# Algorithm for Option Number Selection in Stochastic Paired Comparison Models
## 1 Introduction  
Comparisons in pairs are frequently used in a broad spectrum of problems, across all areas of life. Without claiming to be exhaustive, we mention education [1], management [2], psychology [3], marketing [4], construction [5], social science [6], group decisions [7], engineering [8], medicine [9], economics [10], resource allocation [11], and sports [12] as areas of their application. A paper by [PERSON] and [PERSON] [13] presents more than 150 of the applications that they had been used for before 2003. Paired comparisons play a significant role in decision-making [14, 15]. This method allows for a detailed analysis of the differences and similarities between objects, properties, and aspects, aiding in the selection of the best choice. It is particularly useful in situations where multiple factors must be considered and a clear ranking is necessary [13, 16]. By applying paired comparisons, complex problems can be simplified, leading to more objective decisions. Paired comparisons means that, instead of using a scale, the objects to be evaluated are compared directly to each other, and the decision of the observer (evaluator) is based on the relative merits of the two objects. These opinions are generally more reliable than evaluations based on scales, as it is easier to determine which of two objects, properties, aspects, etc., is 'better','more appropriate','more important', etc., than to describe them with a single number. This is especially true for evaluations based on subjective criteria. In the case of paired comparisons, their data represent the relations between the observed objects. These types of data require methods that are different from conventional statistical procedures. The most popular method based on comparisons in pairs is the Analytic Hierarchy Process (AHP) associated with [PERSON] [17]. This single paper has more than 10 thousand citations; some of these articles provide criticism, see for example [18]. The starting point of the model is a pairwise comparison matrix (PCM), the elements of which show how manytimes one object is stronger than another. The matrix \(A=(a_{ij})\) is reciprocal-symmetric, that is \(a_{i,j}=\frac{1}{a_{i,j}}\). According to [PERSON]'s original model [17], \(a_{i,j}\in\{1/9,1/8,\ldots,1/2,1,2,\ldots,8,9\}\). The observers usually express their decisions verbally by voting for 'equally preferred','moderately preferred','strongly preferred','very strongly preferred', or 'extremely preferred'. These verbal expressions help decision-makers articulate their preferences in a qualitative manner, which are converted to quantitative values; 1, 3, 5, 7, or 9, respectively. Intermediate values are used for comparisons that fall between these levels. We mention that the number of options may be reduced in light of the problem under consideration. Their quantitative results are analyzed using different methods. Usually, every object is compared to every other object; that is, the PCM does not contain missing elements. In this case, the number of pairs under comparison increases quadratically with the number of objects examined. If the number of objects is 10, then the number of pairs equals \(\frac{10.9}{2}=45\), which can be quite overwhelming for an observer. The reliability of their opinions and the measure of the contradictions between their decisions are checked by the inconsistency index of the PCM. The definition of consistency is given in [17] for complete cases; it was recently generalized to incomplete cases in [19]. There are numerous ways to quantify the degree of inconsistency seen [20; 21]. Some research papers present procedures for reducing the level of inconsistency [22; 23; 24; 25]. Most measurements are for complete cases, but they can also be generalized to incomplete cases [26]. The most frequently used measurement is related to the eigenvector method [17; 24], which is the most common evaluation method, but the logarithmic least squares method (LLSM) is also readily used for evaluations, although mainly in incomplete cases [19; 27; 28].