In this paper, paired comparison models with stochastic background are investigated. We focus on the models that allow three options for choice. We estimate all parameters, the strength of the objects and the boundaries of equal decision, by maximum likelihood method. The existence and uniqueness of the estimator are key issues of the evaluation. Although a necessary and sufficient condition for the general case of three options has not been known until now, there are some different sufficient conditions that are formulated in the literature. In this paper, we provide a necessary and sufficient condition for the existence of a maximum and the uniqueness of the argument that maximizes the value, i.e. for the evaluability of the data in models of these types. By computer simulation, we present the efficiency of the condition, comparing it to the previously known sufficient conditions.

翻译：本文研究了具有随机背景的配对比较模型。我们重点关注允许三种选择选项的模型。我们采用最大似然法估计所有参数，包括对象的强度和平等决策的边界。估计量的存在性与唯一性是评估的关键问题。尽管迄今为止，针对三种选项的一般情况的充要条件尚未明确，但文献中已提出一些不同的充分条件。本文为这类模型中最大值的存在性以及使该值最大化的参数唯一性（即数据的可评估性）提供了一个充要条件。通过计算机模拟，我们将该条件与先前已知的充分条件进行比较，展示了其有效性。