We consider the problem of predicting winners in elections, for the case where we are given complete knowledge about all possible candidates, all possible voters (together with their preferences), but where it is uncertain either which candidates exactly register for the election or which voters cast their votes. Under reasonable assumptions, our problems reduce to counting variants of election control problems. We either give polynomial-time algorithms or prove #P-completeness results for counting variants of control by adding deleting candidates voters for Plurality, k-Approval, Approval, Condorcet, and Maximin voting rules. We consider both the general case, where voters' preferences are unrestricted, and the case where voters' preferences are single-peaked.
Another model of predicting election outcomes is that of @cite_0 . They consider a situation where each voter is undecided regarding several possible votes. That is, for each voter we are given several possible preference orders and a probability distribution over these votes. The question is, what is the probability that a designated candidate wins.
@cite_0
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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. Elections involving a large voter population often lead to outcomes that surprise many. A better prediction of the true outcome helps reduce the adverse effect of surprise on the economy of a sizable population. This paper starts from the basic observation that individuals in the underlying population build estimates of the distribution of preferences of the whole population based on their immediate neighbors in the underlying social network. The outcome of the election leads to a surprise if these local estimates contradict the outcome of the election for some fixed voting rule. To get a quantitative understanding, we propose a novel mathematical model of the setting where the individuals in the population and their connections are described by a random graph with connection probabilities that are biased based on the preferences of the individuals. Each individual also has some estimate of the bias in their connections. The connection model is inspired by the homophily effect in social networks. We show that the election outcome leads to a surprise if the discrepancy between the estimated bias and the true bias in the local connections exceeds a certain threshold, and confirm the phenomenon that surprising outcomes are associated only with closely contested elections. We consider large elections with networked voters and compare standard voting rules based on their performance on surprise. Our results show that the rules have different behavior for different parts of the population. It also hints at an impossibility result that any reasonable voting rule will be less surprising for all parts of a population. To attest some of our theoretical predictions, we experiment with the large dataset of UK-EU referendum (a.k.a. Brexit). 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. Multi-winner elections model scenarios where voters must select a fixed-size group of candidates based on their individual preferences. In such scenarios, it is often the case that voters are incentivized to manipulate, i.e. misreport their preferences in order to obtain a better outcome. In this paper, we study the complexity of manipulating multi-winner elections under scoring rules, with a particular focus on the role of tie-breaking rules. We consider both lexicographic tie-breaking rules, which break ties according to a fixed ordering of the candidates, and a natural randomized tie-breaking rule. We describe polynomial-time manipulation algorithms for several special cases of our problem. Specifically, we show that finding a successful manipulation is easy if the underlying voting rule is k-Approval or the number of candidates to be elected is bounded by a constant (for both types of tie-breaking rules), as well as if the manipulator's utility function only takes values in 0, 1 and the ties are broken in the manipulator's favor. 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 present a systematic study of Plurality elections with strategic voters who, in addition to having preferences over election winners, also have secondary preferences, governing their behavior when their vote cannot affect the election outcome. Specifically, we study two models that have been recently considered in the literature: lazy voters, who prefer to abstain when they are not pivotal, and truth-biased voters, who prefer to vote truthfully when they are not pivotal. For both lazy and truth-biased voters, we are interested in their behavior under different tie-breaking rules (lexicographic rule, random voter rule, random candidate rule). Two of these six combinations of secondary preferences and tie-breaking rules have been studied in prior work; for the remaining four, we characterize pure Nash equilibria (PNE) of the resulting strategic games and study the complexity of related computational problems. We then use these results to analyze the impact of different secondary preferences and tie-breaking rules on the election outcomes. Our results extend to settings where some of the voters are non-strategic. 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. We study a generic framework that provides a unified view on two important classes of problems: (i) extensions of the k-median problem where clients are interested in having multiple facilities in their vicinity (e.g., due to the fact that, with some small probability, the closest facility might be malfunctioning and so might not be available for using), and (ii) finding winners according to some appealing multiwinner election rules, i.e., election system aimed for choosing representatives bodies, such as parliaments, based on preferences of a population of voters over individual candidates. Each problem in our framework is associated with a vector of weights: we show that the approximability of the problem depends on structural properties of these vectors. We specifically focus on the harmonic sequence of weights for which the objective function interpreted in a multiwinner election setup reflects to the well-known Proportional Approval Voting (PAV) rule. Our main result is that, due to the specific (harmonic) structure of weights, the problem allows constant factor approximation. This is surprising since the problem can be interpreted as a variant of the k-median problem where we do not assume that the connection costs satisfy the triangle inequality. The algorithm we propose is based on dependent rounding [Srinivasan, FOCS'01] applied to the solution of a natural LP-relaxation of the problem. The rounding process is well known to produce distributions over integral solutions satisfying Negative Correlation (NC), which is usually sufficient for the analysis of approximation guarantees offered by rounding procedures. In our analysis, however, we need to use the fact that the carefully implemented rounding process satisfies a stronger property, called Negative Association (NA), which allows us to apply standard concentration bounds for conditional random variables. Many analyses of plurality-rule elections predict the complete coordination of strategic voting, and hence support for only two candidates. Here I suggest that stable multi-candidate support will arise in equilibrium. A group of voters must partially coordinate behind one of two challenging candidates in order to dislodge a disliked incumbent. In a departure from existing models, the popular support for each challenger is uncertain. This support must be inferred from the private observation of informative signals, such as the social communication of preferences throughout the electorate, or the imperfect observation of opinion polls. The uniquely stable voting equilibrium entails only limited strategic voting and hence incomplete coordination. This is due to the surprising presence of negative feedback: an increase in the degree of strategic voting by others reduces the incentives for an individual to vote strategically. The incentive to vote strategically is lower in relatively marginal elections, after controlling for the distance from contention of a trailing preferred challenger. A calibration of the model applied to the UK General Election of 1997 is consistent with the impact of strategic voting and the reported accuracy of voters understanding of the electoral situation. It suggests that nearly 50 seats may have been lost by the Conservative party due to strategic voting. 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. 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. 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. We consider approval voting elections in which each voter votes for a (possibly empty) set of candidates and the outcome consists of a set of k candidates for some parameter k, e.g., committee elections. We are interested in the min-imax approval voting rule in which the outcome represents a compromise among the voters, in the sense that the maximum distance between the preference of any voter and the outcome is as small as possible. This voting rule has two main drawbacks. First, computing an outcome that minimizes the maximum distance is computationally hard. Furthermore, any algorithm that always returns such an outcome provides incentives to voters to misreport their true preferences. In order to circumvent these drawbacks, we consider approximation algorithms, i.e., algorithms that produce an outcome that approximates the minimax distance for any given instance. Such algorithms can be considered as alternative voting rules. We present a polynomial-time 2-approximation algorithm that uses a natural linear programming relaxation for the underlying optimization problem and deterministically rounds the fractional solution in order to compute the outcome; this result improves upon the previously best known algorithm that has an approximation ratio of 3. We are furthermore interested in approximation algorithms that are resistant to manipulation by (coalitions of) voters, i.e., algorithms that do not motivate voters to misreport their true preferences in order to improve their distance from the outcome. We complement previous results in the literature with new upper and lower bounds on strategyproof and group-strategyproof algorithms. 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. In this paper, we explore the rules that determine intuitive predictions and judgments of confidence and contrast these rules to the normative principles of statistical prediction. Two classes of prediction are discussed: category prediction and numerical prediction. In a categorical case, the prediction is given in nominal form, for example, the winner in an election, the diagnosis of a patient, or a person's future occupation. In a numerical case, the prediction is given in numerical form, for example, the future value of a particular stock or of a student's grade point average. In making predictions and judgments under uncertainty, people do not appear to follow the calculus of chance or the statistical theory of prediction. Instead, they rely on a limited number of heuristics which sometimes yield reasonable judgments and sometimes lead to severe and systematic errors (Kahneman & Tversky, 1972b, 3; Tversky & Kahneman, 1971, 2; 1973, 11). The present paper is concerned with the role of one of these heuristics – representativeness – in intuitive predictions. Given specific evidence (e.g., a personality sketch), the outcomes under consideration (e.g., occupations or levels of achievement) can be ordered by the degree to which they are representative of that evidence. The thesis of this paper is that people predict by representativeness, that is, they select or order outcomes by the degree to which the outcomes represent the essential features of the evidence. We study voting models on graphs. In the beginning, the vertices of a given graph have some initial opinion. Over time, the opinions on the vertices change by interactions between graph neighbours. Under suitable conditions the system evolves to a state in which all vertices have the same opinion. In this work, we consider a new model of voting, called the Linear Voting Model. This model can be seen as a generalization of several models of voting, including among others, pull voting and push voting. One advantage of our model is that, even though it is very general, it has a rich structure making the analysis tractable. In particular we are able to solve the basic question about voting, the probability that certain opinion wins the poll, and furthermore, given appropriate conditions, we are able to bound the expected time until some opinion wins. A person is said to prefer in the stochastic dominance sense one lottery-over-outcomes over another lottery-over-outcomes if the probability of his (at least) first choice being selected in the first lottery is greater than or equal to the analogous probability in the second lottery, the probability of his at least second choice being selected in the first lottery is greater than or equal to the analogous probability in the second lottery, and so on, with at least one strict inequality. This (partial) preference relation is used to define straightforwardness of a social choice function that maps profiles of ordinal preferences into lotteries over outcomes. Given a prior probability distribution on profiles this partial preference ordering (taking into account the additional randomness) is used to induce a partial preference ordering over social choice functions for each individual. These are used in turn to define ex ante Pareto undominated (efficient) social choice functions. The main result is that it is impossible for a social choice function to be both ex ante efficient and straightforward. We also extend the result to cardinal preferences and expected utility evaluations. We show that a voting scheme suggested by Lewis Carroll can be impractical in that it can be computationally prohibitive (specifically, NP-hard) to determine whether any particular candidate has won an election. We also suggest a class of “impracticality theorems” which say that any fair voting scheme must, in the worst-case, require excessive computation to determine a winner. We introduce a class of voting rules called generalized scoring rules. Under such a rule, each vote generates a vector of k scores, and the outcome of the voting rule is based only on the sum of these vectors---more specifically, only on the order (in terms of score) of the sum's components. This class is extremely general: we do not know of any commonly studied rule that is not a generalized scoring rule. We then study the coalitional manipulation problem for generalized scoring rules. We prove that under certain natural assumptions, if the number of manipulators is O(np) (for any p 1 2) and o(n), then the probability that a random profile is manipulable (to any possible winner under the voting rule) is 1--O(e--Ω(n2p--1)). We also show that common voting rules satisfy these conditions (for the uniform distribution). These results generalize earlier results by Procaccia and Rosenschein as well as even earlier results on the probability of an election being tied. We theoretically and experimentally study voter behavior in a setting characterized by plurality rule and mandatory voting, where voters choose from three options. We are interested in the occurrence of strategic voting in an environment where Condorcet cycles may occur. In particular, we focus on how information about the distribution of preferences affects strategic behavior. We also vary the relative importance of the second preferred option to investigate how this affects the strategic vote. Quantal response equilibrium analysis is used to analyze the game and proves to be a good predictor for the experimental data. Our results indeed show that strategic voting arises, the extent of which depends on (i) the availability of information; (ii) the relative importance of the intermediate candidate; (iii) the electorate's relative support for one's preferred candidate; and (iv) the relative position of the plurality-supported candidate in a voter's preference ordering. Our results show that information serves as a coordination device where strategic voting does not harm the plurality-preferred candidate's chances of winning.