We consider multi-agent systems where agents' preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents' preferences is common in many real-life settings due to privacy issues or an ongoing elicitation process. In addition, there may be uncertainty about how the preferences are aggregated. For example, the agenda (a tree whose leaves are labelled with the decisions being compared) may not yet be known or fixed. We therefore study how to determine collectively optimal decisions (also called winners) when preferences may be incomplete, and when the agenda may be uncertain. We show that it is computationally easy to determine if a candidate decision always wins, or may win, whatever the agenda. On the other hand, it is computationally hard to know wheth er a candidate decision wins in at least one agenda for at least one completion of the agents' preferences. These results hold even if the agenda must be balanced so that each candidate decision faces the same number of majority votes. Such results are useful for reasoning about preference elicitation. They help understand the complexity of tasks such as determining if a decision can be taken collectively, as well as knowing if the winner can be manipulated by appropriately ordering the agenda.
The most related work is @cite_7 Like our paper, this considers the computational complexity of determining winners for sequential majority voting. However, they start from an incomplete majority graph which throws away information about individual votes, whilst we start from an incomplete profile.
@cite_7
2066051963 196507789 2168205737 1497863661
In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters' preferences becomes a Condorcet winner—a candidate who beats all other candidates in pairwise majority-rule elections. Bartholdi, Tovey, and Trick provided a lower bound—NP-hardness—on the computational complexity of determining the election winner in Carroll's system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carroll's system is complete for parallel access to NP, that is, it is complete for Theta_ 2 p for which it becomes the most natural complete problem known. It follows that determining the winner in Carroll's elections is not NP-complete unless the polynomial hierarchy collapses. In computational social choice, one important problem is to take the votes of a subelectorate (subset of the voters), and summarize them using a small number of bits. This needs to be done in such a way that, if all that we know is the summary, as well as the votes of voters outside the subelectorate, we can conclude which of the m alternatives wins. This corresponds to the notion of compilation complexity, the minimum number of bits required to summarize the votes for a particular rule, which was introduced by [IJCAI-09]. We study three different types of compilation complexity. The first, studied by , depends on the size of the subelectorate but not on the size of the complement (the voters outside the subelectorate). The second depends on the size of the complement but not on the size of the subelectorate. The third depends on both. We first investigate the relations among the three types of compilation complexity. Then, we give upper and lower bounds on all three types of compilation complexity for the most prominent voting rules. We show that for l-approval (when l ≤ m 2), Borda, and Bucklin, the bounds for all three types are asymptotically tight, up to a multiplicative constant; for l-approval (when l > m 2), plurality with runoff, all Condorcet consistent rules that are based on unweighted majority graphs (including Copeland and voting trees), and all Condorcet consistent rules that are based on the order of pairwise elections (including ranked pairs and maximin), the bounds for all three types are asymptotically tight up to a multiplicative constant when the sizes of the subelectorate and its complement are both larger than m1+∊ for some ∊ > 0. We study the complexity of (approximate) winner determination under the Monroe and Chamberlin-Courant multiwinner voting rules, which determine the set of representatives by optimizing the total satisfaction or dissatisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions in the utilitarian case or as the (dis)satisfaction of the worst off voter in the egalitarian case. We provide good approximation algorithms for the satisfaction-based utilitarian versions of the Monroe and Chamberlin-Courant rules, and inapproximability results for the dissatisfaction-based utilitarian versions of these rules and also for all egalitarian cases. Our algorithms are applicable and particularly appealing when voters submit truncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic preference data. These experiments show that our simple and fast algorithms can, in many cases, find near-perfect solutions. We study the computational complexity of the counting version of the POSSIBLE-WINNER problem for elections. In the POSSIBLE-WINNER problem we are given a profile of voters, each with a partial preference order, and ask if there are linear extensions of the votes such that a designated candidate wins. We also analyze a special case of POSSIBLE-WINNER, the MANIPULATION problem. We provide polynomial-time algorithms for counting manipulations in a class of scoring protocols and in several other voting rules. We show #P-hardness of the counting variant of POSSIBLE-WINNER for plurality and veto and give a simple yet general and practically useful randomized algorithm for a variant of POSSIBLE-WINNER for all voting rules for which a winner can be computed in polynomial time.