# Algorithm for Option Number Selection in Stochastic Paired Comparison Models
## 1 Introduction
Not only can the evaluation method change, but so can the way the PCM is constructed. Maintaining its reciprocal-symmetric property, the matrix can also be formed in another way than how [PERSON] constructed it. In [29], the authors define the elements of the PCM as the ratio of the number of wins to losses. In a socio-economic study [30], the author used the ratio of certain economic characteristics as the elements of the matrix. In this case, the number of possible options that require a decision is infinitely many. Both the eigenvector method (with some modification) and LLSM work for these PCM cases under the same conditions: the PCM matrix has to be irreducible, i.e., the representing graph must be connected [27; 28]. We note that other methods, such as the Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE) and Elimination and Choice Translating Reality (ELECTREE, ELECTRE) are often applied, mainly when focusing on multi-criteria decision making [31; 32; 33]. Another branch of the models connected to comparisons in pairs are the Thurstone motivated models, which have stochastic background. Based on [PERSON]'s brilliant idea, these models envision the latent probabilistic variables behind each object [34]. The starting points were psychological problems but, based on Thurstone motivated models, it is also possible to solve the ranking and rating of objects related to other research questions [35; 36; 37; 38]. [PERSON] assumed the latent variables had a Gaussian distribution, while [PERSON] and [PERSON] assumed a logistic distribution [39]. In [34; 39], two options were allowed for the decisions of the observers, but the models were generalized to three options, allowing ties [40; 41]. Models allowing more than three options are presented in [42]. The ranking and rating of these objects are determined by the expectation of latent random variables. There are numerous possibilities for estimating their parameters. In the case of a Gaussian distribution, least squares methods are usually applied, but in cases of logistic distribution, maximum likelihood estimations are widespread. One publication [42] also applied least squares estimations based on relative frequencies and a cumulative distribution function. In [43], a general model for applying a Gaussian distribution and allowing more than two options is presented. The parameter estimation method is the maximum likelihood method, as it contains many inherent possibilities. The maximum likelihood estimation (MLE) method is based on the maximization of a multi-variate function, and it requires the clarification of the conditions under which the maximum exists and is unique. In [43], the authors provide a sufficient condition for its existence and uniqueness in the case of a Gaussian distribution, and in [44] they generalize this for the case of general strictly log-concave distributions. Previously known conditions are used in [45], assuming a logistic distribution in the case of the two-option model, and in [46] they are used to investigate a modified model in the case of three choices. This condition, in the case of two options, is a necessary and sufficient condition. The condition given by [PERSON] is a sufficient condition but not necessary. It has been significantly generalized in [47] for the case of two and three options within choices, and the new system of conditions is a generalization of the set of conditions in [44]. The general conditions are met much more frequently [47], making them more useful for both theoretical research and practical applications. This set of conditions and its generalization for more than three options are used in this paper during comparisons of option numbers from a theoretical perspective. During this research, the applied number of options is usually two or four, but there are cases when further numbers of options are allowed: four options are used in [43] for evaluating women tennis players, five options are used in [48], six options are used in [38], and seven options are used in [49].