# Algorithm for Option Number Selection in Stochastic Paired Comparison Models
## 2 The Investigated Models  
Let the objects to be rated and ranked be denoted by the numbers \(1,2,\ldots,n\). The models with a stochastic background assume that the current performances of the objects are random variables denoted by \(\xi_{i}\), \(i=1,2,\ldots,n\). Let the expectations be \(\mathrm{E}(\xi_{i})=m_{i}\), which expresses the expected strengths of the object \(i\). Decisions comparing two objects \(i\) and \(j\) are related to the difference between these latent random variables, i.e., \(\xi_{i}-\xi_{j}\). We can separate the expectations as follows:  
\[\xi_{i}-\xi_{j}=m_{i}-m_{j}+\eta_{i,j} \tag{1}\]  
where \(\eta_{i,j}\) are independent identically distributed random variables with the cumulative distribution function \(F\). \(F\) is supposed to be three times continuously differentiable, \(0<F<1\), its probability density function is symmetric to zero, i.e., \(f(-x)=f(x)\), \(x\in\mathbb{R}\), and its logarithm is strictly concave. If \(\eta_{i,j}\) are normally distributed, then we use the nomination of the Thurstone model, if their distribution is logistic, we use the Bradley-Terry model. These models were originally defined for two-option choices, 'worse' or 'better', and then were generalized for three options: 'worse', 'better', or 'equal'. In the case of the two-option model, the differences are compared to zero: the decision 'better'/'worse' indicates whether the difference is positive/negative, respectively. In the three-option model, the value of the difference is compared to the parameter \(0<d\); if it is in \([-d,d]\), we can consider it a tie. More than three, (\(s\)), options can be allowed too, as follows: The options within choices are denoted by \(C_{k}\), \(k=1,2,\ldots,s\) (\(s=5\):'much worse','slightly worse', 'equal','slightly better','much better'). The real line \(\mathbb{R}\) is divided into disjoint sub-intervals \(I_{1}\), \(I_{2}\),..., \(I_{s}\). \(I_{1}=(-\infty,d_{1})\), \(I_{2}=[d_{1},d_{2})\),..., \(I_{s}=[d_{s-1},\infty)\). Decisions have a kind of symmetry, i.e., if \(i\) is 'better'/'much better'/...than \(j\), then \(j\) is worse'/'much worse'/...than \(i\). This means that \(d_{s-1}=-d_{1}\) and, in general, \(d_{s-i}=-d_{i}\), \(i=1,2,\ldots,[s/2]\). The figure belonging to the model allowing \(s\) options can be seen in Figure 1.  
If \(s=2\), then no parameter is  
The maximum likelihood estimation of the parameters \(\underline{m}=(m_{1},\ldots,m_{n})\) and \(\underline{dv}=(-d_{1},\ldots,-d_{[(s-1)/2]})\) is the argument for the maximal value of (8); that is  
\[(\widehat{\underline{m}}^{(s)},\widehat{\underline{dv}}^{(s)})=\mathop{\arg \max}_{\underline{m}\in\mathbb{R}^{n},\ 0<-d_{[\frac{s-1}{2}]}<\cdots<-d_{1}}L(A^{(s)}|\underline{m},\underline{dv}). \tag{9}\]  
We note that the probabilities (2), (3), (4), (5), and (6) depend only on the differences between their expectations; therefore, one coordinate of the parameter vector \(\underline{m}\), for example \(m_{1}\), can be fixed at zero or the constraint \(\sum_{i=1}^{n}m_{i}=0\) can be assumed. Naturally, the maximum value is not necessarily always attained, but some conditions for the data can guarantee the existence and uniqueness of a maximizer. These conditions are different in the case of different options. Hereinafter, data will be referred to as evaluable in an \(s\)-option model if the maximum likelihood estimate of the parameters (9) exists and is unique.