Picking the best alternative in a given set is a well-studied problem at the core of social choice theory. In some applications, one can assume that there is an objectively correct way to compare the alternatives, which, however, cannot be observed directly, and individuals' preferences over the alternatives (votes) are noisy estimates of this ground truth. The goal of voting in this case is to estimate the ground truth from the votes. In this paradigm, it is usually assumed that the ground truth is a ranking of the alternatives by their true quality. However, sometimes alternatives are compared using not one but multiple quality parameters, which may result in cycles in the ground truth as well as in the preferences of the individuals. Motivated by this, we provide a formal model of voting with possibly intransitive ground truth and preferences, and investigate the maximum likelihood approach for picking the best alternative in this case. We show that the resulting framework leads to polynomial-time algorithms, and also approximates the corresponding NP-hard problems in the classic framework.
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