Transitive permutation groups and equipotent voting rules

Let F a two-alternative voting rule and G_F the subgroup of permutations of the voters under which F is invariant. Group theoretic properties of G_F provide information about the voting rule F. In particular, sets of imprimitivity of G_F describe the ‘committee decomposition’ structure of F and permutation group transitivity of G_F (equipotency) is shown to be closely connected with equal distribution of power among the voters. If equipotency replaces anonymity in the hypotheses of May's theorem, voting rules other than simple majority are possible. By combining equipotency with two additional social choice conditions a new characterization of simple majority rule is obtained. Equipotency is proposed as an important alternative to the more restrictive anonymity as a fairness criterion in social choice.     

On eligibility by the de Borda voting rules

A criterion for eligibility of a candidate by the set of scoring rules (de Borda voting rules) is given. This criterion generalizes a necessary (but not sufficient) condition of eligibility in Moulin (Axioms of cooperative decision making, Cambridge University Press, Cambridge, 1988). The author thanks Profs. H. Moulin and A. V. Shapovalov for illuminating discussions.     

Asymptotic vulnerability of positional voting rules to coalitional manipulation

Voting rule performances are sometimes evaluated according to their respective resistances to allow profitable misrepresentation of individual preferences. This seems to be a hard task when scoring systems with possibly non integer weights are involved. In this paper, it is shown how one can still obtain asymptotic results in these settings. Our analysis for three-candidate elections provides a characterization of unstable voting situations at which a positional voting rule is manipulable by some coalition not larger than an arbitrary proportion of the electorate. This allows us to address a conjecture by Pritchard and Wilson (2007). That is, under the Impartial Anonymous Culture (IAC), the plurality rule asymptotically minimizes the vulnerability to coalitional manipulation when the size of the manipulating coalition is unrestricted. This later result is no longer valid when only manipulation by small coalitions is considered: now, the Borda rule tends to outperform other rules. Furthermore, the vulnerability of a positional voting rule to coalitional manipulation is not affected by increasing the size of the manipulating coalition from 0.5 to 1.     

A comparison of three linear rules for classification

This paper compares three different linear procedures for classification: the normal one, the canonical one and a distribution-free one recently described by Heuchenne. The study is mainly conducted using a simulation which makes it possible to compute the probabilities of correct allocation of the three methods in 3888 different cases. The normal rule looks slightly better than Heuchenne's, which looks clearly better than the canonical one. Finally, inference on Heuchenne's method is examined and conditions under which this method is optimal are given.     

An experimental study of voting rules and polls in three-candidate elections

We report the results of elections conducted in a laboratory setting, modelled on a threecandidate example due to Borda. By paying subjects conditionally on election outcomes, we create electorates with (publicly) known preferences. We compare the results of experiments with and without non-binding pre-election polls under plurality rule, approval voting, and Borda rule. We also refer to a theory of voting “equilibria,” which makes sharp predictions concerning individual voter behavior and election outcomes. We find that Condorcet losers occasionally win regardless of the voting rule or presence of polls. Duverger's law (which asserts the predominance of two candidates) appears to hold under plurality rule, but close three-way races often arise under approval voting and Borda rule. Voters appear to poll and vote strategically. In elections, voters usually cast votes that are consistent with some strategic equilibrium. By the end of an election series, most votes are consistent with a single equilibrium, although that equilibrium varies by experimental group and voting rule.     

A theoretic study of a distance-based regression model

The distance-based regression model has many applications in analysis of multivariate response regression in various ?elds, such as ecology, genomics, genetics, human microbiomics, and neuroimaging. It yields a pseudo F test statistic that assesses the relation between the distance(dissimilarity) of the subjects and the predictors of interest. Despite its popularity in recent decades, the statistical properties of the pseudo F test statistic have not been revealed to our knowledge. This study derives the asymptotic properties of the pseudo F test statistic using spectral decomposition under the matrix normal assumption, when the utilized dissimilarity measure is the Euclidean or Mahalanobis distance. The pseudo F test statistic with the Euclidean distance has the same distribution as the quotient of two Chi-squared-type mixtures. The denominator and numerator of the quotient are approximated using a random variable of the form ξχ_d~2+ η, and the approximate error bound is given. The pseudo F test statistic with the Mahalanobis distance follows an F distribution.In simulation studies, the approximated distribution well matched the "exact" distribution obtained by the permutation procedure. The obtained distribution was further validated on H1N1 in?uenza data, aging human brain data, and embryonic imprint data.     

A distance-based statistical analysis of fuzzy number-valued data

Real-life data associated with experimental outcomes are not always real-valued. In particular, opinions, perceptions, ratings, etc., are often assumed to be vague in nature, especially when they come from human valuations. Fuzzy numbers have extensively been considered to provide us with a convenient tool to express these vague data. In analyzing fuzzy data from a statistical perspective one finds two key obstacles, namely, the nonlinearity associated with the usual arithmetic with fuzzy data and the lack of suitable models and limit results for the distribution of fuzzy-valued statistics. These obstacles can be frequently bypassed by using an appropriate metric between fuzzy data, the notion of random fuzzy set and a bootstrapped central limit theorem for general space-valued random elements. This paper aims to review these ideas and a methodology for the statistical analysis of fuzzy number data which has been developed along the last years.     

A comparison of pivot selection rules for primal simplex based network codes

This note reports a comparative study of several approaches for selecting the incoming arc in primal simplex specializations for network problems. Our tests establish that a recent method improves on two established choice rules found effective in earlier studies.     

Nervousness in inventory management: Comparison of basic control rules

Using a rolling horizon planning framework in inventory control leads to nervousness in the planning system caused by instability of order release decisions in successive planning cycles. For a single-stage inventory system with arbitrary stochastic demand it is shown analytically, how planning stability is affected by policy parameters if (s, nQ), (s, S), and (T, S) control rules are applied. It turns out that the reorder point s does not influence stability whereas the lot size determining parameters Q, S − s, and T can have a considerable impact. However, this influence turns out to be quite different for different measures of stability regarding order setup or order quantity deviations, respectively. Using these planning stability results the priority of the different control rules under the aspect of nervousness is discussed.     

A Euclidean distance-based measure of efficiency in data envelopment analysis

Data envelopment analysis (DEA) has been proven as an excellent data-oriented performance evaluation method when multiple inputs and outputs are present in a set of peer decision-making units (DMUs). Several efficiency measures have been proposed in the DEA literature, see, for instances, radial efficiency measure of Charnes et al. (CCR)(A. Charnes. W.W. Cooper, and E. Rhodes, 1978. Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2, 429–444), Russell graph measure (J.T. Russell, and R. Sirvant. 1999. An enhanced DEA Russell graph efficiency measure. Eur. J. Oper. Res. 115, pp. 596–607) and slack-based measure of Tone (K. Tone, 2001. A slack-based measure of efficiency in DEA. Eur. J. Oper. Res. 130, p. 498–509). In this article, we will propose an Euclidean distance-based measure of efficiency. Then, in order to discriminate the performance of efficient DMUs, an alternative super-efficiency DEA model is proposed. The applicability of the models developed is illustrated in the context of the analysis of gas companies performance.     

A general framework for distance-based consensus in ordinal ranking models

The problem of aggregating a set of ordinal rankings of n alternatives has given rise to a number of consensus models. Among the most common of these models are those due to Borda and Kendall, which amount to using average ranks, and the ℓ¹ and ℓ² distance models. A common criticism of these approaches is their use of ordinal rank position numbers directly as the values of being ranked at those levels. This paper presents a general framework for associating value or worth with ordinal ranks, and develops models for deriving a consensus based on this framework. It is shown that the ℓ^p distance models using this framework are equivalent to the conventional ordinal models for any p ⩾ 1. This observation can be seen as a form of validation of the practice of using ordinal data in a manner for which it was presumably not designed. In particular, it establishes the robustness of the simple Borda, Kendall and median ranking models.     

Obnoxious-facility location and data-envelopment analysis: A combined distance-based formulation

We have broadened the classic anti-center models to include generalized-distance measures. It includes in the inherent proximity measures other cost/benefit metrics. Unlike classic data envelopment analysis (DEA), the combined location/DEA model proposed here assumes disposability of input/output's only. It represents a more flexible formulation. The locations of multiple sites are analyzed using a binary integer program, while evaluation is performed by the full strength of a DEA model. Through a case study, we show how location and DEA models can be used together to more realistically characterize a siting decision.     

Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

The Lisbon voting system of the Council of the European Union, which became effective in November 2014, cannot be represented as the intersection of six or fewer weighted games, i.e., its dimension is at least 7. This sets a new record for real-world voting bodies. A heuristic combination of different discrete optimization methods yields a representation as the intersection of 13,368 weighted games. Determination of the exact dimension is posed as a challenge to the community. The system’s Boolean dimension is proven to be 3.     

A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is a mesh-free procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.     

A sequential distance-based approach for imputing missing data: Forward Imputation

Missing data recurrently affect datasets in almost every field of quantitative research. The subject is vast and complex and has originated a literature rich in very different approaches to the problem. Within an exploratory framework, distance-based methods such as nearest-neighbour imputation (NNI), or procedures involving multivariate data analysis (MVDA) techniques seem to treat the problem properly. In NNI, the metric and the number of donors can be chosen at will. MVDA-based procedures expressly account for variable associations. The new approach proposed here, called Forward Imputation, ideally meets these features. It is designed as a sequential procedure that imputes missing data in a step-by-step process involving subsets of units according to their “completeness rate”. Two methods within this context are developed for the imputation of quantitative data. One applies NNI with the Mahalanobis distance, the other combines NNI and principal component analysis. Statistical properties of the two methods are discussed, and their performance is assessed, also in comparison with alternative imputation methods. To this purpose, a simulation study in the presence of different data patterns along with an application to real data are carried out, and practical hints for users are also provided.     

A distance-based point-reassignment heuristic for the k-hyperplane clustering problem

We consider the k-Hyperplane Clustering problem where, given a set of m   points in Rⁿ${\mathbb{R}}^n$, we have to partition the set into k subsets (clusters) and determine a hyperplane for each of them, so as to minimize the sum of the squares of the Euclidean distances between the points and the hyperplane of the corresponding clusters. We give a nonconvex mixed-integer quadratically constrained quadratic programming formulation for the problem. Since even very small-size instances are challenging for state-of-the-art spatial branch-and-bound solvers like Couenne, we propose a heuristic in which many “critical” points are reassigned at each iteration. Such points, which are likely to be ill-assigned in the current solution, are identified using a distance-based criterion and their number is progressively decreased to zero. Our algorithm outperforms the best available one proposed by Bradley and Mangasarian on a set of real-world and structured randomly generated instances. For the largest instances, we obtain an average improvement in the solution quality of 54%.     

A three-dimensional voting system in Hong Kong

The voting system of the Legislative Council of Hong Kong (Legco) is sometimes unicameral and sometimes bicameral, depending on whether the bill is proposed by the Hong Kong government. Therefore, although without any representative within Legco, the Hong Kong government has certain degree of legislative power – as if there is a virtual representative of the Hong Kong government within the Legco. By introducing such a virtual representative of the Hong Kong government, we show that Legco is a three-dimensional voting system. We also calculate two power indices of the Hong Kong government through this virtual representative and consider the C-dimension and the W-dimension of Legco. Finally, some implications of this Legco model to the current constitutional reform in Hong Kong will be given.     

A core of voting games with improved foresight

This paper considers voting situations in which the vote takes place iteratively. If a coalition replaces the status quo a with a contestant b, then b becomes the new status quo, and the vote goes on until a candidate is reached that no winning coalition is willing to replace. It is well known that the core, that is, the set of undominated alternatives, may be empty. To alleviate this problem, Rubinstein [Rubinstein, A., 1980. Stability of decision systems under majority rule. Journal of Economic Theory 23, 150–159] assumes that voters look forward one vote before deciding to replace an alternative by a new one. They will not do so if the new status quo is going to be replaced by a third that is less interesting than the first. The stability set, that is, the set of undominated alternatives under this behavior, is always non-empty when preferences are strict. However, this is not necessarily the case when voters’ indifference is allowed. Le Breton and Salles [Le Breton, M., Salles, M., 1990. The stability set of voting games: Classification and generecity results. International Journal of Game Theory 19, 111–127], Li [Li, S., 1993. Stability of voting games. Social Choice and Welfare 10, 51–56] and Martin [Martin, M., 1998. Quota games and stability set of order d. Economic Letters 59, 145–151] extend the sophistication of the voters by having them look d votes forward along the iterative process. For d sufficiently large, the resulting set of undominated alternatives is always non-empty even if indifference is allowed. We show that it may be unduly large. Next, by assuming that other voters along a chain of votes are also rational, that is, they also look forward to make sure that the votes taking place later on will not lead to a worst issue for them, we are able to reduce the size of this set while insuring its non-emptiness. Finally, we show that a vote with sufficient foresight satisfies a no-regret property, contrarily to the classical core and the stability set.