# Algorithm for Option Number Selection in Stochastic Paired Comparison Models
## 3 Comparison of Models Allowing Different Options in Choices from a Theoretical Aspect  
### The Case of \(s=2\) and \(s+2=4\)  
The early models of paired comparisons allowed for only two options: 'worse' or 'better' [34; 39]. [PERSON] himself argued that, based on experience, people are able to make decisions between these two options and there is not necessarily a need for more. In some sports, such as tennis or basketball, matches can only end in a win or a loss. However, knowing the result, we can sometimes add nuance: for example, the victory was decisive, or one side barely defeated the other. In other words, there may be justification for additional decision options, such as a narrow victory ('slightly better'/'slightly defeated') or a decisive victory ('much better'/'much worse'). These models, which allow for and process more information, could yield more nuanced results. The data's structure is contained in the data matrices but the data matrices \(A^{(4)}\) and \(A^{(2)}\) can be transformed into each other. We follow the following natural transformation rules:  
\[A^{(2)}_{i,j,1}=A^{(4)}_{i,j,1}+A^{(4)}_{i,j,2} \tag{10}\]  
\[A^{(2)}_{i,j,2}=A^{(4)}_{i,j,3}+A^{(4)}_{i,j,4} \tag{11}\]  
This means that if we want to transform the four-option model into a two-option model, we merge the options'slightly better' and'much better' into the 'better' option. In the opposite case, we split the option 'better' into two parts, and the numbers of decisions will be divided into two parts. We require that \(0<A^{(4)}_{i_{1},j_{1},3}\) and \(0<A^{(4)}_{i_{2},j_{2},4}\) for at least one pair of the indices \((i_{1},j_{1})\) and \((i_{2},j_{2})\), respectively. This requirement ensures that the data \(A^{(4)}_{i,j,k}\) actually belong to the four-option model. Now, we compare these models based on their evaluability. First, we track which theoretical cases are possible:  
1. Data are evaluable only in the two-option model. 2. Data are evaluable only in the four-option model. 3. Data are evaluable in both models. 4. Data are not evaluable in either model. (Case A) We can prove that this theoretical case may not happen. For this, we summarize the conditions for evaluability in the case of the two-option and the four-option models. For this, we need two graph definitions. **Definition 1** (The graph belonging to \(A^{(2)}\)). : _The nodes are the objects. There is a directed edge from \(i\) to \(j\) (denoted by \(\rightarrow\)) if \(0<A^{(2)}_{i,j,2}\), i.e., there is a decision according to which object \(i\) is 'better' than object \(j\). This graph will be denoted by \(GC^{(A^{(2)})}\)._  
**Definition 2** (The graph belonging to \(A^{(4)}\)). : _The nodes are the objects. Two types of directed edges are contained in the graph. There is a directed edge from \(i\) to \(j\) (denoted by \(\rightarrow\) and called the'slightly better' edge) if \(0<A^{(4)}_{i,j,3}\), i.e., there is a decision according to which object \(i\) is'slightly better' than object \(j\). The other type of directed edge is defined as follows: there is a double directed edge from \(i\) to \(j\), denoted by \(\rightarrow\) and called the'much better edge', if \(0<A^{(4)}_{i,j,4}\). In other words, there exists a decision according to which object \(i\) is'much better' than object \(j\). This graph will be denoted by \(\textsc{GC}^{(A^{(4)})}\)._  
In the case of the two-option model, the necessary and sufficient condition for evaluability, given by Ford, for logistic distributions and generalized for strictly log-concave distributions in [47] is the following:  
* C.2.1. For every nonempty partition of \(S\) and \(\overline{S}\) of the set {1, 2, \(\ldots\), \(n\)} (\(S\cup\overline{S}=\{1,2,\ldots,n\}\), \(S\cap\overline{S}=\emptyset\)), there are two (not necessarily different) index pairs \((i_{1},j_{1})\) and \((i_{2},j_{2})\), \(i_{1},i_{2}\in S,j_{1},j_{2}\in\overline{S}\), for which \(0<A^{(2)}_{i_{1},j_{1},1}\) and \(0<A^{(2)}_{i_{2},j_{2},2}\). In other words, these subsets are connected in both directions.