As for social choice, all alternatives are ranked by agents to form preferences as linear orders. However, in applications, sometimes some alternatives cannot be ranked, or it is unnecessary to rank them, which leads to unranked alternatives. Hence, without loss of generality, by dividing the set of alternatives into three ranked and unranked subsets, including top-k alternatives, intermediate-r alternatives, and last-l alternatives, the Mallows model on ranked and unranked preferences can be analyzed systematically. Technically, a repeated insertion model is adopted during sampling, and probability distributions are derived for ranked and unranked preferences of alternatives. Experimental results verify the accuracy of the probability distributions for different ranked and unranked preferences of alternatives. Furthermore, in order to solve the preference completion problem where agents have multiple partial rankings, a fuzzy preference completion algorithm, Fuzzy-Multi-Rankings, is proposed, which introduces a fuzzy ranking to complete the target agent’s preference in addition to the traditional nearest-neighbor-based methods. Based on the three ranked and unranked preferences, seven cases can be classified and analyzed for fuzzy preference completion. Experiments on the synthetic datasets and MovieLens dataset confirm the effectiveness and efficiency of our proposed Fuzzy-Multi-Rankings algorithm and also verify the accuracy of the evaluated probability distributions for the proposed seven cases.
