We propose the PeerRank method for peer assessment. This constructs a grade for an agent based on the grades proposed by the agents evaluating the agent. Since the grade of an agent is a measure of their ability to grade correctly, the PeerRank method weights grades by the grades of the grading agent. The PeerRank method also provides an incentive for agents to grade correctly. As the grades of an agent depend on the grades of the grading agents, and as these grades themselves depend on the grades of other agents, we define the PeerRank method by a fixed point equation similar to the PageRank method for ranking web-pages. We identify some formal properties of the PeerRank method (for example, it satisfies axioms of unanimity, no dummy, no discrimination and symmetry), discuss some examples, compare with related work and evaluate the performance on some synthetic data. Our results show considerable promise, reducing the error in grade predictions by a factor of 2 or more in many cases over the natural baseline of averaging peer grades.
Another related problem is selection from the selectors'' @cite_3 . The goal here is to select a subset of @math agents from a group (e.g. to select a subcommittee). The problem of awarding a prize from a group of peers can be seen as the special case of @math . As approval voting is not impartial, Alon, Fischer, Procaccia and Tennenholtz look for impartial rules that approximate approval voting (that is, guarantee that the total approval scores of the @math winners are within a fixed fraction of the optimal answer). Again, a difference with this work is that we are not trying to achieve or approximate impartiality.
@cite_3
2093375346
A group of peers must choose one of them to receive a prize; everyone cares only about winning, not about who gets the prize if someone else. An award rule is impartial if one's message never influences whether or not one wins the prize. We explore the consequences of impartiality when each agent nominates a single (other) agent for the prize. @PARASPLIT On the positive side, we construct impartial nomination rules where both the influence of individual messages and the requirements to win the prize are not very different across agents. Partition the agents in two or more districts, each of size at least 3, and call an agent a local winner if he is nominated by a majority of members of his own district; the rule selects a local winner with the largest support from nonlocal winners, or a fixed default agent in case there is no local winner. @PARASPLIT On the negative side, impartiality implies that ballots cannot be processed anonymously as in plurality voting. Moreover, we cannot simultaneously guarantee that the winner always gets at least one nomination, and that an agent nominated by everyone else always wins. We examine strategy-proof elections to select a winner amongst a set of agents, each of whom cares only about winning. This impartial selection problem was introduced independently by Holzman and Moulin [5] and [1]. Fischer and Klimm [4] showed that the permutation mechanism is impartial and ( )-optimal, that is, it selects an agent who gains, in expectation, at least half the number of votes of the most popular agent. Furthermore, they showed the mechanism is ( 7 12 )-optimal if agents cannot abstain in the election. We show that a better guarantee is possible, provided the most popular agent receives at least a large enough, but constant, number of votes. Specifically, we prove that, for any e > 0, there is a constant N e (independent of the number n of voters) such that, if the maximum number of votes of the most popular agent is at least N e then the permutation mechanism is (( 3 4 - ) )-optimal. This result is tight. We study computational aspects of three prominent voting rules that use approval ballots to select multiple winners. These rules are proportional approval voting, reweighted approval voting, and satisfaction approval voting. Each rule is designed with the intention to compute a representative winning set. We show that computing the winner for proportional approval voting is NP-hard, closing an open problem (Kilgour, 2010). As none of the rules we examine are strategy-proof, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents. In many settings, we show that it is NP-hard for an agent or agents to compute how best to vote given a fixed set of approval ballots of the other agents. This paper is an axiomatic study of consistent approval-based multi-winner rules, i.e., voting rules that select a fixed-size group of candidates based on approval ballots. We introduce the class of counting rules, provide an axiomatic characterization of this class and, in particular, show that counting rules are consistent. Building upon this result, we axiomatically characterize three important consistent multi-winner rules: Proportional Approval Voting, Multi-Winner Approval Voting and the Approval Chamberlin--Courant rule. Our results demonstrate the variety of multi-winner rules and illustrate three different, orthogonal principles that multi-winner voting rules may represent: individual excellence, diversity, and proportionality. We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show that computing the winner for proportional approval voting is NP-hard, closing a long standing open problem. As none of the rules are strategyproof, even for dichotomous preferences, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents. In many settings, we show that it is NP-hard for an agent or agents to compute how best to vote given a fixed set of approval ballots from the other agents. We consider the special case of approval voting when the set of agents and the set of alternatives coincide. This captures situations in which the members of an organization want to elect a president or a committee from their ranks, as well as a variety of problems in networked environments, for example in internet search, social networks like Twitter, or reputation systems like Epinions. More precisely, we look at a setting where each member of a set of n agents approves or disapproves of any other member of the set and we want to select a subset of k agents, for a given value of k, in a strategyproof and approximately efficient way. Here, strategyproofness means that no agent can improve its own chances of being selected by changing the set of other agents it approves. A mechanism is said to provide an approximation ratio of α for some α ≥ 1 if the ratio between the sum of approval scores of any set of size k and that of the set selected by the mechanism is always at most α. We show that for k ∈ 1, 2,..., n − 1 , no deterministic strategyproof mechanism can provide a finite approximation ratio. We then present a randomized strategyproof mechanism that provides an approximation ratio that is bounded from above by four for any value of k, and approaches one as k grows. We study the problem of selecting a member of a set of agents based on impartial nominations by agents from that set. The problem was studied previously by and by Holzman and Moulin and has important applications in situations where representatives are selected from within a group or where publishing or funding decisions are made based on a process of peer review. Our main result concerns a randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. Subject to impartiality, this is best possible. In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations. We prove the following theorem: if we fix the number of alternatives, generate @math votes i.i.d. according to a distribution @math , and let @math go to infinity, then for any @math , with probability at least @math , the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) @math , (3) @math , and (4) @math . This theorem holds for any set of vote operations, any individual vote distribution @math , and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also show that many well-studied types of strategic behavior fall under our framework, including (but not limited to) constructive destructive manipulation, bribery, and control by adding deleting votes, margin of victory, and minimum manipulation coalition size. Therefore, our main theorem naturally applies to these problems. 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. The margin of victory of an election is a useful measure to capture the robustness of an election outcome. It also plays a crucial role in determining the sample size of various algorithms in post election audit, polling etc. In this work, we present efficient sampling based algorithms for estimating the margin of victory of elections. More formally, we introduce the problem, where given an election @math on @math voters, the goal is to estimate the margin of victory @math of @math within an additive factor of @math . We study the problem for many commonly used voting rules including scoring rules, approval, Bucklin, maximin, and Copeland @math We observe that even for the voting rules for which computing the margin of victory is NP-Hard, there may exist efficient sampling based algorithms, as observed in the cases of maximin and Copeland @math voting rules. to the Handbook on Approval Voting.- History of Approval Voting.- Acclamation Voting in Sparta: An Early Use of Approval Voting.- Going from Theory to Practice: The Mixed Success of Approval Voting.- Axiomatic Theory.- Collective Choice for Simple Preferences.- Axiomatizations of Approval Voting.- Committees.- Approval Balloting for Multi-winner Elections.- Does Choosing Committees from Approval Balloting Fulfill the Electorate's Will?.- Strategic Voting.- The Basic Approval Voting Game.- Approval Voting in Large Electorates.- Computational Aspects of Approval Voting.- Probabilistic Exercises.- On the Condorcet Efficiency of Approval Voting and Extended Scoring Rules for Three Alternatives.- Behavioral Heterogeneity Under Approval and Plurality Voting.- In Silico Voting Experiments.- Experiments.- Laboratory Experiments on Approval Voting.- Framed Field Experiments on Approval Voting: Lessons from the 2002 and 2007 French Presidential Elections.- Approval Voting in Germany: Description of a Field Experiment.- Electoral Competition.- Classical Electoral Competition Under Approval Voting.- Policy Moderation and Endogenous Candidacy in Approval Voting Elections.- Meaning for Individual and Society.- Describing Society Through Approval Data.- Approval as an Intrinsic Part of Preference. 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. We study sincere-strategy preference-based approval voting (SP-AV), a system proposed by Brams and Sanver [1] and here adjusted so as to coerce admissibility of the votes (rather than excluding inadmissible votes a priori), with respect to procedural control. In such control scenarios, an external agent seeks to change the outcome of an election via actions such as adding deleting partitioning either candidates or voters. SP-AV combines the voters' preference rankings with their approvals of candidates, where in elections with at least two candidates the voters' approval strategies are adjusted – if needed – to approve of their most-preferred candidate and to disapprove of their least-preferred candidate. This rule coerces admissibility of the votes even in the presence of control actions, and hybridizes, in effect, approval with pluralitiy voting. We prove that this system is computationally resistant (i.e., the corresponding control problems are NP-hard) to 19 out of 22 types of constructive and destructive control. Thus, SP-AV has more resistances to control than is currently known for any other natural voting system with a polynomial-time winner problem. In particular, SP-AV is (after Copeland voting, see [2, 3]) the second natural voting system with an easy winner-determination procedure that is known to have full resistance to constructive control, and unlike Copeland voting it in addition displays broad resistance to destructive control (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 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. We consider the problem of manipulation of elections using positional voting rules under impartial culture voter behaviour. We consider both the logical possibility of coalitional manipulation, and the number of voters who must be recruited to form a manipulating coalition. It is shown that the manipulation problem may be well approximated by a very simple linear program in two variables. This permits a comparative analysis of the asymptotic (large-population) manipulability of the various rules. It is seen that the manipulation resistance of positional rules with 5 or 6 (or more) candidates is quite different from the more commonly analyzed three- and four-candidate cases. In cooperative multiagent systems an alternative that maximizes the social welfare-the sum of utilities-can only be selected if each agent reports its full utility function. This may be infeasible in environments where communication is restricted. Employing a voting rule to choose an alternative greatly reduces the communication burden, but leads to a possible gap between the social welfare of the optimal alternative and the social welfare of the one that is ultimately elected. Procaccia and Rosenschein (2006) [13] have introduced the concept of distortion to quantify this gap. In this paper, we present the notion of embeddings into voting rules: functions that receive an agent@?s utility function and return the agent@?s vote. We establish that very low distortion can be obtained using randomized embeddings, especially when the number of agents is large compared to the number of alternatives. We investigate our ideas in the context of three prominent voting rules with low communication costs: Plurality, Approval, and Veto. Our results arguably provide a compelling reason for employing voting in cooperative multiagent systems. We consider elections where the chair replaces either candidates or votes, with the goal of making a specific candidate win (constructive control) or lose (destructive control). We call this replacement control'' and study its computational complexity for several scoring rules (plurality, veto, Borda, k-approval), as well as for approval voting. 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 Approval Voting systems where each voter decides on a subset of candidates he she approves. We focus on the optimization problem of finding the committee of fixed size k, minimizing the maximal Hamming distance from a vote. In this paper we give a PTAS for this problem and hence resolve the open question raised by [AAAI’10]. The result is obtained by adapting the techniques developed by [JACM’02] originally used for the less constrained Closest String problem. The technique relies on extracting information and structural properties of constant size subsets of votes. The margin of victory of an election, defined as the smallest number k such that k voters can change the winner by voting differently, is an important measurement for robustness of the election outcome. It also plays an important role in implementing efficient post-election audits, which has been widely used in the United States to detect errors or fraud caused by malfunctions of electronic voting machines. In this paper, we investigate the computational complexity and (in)approximability of computing the margin of victory for various voting rules, including approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also prove a dichotomy theorem, which states that for all continuous generalized scoring rules, including all voting rules studied in this paper, either with high probability the margin of victory is Θ(√n), or with high probability the margin of victory is Θ(n), where n is the number of voters. Most of our results are quite positive, suggesting that the margin of victory can be efficiently computed. This sheds some light on designing efficient post-election audits for voting rules beyond the plurality rule.