# Algorithm for Option Number Selection in Stochastic Paired Comparison Models
## 3 Comparison of Models Allowing Different Options in Choices from a Theoretical Aspect
### Comparison of Models Allowing Different Options within Choices: The General Case  
The comparison of three-option and five-option cases is included in the conference paper [50]; therefore, we do not provide details here. Instead, in this section, we turn to the general case, which includes the analysis in Section 3.1 and the analysis of the cases \(s=3\) and \(s+2=5\). Data will be denoted by \(A^{(s)}_{i,j,k}\), (\(k=1,2,\ldots,s\)) and \(A^{(s+2)}_{i,j,k}\), (\(k=1,2,\ldots,s+2\)) in the \(s\) and (\(s+2\))-option models, respectively. Data conversion is performed using the following formulas:  
\[A^{(s)}_{i,j,1}=A^{(s+2)}_{i,j,1}+A^{(s+2)}_{i,j,2} \tag{14}\]  
\[A^{(s)}_{i,j,s}=A^{(s+2)}_{i,j,s+1}+A^{(s+2)}_{i,j,s+2} \tag{15}\]  
and  
\[A^{(s)}_{i,j,k}=A^{(s+2)}_{i,j,k+1},k=2,3,\ldots,s-1 \tag{16}\]  
This means that the two outermost decisions are merged and the 'internal' decisions remain unchanged. The appropriate graph definition is the following:  
**Definition 3** (The graph belonging to \(A^{(s^{*})}\)). : _The nodes are the objects. Next to the edges belonging to ties are two types of directed edge: The first one is a'simple' directed edge from \(i\) to \(j\) (denoted by \(\rightarrow\) and called the 'better to some extent' edge) that exists if \(0<A^{(s^{*})}_{i,j,k}\)\(k=[\frac{s^{*}+1}{2}]+1,\ldots,s^{*}-1\); i.e., there is a decision according to which object \(i\) is 'better to some extent' than object \(j\). The other is an 'extremely better' directed edge: it exists from \(i\) towards \(j\) if \(0<A^{(s^{*})}_{i,j,s^{*}}\) and it is denoted by \(\rightarrow\). Expressed in words, there is an extreme decision according to which \(i\) exceeds \(j\). This graph will be denoted by \(GC^{(A^{(s^{*})})}\). We substitute \(s^{*}=s\) and \(s^{*}=s+2\)._  
The set of conditions that guarantees data evaluability in the model allowing \(s^{*}\) options is as follows:  
* C.s*.1. For every value of \(k=1,2,\ldots,s^{*}\) there exists an index pair \(i_{k},j_{k}\) for which \(0<A^{(s^{*})}_{i_{k},j_{k},k}\). We note that this condition is necessary. If it does not hold, the number of options has to be reduced. * C.s*.2. For every nonempty partition of \(S\) and \(\overline{S}\) (\(S\cup\overline{S}=\{1,2,\ldots,n\}\), \(S\cap\overline{S}=\emptyset\)), 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 \[0<A^{(s^{*})}_{i_{1},j_{1},1}\text{ and }0<A^{(s^{*})}_{i_{2},j_{2},s^{*}}\] (17) are satisfied or there exists such an index pair \((i_{3},j_{3})\) for which \[0<A^{(s^{*})}_{i_{3},j_{3},k},\text{ for some }k=2,3,\ldots,s^{*}-1\] (18)  
Figure 4: Comparison results which can be evaluated in neither model (objects to be compared are denoted by 1, 2, and 3). is satisfied. Expressed in words, the \(S\) and \(\overline{S}\) are connected either by a non-extreme decision or by two extreme decisions made back and forth. We note that this condition is also a necessary condition, not just sufficient. * C.s*.3. There exists a cycle in the graph \(GC^{(A^{(s^{*})})}\) along the 'better to some extent' and 'extremely better' edges which contains at least one 'extremely better' edge. In other words, there exists a cycle \((i_{1},i_{2},\ldots,i_{h},i_{1})\) for which \(i_{1},i_{2},\ldots,i_{h}\) are different, and \(0<A^{(s^{*})}_{i_{1},i_{2}+1,k_{1}}\), where \(k_{l}\in\{[\frac{s^{*}+1}{2}]+1,\ldots,s^{*}\}\), and there exists at least one index pair in the cycle for which \(k_{l}=s^{*}\). We note that it may be the case that this cycle contains only two different nodes. As a generalization of Theorem 4 in [47], one can prove that conditions C.s*.1., C.s*.2., and C.s*.3. guarantee the evaluability of the data matrix \(A^{(s^{*})}\) in the \(s^{*}\)-option model. We will apply this statement for \(s^{*}=s\) and \(s^{*}=s+2\). Comparison Based on Evaluability  
1. Data are evaluable only in the \(s\)-option model. 2. Data are evaluable only in the (\(s\)+2)-option model. 3. Data are evaluable in both models. 4.