Computing the Extremal Possible Ranks with Incomplete Preferences
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In an election via a positional scoring rule, each candidate receives from each voter a score that is determined only by the position of the candidate in the voter's total ordering of the candidates. A winner (respectively, unique winner) is a candidate who receives a score not smaller than (respectively, strictly greater than) the remaining candidates. When voter preferences are known in an incomplete manner as partial orders, a candidate can be a possible/necessary (unique) winner based on the possibilities of completing the partial votes. The computational problems of determining the possible and necessary winners and unique winners have been studied in depth, culminating in a full classification of the class of "pure" positional scoring rules into tractable and intractable ones for each problem. The above problems are all special cases of reasoning about the range of possible positions of a candidate under different tie breakers. Determining this range, and particularly the extremal positions, arises in every situation where the ranking plays an important role in the outcome of an election, such as in committee selection, primaries of political parties, and staff recruiting. Our main result establishes that the minimal and maximal positions are hard to compute (NP-hard) for every positional scoring rule, pure or not. Hence, none of the tractable variants of necessary/possible winner determination remain tractable for extremal position determination. We do show, however, that tractability can be retained when reasoning about the top-k and the bottom-k positions for a fixed k. We also study the complexity of determining the extremal positions in non-positional voting rules that are nevertheless based on assignments of scores, including the Copeland, Bucklin and Maximin rules.
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