Discriminating DEA efficient candidates by considering their least relative total scores

Data envelopment analysis very often identifies more than one candidate in a voting system to be DEA efficient. In order to choose a winner from among the DEA efficient candidates, this paper proposes a new method that discriminates the DEA efficient candidates by considering their least relative total scores. The proposed method is illustrated with two numerical examples and proves to be effective and practical.     

Condorcet winner configurations of linear networks

We consider the problem of locating a fixed number of facilities along a line to serve n players. We model this problem as a cooperative game and assume that any locational configuration can be eventually disrupted through a strict majority of players voting for an alternative configuration. A solution of such a voting location problem is called a Condorcet winner configuration. In this article, we state three necessary and one sufficient condition for a configuration to be a Condorcet winner. Consequently, we propose a fast algorithm which enables us to verify whether a given configuration is a Condorcet winner, and can be efficiently used also for computing the (potentially empty) set of all Condorcet winner configurations.     

Complexity of computing median linear orders and variants

Given a finite set X and a collection Π of linear orders defined on X, computing a median linear order (Condorcet-Kemenyʼs problem) consists in determining a linear order minimizing the remoteness from Π. This remoteness is based on the symmetric distance, and measures the number of disagreements between O and Π. In the context of voting theory, X can be considered as a set of candidates and the linear orders of Π as the preferences of voters, while a linear order minimizing the remoteness from Π can be adopted as the collective ranking of the candidates with respect to the votersʼ opinions. This paper studies the complexity of this problem and of several variants of it: computing a median order, computing a winner according to this method, checking that a given candidate is a winner and so on. We try to locate these problems inside the polynomial hierarchy.     

Three new models for preference voting and aggregation

Preference voting and aggregation require the determination of the weights associated with different ranking places. This paper proposes three new models to assess the weights. Two of them are linear programming (LP) models which determine a common set of weights for all the candidates considered and the other is a nonlinear programming (NLP) model that determines the most favourable weights for each candidate. The proposed models are examined with two numerical examples and it is shown that the proposed models cannot only choose a winner, but also give a full ranking of all the candidates.     

Condorcet Winners for Public Goods

In this work, we consider a public facility allocation problem decided through a voting process under the majority rule. A location of the public facility is a majority rule winner if there is no other location in the network where more than half of the voters would have been closer to than the majority rule winner. We develop fast algorithms for interesting cases with nice combinatorial structures. We show that the computing problem and the decision problem in the general case, where the number of public facilities is more than one and is considered part of the input size, are all NP-hard. Finally, we discuss majority rule decision making for related models.     

Approval voting: three examples

In this paper we discuss three examples of approval voting games. The first one illustrates that a stronger solution concept than perfection is needed for a strategic analysis of this type of games. The second example shows that sophisticated voting can imply that the Condorcet winner gets no vote. The third example shows the possibility of insincere voting being a stable equilibrium.      

Approximations of quasi-stationary distributions for markov chains

We consider a simple and widely used method for evaluating quasi-stationary distributions of continuous time Markov chains. The infinite state space is replaced by a large, but finite approximation, which is used to evaluate a candidate distribution. We give some conditions under which the method works, and describe some important pitfalls.     

Inequality in population weights,majority threshold and the inversion probability in the case of three states

Two-tier voting systems are prone to majority inversions, when the outcome of an election is not backed by a majority of popular vote. We study the inversion probability in a model with two candidates, three states and uniformly distributed fractions of supporters for each candidate. We show that the inversion probability in a two-tier voting system with three states eventually decreases with a majority threshold in the states and increases with the inequality in the size distribution of the states.     

Choice from a Three-Element Set: Some Lessons of the 2000 Presidential Campaign in the United States

We consider the behavior of four choice rules—plurality voting, approval voting, Borda count, and self-consistent choice—when applied to choose the best option from a three-element set. It is assumed that the two main options are preferred by a large majority of the voters, while the third option gets a very small number of votes and influences the election outcome only when the two main options receive a close number of votes. When used to rate the main options, Borda count and self-consistent choice contain terms that allow both for the strength of preferences of the voters and the rating of the main candidates by voters who vote for the third option. In this way, it becomes possible to determine more reliably the winner when plurality voting or approval voting produce close results.     

A Bayesian approach to a sequential three‐decision problem: Voting and the uses of information in electoral choice*

This paper investigates a situation of decision‐making under risk in which an individual must select one of three actions. Substantively, an electoral example in which a citizen must decide how and whether to vote in a two‐candidate election is used to illustrate the argument. Only the value of the consequences of one action, which is to abstain from voting, is known. The expected values of voting for either candidate must be estimated based upon a sample of information. Specifically, we are interested in how one may decide when to stop gathering information and the behavioral consequences of that choice for the voting decision. The dependence of the voting decision on the original ambiguity and magnitude of the expected utilities and on the costs of information is also explored.     

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.     

Optimal location of candidates in ideological space

The strong point is the point which minimizes the probability that a candidate be defeated in a two-party election. The center of power is the weighted average of voters' positions in a spatial voting game, where the weights are given by Shapley's modified value. We show that, under very general conditions, the strong point and the center of power coincide.     

An optical model for implementing Parrondo’s game and designing stochastic game with long-term memory

An optical model for a photon propagating through a designed array of beam splitters is developed to give a physical implementation of Parrondo’s game and Parrondo’s history-dependent game. The winner in this optical model is a photon passed the beam splitter. The loser is a photon being reflected by the beam splitter. The optical beam splitter is the coin-tosser. We designed new games with long-term memory by using this optical diagram method. The optical output of the combined game of two losing games could be a win, or a loss, or an oscillation between win and loss. The modern technology to implement this optical model is well developed. A circularly polarized photon is a possible candidate for this physical implementation in laboratory.     

The Borda rule and Pareto stability: a further comment

In their paper “The Borda rule and Pareto stability: a comment” published in 1979 by Econometrica, Farkas and Nitzan revealed the “intimate relationship” between the Borda rule and the Pareto criterion. The idea was the following: in a profile of total orders, when there is a candidate who obviously wins under unanimous agreement of the voters, that candidate should be in the choice set. In a profile where there is no obvious winner, the candidates that are the closest to unanimity should be chosen. According to this principle, they defined a choice rule called “closeness to unanimity” and they showed that it is equivalent to the Borda rule. In our paper, we give an equivalent result for a ranking rule. Then we try to obtain similar results when aggregating profiles of tournaments, weak orders, semiorders, fuzzy relations, … and we show that the definition of an obvious winner is no more obvious.     

Decay,growth, continuous dependence and uniqueness results in generalized heat conduction theories

Decay, growth, continuous dependence and backward uniqueness results are established for a system of equations which has been proposed as a possible candidate to describe heat conduction with space correlation.     

Guarantees for the success frequency of an algorithm for finding Dodgson-election winners

In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a Dodgson election is known to be complete for the Θ ₂ ^p level of the polynomial hierarchy. This implies that unless P=NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates—although the number of voters may still be polynomial in the number of candidates—a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner.     

Extensions of the Young and Levenglick result about the inconsistency of Condorcet voting correspondences

The Young’s Consistency property means that when some candidates are chosen as winners by two disjoint electorates, those and only those candidates are chosen in the aggregated electorate. We define two new properties requiring that, when a candidate x is elected in a situation and a new electorate is added for which x is a very good candidate, x will remain elected. These properties, weaker than Young’s Consistency, lead to new impossibility results strengthening the Young and Levenglick result about the inconsistency of Condorcet Voting Correspondences. Also we adapt the Young and Levenglick result to the k-choice voting function context.     

Strong and coalition-proof political equilibria under plurality and runoff rule

We analyze the effects of voter coordination on the equilibrium of voting games. Specifically, we analyze the concepts of strong equilibrium and coalition-proof equilibrium for plurality rule and runoff rule elections. We characterize these equilibria in the three candidate case, and provide results for important special cases in the general multicandidate case.      

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.     

Semi-discrete Schwarz waveform relaxation algorithms for reaction diffusion equations

For time dependent problems, the Schwarz waveform relaxation (SWR) algorithm can be analyzed both at the continuous and semi-discrete level. For consistent space discretizations, one would naturally expect that the semi-discrete algorithm performs as predicted by the continuous analysis. We show in this paper for the reaction diffusion equation that this is not always the case. We consider two space discretization methods—the 2nd-order central finite difference method and the 4th-order compact finite difference method, and for each method we show that the semi-discrete SWR algorithm with Dirichlet transmission condition performs as predicted by the continuous analysis. However, for Robin transmission condition the semi-discrete SWR algorithm performs worse than predicted by the continuous analysis. For each type of transmission conditions, we show that the convergence factors of the semi-discrete SWR algorithm using the two space discretization methods are (almost) equal. Numerical results are presented to validate our conclusions.