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
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Voting is a way to aggregate individual voters' preferences. Traditionally a voter's preference is represented by a total order on the set of candidates. However, sometimes one may not have complete information about a voter's preference, and in this case, can only model a voter's preference as a partial order. Given this framework, there has been work on computing the possible and necessary winners of a (partial) profile. In this paper, we take a step further, look at sets of questions to ask in order to determine the outcome of such a partial profile. Specifically, we call a set of questions a deciding set for a candidate if the outcome of the vote for the candidate is determined no matter how the questions are answered by the voters, and a possible winning (losing) set if there is a way to answer these questions to make the candidate a winner (loser) of the vote. We discuss some interesting properties about these sets of queries, prove some complexity results about them under some well-known voting rules such as plurality and Borda, and consider their application in vote elicitation. We consider manipulation problems when the manipulator only has partial information about the votes of the non-manipulators. Such partial information is described by an information set, which is the set of profiles of the non-manipulators that are indistinguishable to the manipulator. Given such an information set, a dominating manipulation is a non-truthful vote that the manipulator can cast which makes the winner at least as preferable (and sometimes more preferable) as the winner when the manipulator votes truthfully. When the manipulator has full information, computing whether or not there exists a dominating manipulation is in P for many common voting rules (by known results). We show that when the manipulator has no information, there is no dominating manipulation for many common voting rules. When the manipulator's information is represented by partial orders and only a small portion of the preferences are unknown, computing a dominating manipulation is NP-hard for many common voting rules. Our results thus throw light on whether we can prevent strategic behavior by limiting information about the votes of other voters. While voting schemes provide an effective means for aggregating preferences, methods for the effective elicitation of voter preferences have received little attention. We address this problem by first considering approximate winner determination when incomplete voter preferences are provided. Exploiting natural scoring metrics, we use max regret to measure the quality or robustness of proposed winners, and develop polynomial time algorithms for computing the alternative with minimax regret for several popular voting rules. We then show how minimax regret can be used to effectively drive incremental preference vote elicitation and devise several heuristics for this process. Despite worst-case theoretical results showing that most voting protocols require nearly complete voter preferences to determine winners, we demonstrate the practical effectiveness of regret-based elicitation for determining both approximate and exact winners on several real-world data sets. Usually a voting rule requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a voting rule, a profile of partial orders, and an alternative (candidate) c, two important questions arise: first, is it still possible for c to win, and second, is c guaranteed to win? These are the possible winner and necessary winner problems, respectively. Each of these two problems is further divided into two sub-problems: determining whether c is a unique winner (that is, c is the only winner), or determining whether c is a co-winner (that is, c is in the set of winners). We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We completely characterize the complexity of possible necessary winner problems for the following common voting rules: a class of positional scoring rules (including Borda), Copeland, maximin, Bucklin, ranked pairs, voting trees, and plurality with runoff. This paper presents a new approach to identifying and eliminating mislabeled training instances for supervised learning. The goal of this approach is to improve classification accuracies produced by learning algorithms by improving the quality of the training data. Our approach uses a set of learning algorithms to create classifiers that serve as noise filters for the training data. We evaluate single algorithm, majority vote and consensus filters on five datasets that are prone to labeling errors. Our experiments illustrate that filtering significantly improves classification accuracy for noise levels up to 30 . An analytical and empirical evaluation of the precision of our approach shows that consensus filters are conservative at throwing away good data at the expense of retaining bad data and that majority filters are better at detecting bad data at the expense of throwing away good data. This suggests that for situations in which there is a paucity of data, consensus filters are preferable, whereas majority vote filters are preferable for situations with an abundance of data. Preferences can be aggregated using voting rules. We consider here the family of rules which perform a sequence of pairwise majority comparisons between two candidates. The winner thus depends on the chosen sequence of comparisons, which can be represented by a binary tree. We address the difficulty of computing candidates that win for some trees, and then introduce and study the notion of fair winner, i.e. candidates who win in a balanced tree. We then consider the situation where we lack complete informations about preferences, and determine the computational complexity of computing winners in this case. This paper establishes the computational complexity status for a problem of deciding on the quality of a committee. Starting with individual preferences over alternatives, we analyse when it can be determined efficiently if a given committee C satisfies a weak (resp. strong) Condorcet criterion–i.e., if C is at least as good as (resp. better than) every other committee in a pairwise majority comparison. Scoring functions used in classic voting rules are adapted for these comparisons. In particular, we draw the sharp separation line between computationally tractable and intractable instances with respect to different voting rules. Finally, we show that deciding if there exists a committee which satisfies the weak (resp. strong) Condorcet criterion is computationally hard. We suggest a new model for strategic voting based on local dominance, where voters consider a set of possible outcomes without assigning probabilities to them. We prove that voting equilibria under the Plurality rule exist for a broad class of local dominance relations. Furthermore, we show that local dominance-based dynamics quickly converge to an equilibrium if voters start from the truthful state, and we provide weaker convergence guarantees in more general settings. Using extensive simulations of strategic voting on generated and real profiles, we show that emerging equilibria replicate widely known patterns of human voting behavior such as Duverger's law, and that they generally improve the quality of the winner compared to non-strategic voting. Parallelizing Monte Carlo tree search (MCTS) has been considered to be a way to improve the strength of Computer Go programs. In this paper, we analyze the performance of two root parallelization methods: the standard strategy based on average selection and our new strategy based on majority voting. As a starting code base, we used Fuego, which is one of the best programs available. Our experimental results with 64 central processing unit (CPU) cores show that majority voting outperforms average selection. Additionally, we show through an extensive analysis that root parallelization has limitations. 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. Uncertainty arises in preference aggregation in several ways. There may, for example, be uncertainty in the votes or the voting rule. Such uncertainty can introduce computational complexity in determining which candidate or candidates can or must win the election. In this paper, we survey recent work in this area and give some new results. We argue, for example, that the set of possible winners can be computationally harder to compute than the necessary winner. As a second example, we show that, even if the unknown votes are assumed to be single-peaked, it remains computationally hard to compute the possible and necessary winners, or to manipulate the election. 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. A possible winner of an election is a candidate that has, in some kind of incomplete-information election, the possibility to win in a complete extension of the election. The first type of problem we study is the Possible co-Winner with respect to the Addition of New Candidates (PcWNA) problem, which asks, given an election with strict preferences over the candidates, is it possible to make a designated candidate win the election by adding a limited number of new candidates to the election? In the case of unweighted voters we show NP-completeness of PcWNA for a broad class of pure scoring rules. We will also briefly study the case of weighted voters. The second type of possible winner problem we study is Possible Winner co-Winner under Uncertain Voting System (PWUVS and PcWUVS). Here, uncertainty is present not in the votes but in the election rule itself. For example, PcWUVS is the problem of whether, given a set C of candidates, a list of votes over C, a distinguished candidate c ∈ C, and a class of election rules, there is at least one election rule from this class under which c wins the election. We study these two problems for a class of systems based on approval voting, the family of Copelandα elections, and a certain class of scoring rules. Our main result is that it is NP-complete to determine whether there is a scoring vector that makes c win the election, if we restrict the set of possible scoring vectors for an m-candidate election to those of the form (α1,..., αm−4, x1, x2, x3, 0), with xi = 1 for at least one i ∈ 1, 2, 3 . We consider a well-known distributed colouring game played on a simple connected graph: initially, each vertex is coloured black or white; at each round, each vertex simultaneously recolours itself by the colour of the simple (strong) majority of its neighbours. A set of vertices M is said to be a dynamo, if starting the game with only the vertices of M coloured black, the computation eventually reaches an all-black configuration.The importance of this game follows from the fact that it models the spread of faults in point-to-point systems with majority-based voting; in particular, dynamos correspond to those sets of initial failures which will lead the entire system to fail. Investigations on dynamos have been extensive but restricted to establishing tight bounds on the size (i.e., how small a dynamic monopoly might be).In this paper we start to study dynamos systematically with respect to both the size and the time (i.e., how many rounds are needed to reach all-black configuration) in various models and topologies.We derive tight tradeoffs between the size and the time for a number of regular graphs, including rings, complete d-ary trees, tori, wrapped butterflies, cube connected cycles and hypercubes. In addition, we determine optimal size bounds of irreversible dynamos for butterflies and shuffle-exchange using simple majority and for DeBruijn using strong majority rules. Finally, we make some observations concerning irreversible versus reversible monotone models and slow complete computations from minimal dynamos. We study the control complexity of fallback voting. Like manipulation and bribery, electoral control describes ways of changing the outcome of an election; unlike manipulation or bribery attempts, control actions---such as adding deleting partitioning either candidates or voters---modify the participative structure of an election. Via such actions one can try to either make a favorite candidate win ("constructive control") or prevent a despised candidate from winning ("destructive control"). Computational complexity can be used to protect elections from control attempts, i.e., proving an election system resistant to some type of control shows that the success of the corresponding control action, though not impossible, is computationally prohibitive. We show that fallback voting, an election system combining approval with majority voting, is resistant to each of the common types of candidate control and to each common type of constructive control. Among natural election systems with a polynomial-time winner problem, only plurality and sincere-strategy preference-based approval voting (SP-AV) were previously known to be fully resistant to candidate control, and only Copeland voting and SP-AV were previously known to be fully resistant to constructive control. However, plurality has fewer resistances to voter control, Copeland voting has fewer resistances to destructive control, and SP-AV (which like fallback voting has 19 out of 22 proven control resistances) is arguably less natural a system than fallback voting. Schulze voting is a recently introduced voting system enjoying unusual popularity and a high degree of real-world use, with users including the Wikimedia foundation, several branches of the Pirate Party, and MTV. It is a Condorcet voting system that determines the winners of an election using information about paths in a graph representation of the election. We resolve the complexity of many electoral control cases for Schulze voting. We find that it falls short of the best known voting systems in terms of control resistance, demonstrating vulnerabilities of concern to some prospective users of the system. The Borda voting rule is a positional scoring rule where, for m candidates, for every vote the first candidate receives m- 1 points, the second m- 2 points and so on. A Borda winner is a candidate with highest total score. It has been a prominent open problem to determine the computational complexity of UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA: Can one add a certain number of additional votes (called manipulators) to an election such that a distinguished candidate becomes a winner? We settle this open problem by showing NP-hardness even for two manipulators and three input votes. Moreover, we discuss extensions and limitations of this hardness result. Approval voting is a well-known voting procedure for single-winner elections. Voters approve of as many candidates as they like, and the candidate with the most approvals wins (Brams and Fishburn 1978, 1983, 2005). But Merrill and Nagel (1987) point out that there are many ways to aggregate approval votes to determine a winner, justifying a distinction between approval balloting, in which each voter submits a ballot that identifies the candidates the voter approves of, and approval voting, the procedure of ranking the candidates according to their total numbers of approvals. Existing gesture segmentations use the backward spotting scheme that first detects the end point, then traces back to the start point and sends the extracted gesture segment to the hidden Markov model (HMM) for gesture recognition. This makes an inevitable time delay between the gesture segmentation and recognition and is not appropriate for continuous gesture recognition. To solve this problem, we propose a forward spotting scheme that executes gesture segmentation and recognition simultaneously. The start and end points of gestures are determined by zero crossing from negative to positive (or from positive to negative) of a competitive differential observation probability that is defined by the difference of observation probability between the maximal gesture and the non-gesture. We also propose the sliding window and accumulative HMMs. The former is used to alleviate the effect of incomplete feature extraction on the observation probability and the latter improves the gesture recognition rate greatly by accepting all accumulated gesture segments between the start and end points and deciding the gesture type by a majority vote of all intermediate recognition results. We use the predetermined association mapping to determine the 3D articulation data, which reduces the feature extraction time greatly. We apply the proposed simultaneous gesture segmentation and recognition method to recognize the upper-body gestures for controlling the curtains and lights in a smart home environment. Experimental results show that the proposed method has a good recognition rate of 95.42 for continuously changing gestures. We show how computational complexity might protect the integrity of social choice. We exhibit a voting rule that efficiently computes winners but is computationally resistant to strategic manipulation. It is NP-complete for a manipulative voter to determine how to exploit knowledge of the preferences of others. In contrast, many standard voting schemes can be manipulated with only polynomial computational effort.