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.
Pini prove that computing the possible and necessary winners for the STV rule is NP-hard @cite_6 . They show it is NP-hard even to approximate these sets within some constant factor in size. They also give a preference elicitation procedure which focuses just on the set of possible winners.
@cite_6
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Many NP-hard winner determination problems admit polynomial-time algorithms when restricting inputs to be single-peaked. Commonly, such algorithms employ dynamic programming along the underlying axis. We introduce a new technique: carefully chosen integer linear programming (IP) formulations for certain voting problems admit an LP relaxation which is totally unimodular if preferences are single-peaked, and which thus admits an integral optimal solution. This technique gives fast algorithms for finding optimal committees under the PAV and Chamberlin-Courant voting rules under single-peaked preferences, as well as for certain OWA-based rules. Under single-crossing preferences, Young scores can also be calculated. An advantage of this technique is that no special-purpose algorithm needs to be used to exploit structure in the input preferences: any standard IP solver will terminate in the first iteration if the input is single-peaked, and will continue to work otherwise. We study the manipulation of voting schemes, where a voter lies about their preferences in the hope of improving the election's outcome. All voting schemes are potentially manipulable. However, some, such as the Single Transferable Vote (STV) scheme used in Australian elections, are resistant to manipulation because it is NP-hard to compute the manipulating vote(s). We concentrate on STV and some natural generalisations of it called Scoring Elimination Protocols. We show that the hardness result for STV is true only if both the number of voters and the number of candidates are unbounded---we provide algorithms for a manipulation if either of these is fixed. This means that manipulation would not be hard in practice when either number is small. Next we show that the weighted version of the manipulation problem is NP-hard for all Scoring Elimination Protocols except one, which we provide an algorithm for manipulating. Finally we experimentally test a heuristic for solving the manipulation problem and conclude that it would not usually be effective. We investigate the extent to which it is possible to compute the probability of a particular candidate winning an election, given imperfect information about the preferences of the electorate. We assume that for each voter, we have a probability distribution over a set of preference orderings. Thus, for each voter, we have a number of possible preference orderings - we do not know which of these orderings actually represents the preferences of the voter, but for each ordering, we know the probability that it does. For the case where the number of candidates is a constant, we are able to give a polynomial time algorithm to compute the probability that a given candidate will win. We present experimental results obtained with an implementation of the algorithm, illustrating how the [email protected]?s performance in practice is better than its predicted theoretical bound. However, when the number of candidates is not bounded, we prove that the problem becomes #P-hard for the Plurality, k-approval, Borda, Copeland, and Bucklin voting rules. We further show that even evaluating if a candidate has any chance of winning is NP-complete for the Plurality voting rule in the case where voters may have different weights. With unweighted voters, we give a polynomial algorithm for Plurality, and show that the problem is hard for many other voting rules. Finally, we give a Monte Carlo approximation algorithm for computing the probability of a candidate winning in any settings, with an error that is as small as desired. We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.