A Consistent Extension of Condorcet’s Election Principle

Economic Papers: A journal of applied economics and policy, 2011

There are three main aggregation techniques designed for progress measurements to help facilitate the benchmarking and ranking of countries according to aggregated dimensions. They are: (i) additive methods; (ii) geometric aggregations; and (iii) non-compensatory multi-criteria analysis. Virtually, all measures utilise one of the first two techniques. This article will critically review these aggregation approaches. In doing so, this article will assert that the Condorcet approach, despite being overlooked by many major institutions, demands strong consideration for aggregating progress measures. The theoretical implications of using the Condorcet method will be explored. An empirical application of the Condorcet model is undertaken on the Resource-Infrastructure-Environment Index to test the validity of this approach as an aggregation technique for progress measures.

Fiscal Studies, 2013

There is a widespread consensus that well-being is a multidimensional notion. To quantify multidimensional well-being, information on the relative weights of the different dimensions is essential. There is, however, considerable disagreement in the literature on the most appropriate weighting scheme to be used. Making use of a recent data set for Flanders, we compare various methods to select a weighting scheme. The results are indeed different such that, for instance, a policymaker would identify different groups of individuals as being worst-off depending on the scheme that is chosen. In order to compare and evaluate the weighting schemes, we simulate the support each scheme would get in a hypothetical voting procedure. Weighting schemes that obtain a higher support reflect better the priorities of the respondents themselves and suffer less from the problem of paternalism. Quite remarkably, the popular equal weighting scheme is found to be the least supported in our data set. de gaer for their valuable comments. All remaining shortcomings are ours.

The American Political Science Review, 1988

Condcrcet's criterion states that an alternative that defeats every other by a simple majority is the socially optimal choice. Condorcet argued that if the object of voting is to determine the “best” decision for society but voters sometimes make mistakes in their judgments, then the majority alternative (if it exists) is statistically most likely to be the best choice. Strictly speaking, this claim is not true; in some situations Bordas rule gives a sharper estimate of the best alternative. Nevertheless, Condorcet did propose a novel and statistically correct rule for finding the most likely ranking of the alternatives. This procedure, which is sometimes known as “Kemeny's rule,” is the unique social welfare function that satisfies a variant of independence of irrelevant alternatives together with several other standard properties.

i-Perception, 2011

We address the topic of "pictorial depth" in cases of pictures that are unlike photographic renderings. The most basic measure of "depth" is no doubt that of depth order. We establish depth order through the pairwise depth-comparison method, involving all pairs from a set of 49 fiducial points. The pictorial space for this study was evoked by a capriccio (imaginary landscape) by Francesco Guardi (1712-1793). In such a drawing pictorial space is suggested by the artist through a small set of conventional depth cues. As a result typical Western observers tend to agree largely in their visual awareness when looking at such art. We rank depths for locations that are not on a single surface and far apart in pictorial space. We find that observers resolve about 40 distinct depth layers and agree largely in this. From a previous experiment we have metrical data for the same observers. The rank correlations between the results are high. Perhaps surprisingly, we find no c...

Journal of Classification, 1994

The plurality rule for molecular sequences is a consensus function gO which maps each profile of length k (i.e., each sequence of k bases appearing at an aligned position of k molecules) to a set of consensus results (i.e., ambiguity codes) that are descriptive summaries of the profile. Since the plurality rule go is a median procedure, its results are solutions of an optimization problem that is well known and intensively studied in the theory of social choice. However, go fails to satisfy properties of consistency that enable users to understand go's behavior for long profiles according to its behavior for short profiles. Because consistency is a desirable feature of consensus functions, we explore the boundaries of its applicability to go. We distinguish between two types of consistency, weak and strong, and we apply them to profiles and consensus results as well as to go itself. For go we obtain simple characterizations of weakly consistent profiles, of weakly consistent results, and of strongly consistent results.

Life, 2016

The existence of multiple copies of genes is a well-known phenomenon. A gene family is a set of sufficiently similar genes, formed by gene duplication. In earlier works conducted on a limited number of completely sequenced and annotated genomes it was found that size of gene family and size of genome are positively correlated. Additionally, it was found that several atypical microbes deviated from the observed general trend. In this study, we reexamined these associations on a larger dataset consisting of 1484 prokaryotic genomes and using several ranking approaches. We applied ranking methods in such a way that genomes with lower numbers of gene copies would have lower rank. Until now only simple ranking methods were used; we applied the Kemeny optimal aggregation approach as well. Regression and correlation analysis were utilized in order to accurately quantify and characterize the relationships between measures of paralog indices and genome size. In addition, boxplot analysis was employed as a method for outlier detection. We found that, in general, all paralog indexes positively correlate with an increase of genome size. As expected, different groups of atypical prokaryotic genomes were found for different types of paralog quantities. Mycoplasmataceae and Halobacteria appeared to be among the most interesting candidates for further research of evolution through gene duplication.

Social Indicators Research

In recent times, composite indicators have gained astounding popularity in a wide variety of research areas. Their adoption by global institutions has further captured the attention of the media and policymakers around the globe, and their number of applications has surged ever since. This increase in their popularity has solicited a plethora of methodological contributions in response to the substantial criticism surrounding their underlying framework. In this paper, we put composite indicators under the spotlight, examining the wide variety of methodological approaches in existence. In this way, we offer a more recent outlook on the advances made in this field over the past years. Despite the large sequence of steps required in the construction of composite indicators, we focus particularly on two of them, namely weighting and aggregation. We find that these are where the paramount criticism appears and where a promising future lies. Finally, we review the last step of the robustness analysis that follows their construction, to which less attention has been paid despite its importance. Overall, this study aims to provide both academics and practitioners in the field of composite indices with a synopsis of the choices available alongside their recent advances.

Lecture Notes in Computer Science, 2010

RANK AGGREGATION is important in many areas ranging from web search over databases to bioinformatics. The underlying decision problem KE-MENY SCORE is NP-complete even in case of four input rankings to be aggregated into a "median ranking". We study efficient polynomial-time data reduction rules that allow us to find optimal median rankings. On the theoretical side, we improve a result for a "partial problem kernel" from quadratic to linear size. On the practical side, we provide encouraging experimental results with data based on web search and sport competitions, e.g., computing optimal median rankings for real-world instances with more than 100 candidates within milliseconds. 1 Introduction We investigate the effectiveness of data reduction for computing optimal solutions of the NP-hard RANK AGGREGATION problem. Kemeny's corresponding voting scheme can be described as follows. An election (V, C) consists of a set V of n votes and a set C of m candidates. A vote or a ranking is a total order of all candidates. For instance, in case of three candidates a, b, c, the order c > b > a means that candidate c is the best-liked one and candidate a is the least-liked one. For each pair of votes v, w, the Kendall-Tau distance (also known as the number of inversions) between v and w is defined as KT-dist(v, w) = {c,d}⊆C d v,w (c, d), where d v,w (c, d) is set to 0 if v and w rank c and d in the same order, and is set to 1, otherwise. The score of a ranking l with respect to an election (V, C) is defined as v∈V KT-dist(l, v). A ranking l with a minimum score is called a Kemeny ranking of (V, C) and its score is the Kemeny score of (V, C). The central problem considered in this work is as follows: RANK AGGREGATION: Given an election (V, C), find a Kemeny ranking of (V, C). Its decision variant KEMENY SCORE asks whether there is a Kemeny ranking of (V, C) with score at most some additionally given positive integer k. The RANK AGGREGA-TION problem has numerous applications, ranging from building meta-search engines for the web or spam detection [7] over databases [8] to the construction of genetic maps Supported by the DFG, research project PAWS, NI 369/10.

Mathematical Programming, 2011

We propose a technique that we call HodgeRank for ranking data that may be incomplete and imbalanced, characteristics common in modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct pairwise rankings, represented as

Rationality and Society, 1990

2019

Given a profile of preferences on a set of alternatives, a majority relation is a complete binary relation that agrees with the strict preference of a strict majority of these preferences whenever such strict strict majority is observed. We show that a majority binary relation is, among all conceivable binary relations, the most representative of the profile of preferences from which it emanates. We define "the most representative" to mean "the closest in the aggregate". This requires a definition of what it means for a pair of preferences to be closer to each other then another. We assume that this definition takes the form of a distance function defined over the set of all conceivable preferences. We identify a necessary and sufficient condition for such a distance to be minimized by the preference of the majority. This condition requires the distance to be additive with respect to a plausible notion of compromise between preferences. The well-known Kemeny dist...

Decision Analysis

Rank aggregation is widely used in group decision making and many other applications, where it is of interest to consolidate heterogeneous ordered lists. Oftentimes, these rankings may involve a large number of alternatives, contain ties, and/or be incomplete, all of which complicate the use of robust aggregation methods. In particular, these characteristics have limited the applicability of the aggregation framework based on the Kemeny-Snell distance, which satisfies key social choice properties that have been shown to engender improved decisions. This work introduces a binary programming formulation for the generalized Kemeny rank aggregation problem—whose ranking inputs may be complete and incomplete, with and without ties. Moreover, it leverages the equivalence of two ranking aggregation problems, namely, that of minimizing the Kemeny-Snell distance and of maximizing the Kendall-τ correlation, to compare the newly introduced binary programming formulation to a modified version o...

Mathematics, 2021

Preference aggregation and in particular ranking aggregation are mainly studied by the field of social choice theory but extensively applied in a variety of contexts. Among the most prominent methods for ranking aggregation, the Kemeny method has been proved to be the only one that satisfies some desirable properties such as neutrality, consistency and the Condorcet condition at the same time. Unfortunately, the problem of finding a Kemeny ranking is NP-hard, which prevents practitioners from using it in real-life problems. The state of the art of exact algorithms for the computation of the Kemeny ranking experienced a major boost last year with the presentation of an algorithm that provides searching time guarantee up to 13 alternatives. In this work, we propose an enhanced version of this algorithm based on pruning the search space when some Condorcet properties hold. This enhanced version greatly improves the performance in terms of runtime consumption.

Journal of Ambient Intelligence and Humanized Computing

The time required for solving the ranking aggregation problem using the Kemeny method increases factorially with the number of alternatives to be ranked, which prevents its use when this number is large. Exact algorithms use domain information to discard rankings as possible solution, thus saving runtime. The amount of rankings that can be discarded varies for each profile and cannot be known beforehand. For profiles of rankings with large number of alternatives, the amount of rankings discarded highly affects the feasibility of the computation of Kemeny ranking. How to identify the profiles that are more time-consuming when finding the Kemeny ranking is not trivial. In this work we propose the use of machine learning models to predict how difficult is to obtain the Kemeny ranking in terms of runtime. The results obtained are promising, with values of the area under the curve metric over 80%. Furthermore, it is possible to extract from the proposed models the characteristics of the ...

Group Decision and Negotiation, 2020

Voting paradoxes have played an important role in the theory of voting. They typically say very little about the circumstances in which they are particularly likely or unlikely to occur. They are basically existence findings. In this article we study some well known voting paradoxes under the assumption that the underlying profiles are drawn from the Condorcet domain, i.e. a set of preference profiles where a Condorcet winner exists. The motivation for this restriction is the often stated assumption that profiles with a Condorcet winner are more likely than those without it. We further restrict the profiles by assuming that the starting point of our analysis is that the Condorcet winner coincides with the choice of the voting rule under scrutiny. The reason for making this additional restriction is that—intuitively—the outcomes that coincide with the Condorcet winner make those outcomes stable and, thus, presumably less vulnerable to various voting paradoxes. It will be seen that th...

International Journal of Information Technology & Decision Making, 2016

In this paper, we develop a method based on the idea of pairwise voting to rank projects or candidates and incorporate in the ranking process how strongly the referees/voters feel about the comparisons they make. Voting is a modified form of ranking and all the votes are equally important. However, there are situations similar to voting in which the votes are not just ordinal but each voter expresses an intensity of preference for the different candidates, e.g., ranking projects for funding. We show that our method yields the same results as ordinal voting in large populations when the intensity of preferences becomes extreme. Voting with intensity of preferences does not violate democracy but soften the stand of voters and allows for consideration of the diversity of issues involved in voting.

Public Choice, 2016

This paper expands the illustration and analysis regarding the susceptibility of nine voting procedures to two types of what are generally known as No-Show paradoxes. Following the article by Felsenthal and Tideman (Theory and Decision 75:59-77, 2013), the two paradoxes are denoted as P-TOP and P-BOT paradoxes. According to the P-TOP paradox it is possible that if candidate x has been elected by a given electorate then, ceteris paribus, another candidate, y, may be elected if additional voters join the electorate who rank x at the top of their preference ordering. Similarly, according to the P-BOT paradox it is possible that if candidate y has not been elected by a given electorate then, ceteris paribus, y may be elected if additional voters join the electorate who rank y at the bottom of their preference ordering. Voting procedures that are susceptible to these paradoxes are considered to be afflicted with a particularly serious defect because instead of encouraging voters to participate in an election and vote according to their true preference orderings, they may inhibit voters from participating in an election and thereby undermine the rationale for conducting elections.

Nature Communications, 2020

Eyewitness misidentification accounts for 70% of verified erroneous convictions. To address this alarming phenomenon, research has focused on factors that influence likelihood of correct identification, such as the manner in which a lineup is conducted. Traditional lineups rely on overt eyewitness responses that confound two covert factors: strength of recognition memory and the criterion for deciding what memory strength is sufficient for identification. Here we describe a lineup that permits estimation of memory strength independent of decision criterion. Our procedure employs powerful techniques developed in studies of perception and memory: perceptual scaling and signal detection analysis. Using these tools, we scale memory strengths elicited by lineup faces, and quantify performance of a binary classifier tasked with distinguishing perpetrator from innocent suspect. This approach reveals structure of memory inaccessible using traditional lineups and renders accurate identifications uninfluenced by decision bias. The approach furthermore yields a quantitative index of individual eyewitness performance.

Lecture Notes in Computer Science, 2012

Social Choice and Welfare, 2016

Annals of Operations Research, 2019

Given a ranking of elements of a set and given a disjoint partition of the same set, the ranking does not generally imply a total order of the partition. In this paper, we introduce the Kendallτ partition ranking, a linear order of the subsets of the partition which follows from the given ranking. We prove that, under certain assumptions, the Kendall-τ partition ranking is robust, in the sense that it remains the same when removing subsets of the partition. Then, we give several results (properties) concerning the adequacy of the ranking for ordering the partition and we prove that the integrality gap of the 0-1 problem associated to the Kendall-τ partition ranking tends to 8/7. Additionally, we provide a comparison of the new ranking with respect to the mean and median based scores from a theoretical and empirical point of view. Finally, a real application with data from the Programme for International Student Assessment is presented, where countries are ordered based on their school rankings.