavior have been grouped in different coalitions, which causes confusion. 2. For the feasible strategy \(A_{2}=\left\{a_{3}\right\}\in P\left(V\right),\) the agreement subset \(=\emptyset\), the disagreement subset \(=\left\{u_{6}\right\}\), and the neutral subset \(=\left\{u_{1},u_{2},u_{3},u_{4},u_{5}\right\}.\) This is because, in the information system (Table 1) the attitude of \(u_{6}\) in the column of \(a_{3}\) is in disagreement, that is \"\(-\)\". Also the cases of \(u_{2},u_{3},u_{4}\) and \(u_{5}\) are not different from \(u_{6}\), but according to Reference [4], they should be in neutral, so the agents of the same character are grouped in different coalitions, which is an infirmity of the [PERSON] et al.