Evaluability of paired comparison data in stochastic paired comparison models: Necessary and sufficient condition

There are numerous methods to estimate the strength of different objects based on paired comparisons when the output allows for three options of choices. Given the results of the comparisons, there might exist a value for the strength of the objects that is the uniquely most likely, that is, which maximizes the likelihood of getting exactly these results. Hence, the existence and uniqueness of the estimator are key issues of the evaluation of stochastic paired comparison models. Accordingly, there have been works in the literature that provide sufficient conditions for the existence and uniqueness of such a value. In this paper, we formulate and prove a necessary and sufficient condition for the existence and uniqueness of the argument that maximizes the likelihood. In other words, we provide an easy-to-check condition for the evaluability of the data for the paired comparison models with stochastic background, that is Thurstone-motivated models including three-option Bradley-Terry model and also three-option Davidson model. We prove that the time required to check the conditions is polynomial in the number of objects to evaluate. Through computer simulations, we demonstrate the efficacy of this new set of necessary and sufficient conditions, comparing it against previously established sufficient conditions.