  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,
