Voting power in a system of compound majority rule

In 1973, the United States Supreme Court ruled that a water district's voting scheme that apportioned votes on the basis of the assessed valuation of acreage in the district was constitutional. Among the justifications for the constitutionality of this scheme was the concurrent requirement that legislation be approved by a majority of voters as well as by a majority of weighted votes. However, analysis of this voting scheme in game-theoretic terms indicates that this justification is only partial: when two sets of winning coalitions must form simultaneously in order to pass legislation, the voting power of each voter in the combined system equals the mean of the voting power afforded each voter in each simple system. The results can be generalized to three or more concurrent requirements.     

Asymptotic probability of majority inversion under a general binomial voting model

We obtain limits on the probability of majority inversion when the number of voters tends to infinity, for a binomial voting model specific to each state with different population sizes, and for different voting quotas in the two stages of the voting procedure. The case of weighted votes at the second stage is also discussed. For an important special case where the limit cannot be determined, we provide an exact expression for the inversion probability, but only for unweighted votes.     

On the inverse power index problem

Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley–Shubik and the (normalized) Banzhaf power index.     

Optimal bounds for the no-show paradox via SAT solving

One of the most important desirable properties in social choice theory is Condorcet-consistency, which requires that a voting rule should return an alternative that is preferred to any other alternative by some majority of voters. Another desirable property is participation, which requires that no voter should be worse off by joining an electorate. A seminal result by Moulin (1988) has shown that Condorcet-consistency and participation are incompatible whenever there are at least 4 alternatives and 25 voters. We leverage SAT solving to obtain an elegant human-readable proof of Moulin’s result that requires only 12 voters. Moreover, the SAT solver is able to construct a Condorcet-consistent voting rule that satisfies participation as well as a number of other desirable properties for up to 11 voters, proving the optimality of the above bound. We also obtain tight results for set-valued and probabilistic voting rules, which complement and significantly improve existing theorems.     

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.     

Ranking players by “power” in weighted voting systems: The multi-relation method

A new quantitative definition of the “power index” is proposed for a voter in weighted voting systems (WV-schemes), where the voters a priori do not have equal rights. The proposed “power index” is generated in a self-consistent manner from information about the number of times that a voter enters any winning coalition in the WV-scheme. Explicit formulas are derived for computing the “power index”. The discussion is illustrated with prototype examples. __________ Translated from Prikladnaya Matematika i Informatika, No. 25, pp. 81–98, 2007.     

A new proposed model of restricted data envelopment analysis by correlation coefficients

The concept of efficiency in data envelopment analysis (DEA) is defined as weighted sum of outputs/weighted sum of inputs. In order to calculate the maximum efficiency score, each decision making unit (DMU)’s inputs and outputs are assigned to different weights. Hence, the classical DEA allows the weight flexibility. Therefore, even if they are important, the inputs or outputs of some DMUs can be assigned zero (0) weights. Thus, these inputs or outputs are neglected in the evaluation. Also, some DMUs may be defined as efficient even if they are inefficient. This situation leads to unrealistic results. Also to eliminate the problem of weight flexibility, weight restrictions are made in DEA. In our study, we proposed a new model which has not been published in the literature. We describe it as the restricted data envelopment analysis ((ARIII(COR))) model with correlation coefficients. The aim for developing this new model, is to take into account the relations between variables using correlation coefficients. Also, these relations were added as constraints to the CCR and BCC models. For this purpose, the correlation coefficients were used in the restrictions of input–output each one alone and their combination together. Inputs and outputs are related to the degree of correlation between each other in the production. Previous studies did not take into account the relationship between inputs/outputs variables. So, only with expert opinions or an objective method, weight restrictions have been made. In our study, the weights for input and output variables were determined, according to the correlations between input and output variables. The proposed new method is different from other methods in the literature, because the efficiency scores were calculated at the level of correlations between the input and/or output variables.     

Improving weak efficiency frontiers in the fuzzy data envelopment analysis models

Evaluating the performance of activities or organization by common data envelopment analysis models requires crisp input/output data. However, the precise inputs and outputs of production processes cannot be always measured. Thus, the data envelopment analysis measurement containing fuzzy data, called “fuzzy data envelopment analysis”, has played an important role in the evaluation of efficiencies of real applications. This paper focuses on the fuzzy CCR model and proposes a new method for determining the lower bounds of fuzzy inputs and outputs. This improves the weak efficiency frontiers of the corresponding production possibility set. Also a numerical example illustrates the capability of the proposed method.     

A formal framework for quantifying voter-controlled privacy

Privacy is a necessary requirement for voting. Without privacy, voters can be forced to vote in specific ways, and the forcing party can check their compliance. But offering privacy does not suffice: if a voter can reduce her privacy, an attacker can force her to do so. In this paper, we distinguish various ways that a voter can communicate with the intruder to reduce her privacy and classify them according to their ability to reduce the privacy of a voter. We develop a framework combining knowledge reasoning and trace equivalences to formally model voting protocols and define voter-controlled privacy. Our framework is quantitative, in the sense that it defines a measure for the privacy of a voter. Therefore, the framework can precisely measure the level of privacy for a voter for each of the identified privacy-reduction classes. The quantification allows our framework to capture receipts that reduce, but not nullify, the privacy of the voter.     

On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the ??yea?? voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243?C260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9?voters which do not admit a unique minimum sum integer weight representation.     

A new data envelopment analysis model with fuzzy random inputs and outputs

This paper first presents several formulas for mean chance distributions of triangular fuzzy random variables and their functions, then develops a new class of fuzzy random data envelopment analysis (FRDEA) models with mean chance constraints, in which the inputs and outputs are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. According to the established formulas for the mean chance distributions, we can turn the mean chance constraints into their equivalent stochastic ones. On the other hand, since the objective in the FRDEA model is the expectation about the ratio of the weighted sum of outputs and the weighted sum of inputs for a target decision-making unite (DMU), for general fuzzy random inputs and outputs, we suggest an approximation method to evaluate the objective; and for triangular fuzzy random inputs and outputs, we propose a method to reduce the objective to its equivalent stochastic one. As a consequence, under the assumption that the inputs and the outputs are triangular fuzzy random vectors, the proposed FRDEA model can be reduced to its equivalent stochastic programming one, in which the constraints contain the standard normal distribution function, and the objective is the expectation for a function of the normal random variable. To solve the equivalent stochastic programming model, we design a hybrid algorithm by integrating stochastic simulation and genetic algorithm (GA). Finally, one numerical example is presented to demonstrate the proposed FRDEA modeling idea and the effectiveness of the designed hybrid algorithm.     

Properties of voting with veto power

The article investigates a game-theoretical model with veto power in application to the election of the chairman of the board in a corporation. Alternative voting rules are considered, the most interesting being open voting in a specified order. Conditions are determined when voter 1 can ensure the election of his candidate. A complete solution is obtained for three voters, both in the case of strict preferences and in the case when the preferences of voters 2 and 3 are incompletely defined (contain an uncertainty). The latter case is particularly relevant because it arises for several decision-making rules and is an inseparable part of the real-life election process, when the voters represent several interest groups.     

Dominant strategies and restricted ballots with variable electorate

This paper considers dominant (uniformly best) voting strategies in a variable-electorate model for social choice with restricted ballot sets. A voter can have any weak preference order on the alternatives. A voter who partitions the alternatives into three or more indifference classes never has a dominant strategy. The paper specifies conditions on a social choice function that are necessary and sufficient for a voter who partitions the alternatives into two indifference classes to have a dominant strategy.     

Incoherent majorities: The McGarvey problem in judgement aggregation

Judgement aggregation is a model of social choice where the space of social alternatives is the set of consistent truth-valuations (‘judgements’) on a family of logically interconnected propositions. It is well known that propositionwise majority voting can yield logically inconsistent judgements. We show that, for a variety of spaces, propositionwise majority voting can yield any possible judgement. By considering the geometry of sub-polytopes of the Hamming cube, we also estimate the number of voters required to achieve all possible judgements. These results generalize the classic results of McGarvey (1953) [13] and Stearns (1959) [22].     

Reliability analysis and optimization of weighted voting systems with continuous states input

Weighted voting systems are widely used in many practical fields such as target detection, human organization, pattern recognition, etc. In this paper, a new model for weighted voting systems with continuous state inputs is formulated. We derive the analytical expression for the reliability of the entire system under certain distribution assumptions. A more general Monte Carlo algorithm is also given to numerically analyze the model and evaluate the reliability. This paper further proposes a reliability optimization problem of weighted voting systems under cost constraints. A genetic algorithm is introduced and applied as the optimization technique for the model formulated. A numerical example is then presented to illustrate the ideas.     

Radial DEA models without inputs or without outputs

In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial efficiency score) and one less constraint (the convexity constraint).     

Majority voting leads to unanimity

We consider a situation where society decides, through majority voting in a secret ballot, between the alternatives of ‘reform’ and ‘status quo’. Reform is assumed to create a minority of winners, while being efficient in the Kaldor–Hicks sense. We explore the consequences of allowing binding transfers between voters conditional on the chosen alternative. In particular, we establish conditions under which the winners wish to compensate all losers, thus leading to unanimity for reform, rather than compensating some losers to form a non-maximal majority. The analysis employs concepts from cooperative game theory.      

Conducting secret ballot elections in computer networks: Problems and solutions

We discuss problems related to devising a secret balloting system with the following properties: (1) all eligible voters and they only may vote, (2) all ballots are secret, i.e. do not reveal the identity of the voter, (3) all voters may check whether their ballots have been correctly assigned, (4) the voters may revise their ballots within a predetermined time, and (5) errors in ballot assignment can be corrected within a predetermined time.     

On classifying inputs and outputs in DEA: A revised model

Cook and Zhu [Cook, W.D., Zhu, J., 2007. Classifying inputs and outputs in data envelopment analysis. European Journal of Operational Research 180, 692–699] introduced a new method to determine whether a measure is an input or an output. In practice, however, their method may produce incorrect efficiency scores due to a computational problem as result of introducing a large positive number to the model. This note introduces a revised model that does not need such a large positive number.     

A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. Crisp input and output data are fundamentally indispensable in conventional DEA. However, the observed values of the input and output data in real-world problems are sometimes imprecise or vague. Many researchers have proposed various fuzzy methods for dealing with the imprecise and ambiguous data in DEA. In this study, we provide a taxonomy and review of the fuzzy DEA methods. We present a classification scheme with four primary categories, namely, the tolerance approach, the α-level based approach, the fuzzy ranking approach and the possibility approach. We discuss each classification scheme and group the fuzzy DEA papers published in the literature over the past 20 years. To the best of our knowledge, this paper appears to be the only review and complete source of references on fuzzy DEA.