In this paper paired comparison models with stochastic background are investigated. We focus on the models which allow three options for choice and the parameters are estimated by maximum likelihood method. The existence and uniqueness of the estimator is a key issue of the evaluation. In the case of two options, a necessary and sufficient condition is given by Ford in the Bradley-Terry model. We generalize this statement for the set of strictly log-concave distribution. Although in the case of three options necessary and sufficient condition is not known, there are two different sufficient conditions which are formulated in the literature. In this paper we generalize them, moreover we compare these conditions. Their capacities to indicate the existence of the maximum are analyzed by a large number of computer simulations. These simulations support that the new condition indicates the existence of the maximum much more frequently then the previously known ones,

翻译：本文研究了具有随机背景的成对比较模型，重点关注允许三种选择的模型，并采用极大似然法估计参数。估计量的存在性与唯一性是评估的关键问题。在两种选择的情况下，Ford在Bradley-Terry模型中给出了一个充要条件。我们将这一结论推广到严格对数凹分布族。尽管三种选择情况下的充要条件尚未可知，但文献中提出了两种不同的充分条件。本文对这些条件进行了推广，并进一步比较了它们。通过大量计算机模拟分析了这些条件指示极大值存在的能力。模拟结果表明，新条件指示极大值存在的频率远高于先前已知的条件。