We investigate approval-based committee voting with incomplete information about the approval preferences of voters. We consider several models of incompleteness where each voter partitions the set of candidates into approved, disapproved, and unknown candidates, possibly with ordinal preference constraints among candidates in the latter category. This captures scenarios where voters have not evaluated all candidates and/or it is unknown where voters draw the threshold between approved and disapproved candidates. We study the complexity of some fundamental computational problems for a number of classic approval-based committee voting rules including Proportional Approval Voting and Chamberlin-Courant. These problems include that of determining whether a given set of candidates is a possible or necessary winning committee and whether it forms a committee that possibly or necessarily satisfies representation axioms. We also consider the problem whether a given candidate is possibly or necessarily a member of the winning committee.
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.Tha 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 (DP-complete) 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. ACM Subject ClassificationTheory of computation → Algorithmic game theory and mechanism design Keywords and phrases Positional scoring rules, Incomplete preferences.
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