Two-Stage Election Procedures

A discussion is presented regarding the single-stage election procedures Plurality, Negative Plurality, and Borda Rules; with an emphasis on their Condorcet Efficiency, which measures their propensity to elect the Condorcet Winner. A theoretical analysis shows how the Condorcet Efficiency of these voting rules change as the various measures of group mutual coherence change. Both Plurality and Negative Plurality Rules are found to perform very poorly over identifiable ranges of the defined measures of group mutual coherence. While Borda Rule is not always found to be the most Condorcet Efficient single-stage rule in all scenarios, it never performs poorly under any scenario. These theoretical results are then reinforced by empirically-based analysis to show very solid support for using Borda Rule, despite some concerns that have been expressed that Borda Rule requires voters to rank candidates, while Plurality and Negative Plurality Rules do not.

We compare the Condorcet Efficiencies of the plurality rule, the negative plurality rule, and the Borda rule to examine what might be gained by using a voting rule that requires candidate rankings. Previous analyses have considered (1) voting situations for which the three rules determine different candidates as winners (and the Condorcet Efficiencies might actually differ across the three rules) as well as (2) refined measures of Group Mutual Coherence that assess the presence of underlying models of rational behavior that governs voter preferences. In this paper, we present the final step of this particular line of analysis by analyzing (1) and (2) simultaneously. We conclude that there is a considerable benefit on the basis of Condorcet Efficiency from asking voters to submit candidate rankings in three-candidate elections.

Consider a three-alternative election with n voters and assume that preferences are single-peaked. LetC(λ,n) be the Condorcet efficiency of the rule that assigns 1,λ and 0 points (respectively) to each first, second and third place vote. An exact representation is obtained forC(λ,∞). This relation shows that Borda rule (λ=1/2) is not the most efficient rule. In addition to this result, exact closed form relations are provided forC(0,n),C(1/2,n) andC(1,n). All these relations are obtained by assuming that every admissible configuration of preferences is equally likely to occur.

We use preference data from the 2015 parliament election in the Austrian federal state of Styria to analyze different voting rules. An exit poll right after the election collected data on ordinal and cardinal preferences from approximately 1000 actual voters. Our analysis is threefold. First, we determine the hypothetical social outcomes under different voting rules; second, we investigate the stability of the outcomes under those rules. Finally, we provide a categorization of different types of parties and analyze the impact of certain voting rules (Plurality Rule, Plurality Run Off, Hare System, Condorcet Method, Approval Voting, Borda Rule, Evaluative Voting, and Majority Judgment) on the performances of parties in those scenarios.

This paper combines social choice theory with mathematical optimization by applying various group decision concepts to a classical problem of combinatorial optimization, namely the famous traveling salesperson (sales- man) problem. The aim of the latter is to find a tour through all vertices of a given graph along edges of minimal total cost. In this contribution we replace the measure of additive edge costs by the social acceptance of different edges and the resulting tours. In particular, for four different voting rules, the Borda rule, Approval voting, Plurality rule and Simple Majority rule, we will investi- gate the social acceptance of tours derived from global and local decisions. It will be shown that these two decision approaches can lead to widely varying results.

In collective decision making, where a voting rule is used to take a\ncollective decision among a group of agents, manipulation by one or more agents\nis usually considered negative behavior to be avoided, or at least to be made\ncomputationally difficult for the agents to perform. However, there are\nscenarios in which a restricted form of manipulation can instead be beneficial.\nIn this paper we consider the iterative version of several voting rules, where\nat each step one agent is allowed to manipulate by modifying his ballot\naccording to a set of restricted manipulation moves which are computationally\neasy and require little information to be performed. We prove convergence of\niterative voting rules when restricted manipulation is allowed, and we present\nexperiments showing that most iterative voting rules have a higher Condorcet\nefficiency than their non-iterative version.\n

Standard voting rules usually assume that the preferences of voters are provided in the form of complete rankings over a fixed set of alternatives. This assumption does not hold in applications like recommendation systems where the set of alternatives is extremely large and only partial preferences can be elicited. In this paper, we study the problem of strategic manipulation of voting rules that aggregate voter preferences provided in the form of pairwise comparisons between alternatives. Our contributions are twofold: first, we show that any onto pairwise voting rule is manipulable in principle. Next, we analyze how the computational complexity of manipulation of such rules varies with the structure of the graph induced by the pairs of alternatives that the manipulator is allowed to vote over and the type of the preference relation. Building on natural connections between the pairwise manipulation and sports elimination problems (including a mixed-elimination variant that we introduce in this paper), we show that manipulating pairwise voting rules can be computationally hard even in the single-manipulator setting, a setting where most standard voting rules are known to be easy to manipulate.

The Condorcet efficiency of Borda Rule with anonymous voters

Previous research has developed a representation for the Condorcet efficiency of plurality and negative plurality rules on three candidates for large electorates under the impartial culture condition. This study considers the two stage voting rules on three candidates, plurality elimination and negative plurality elimination. A representation is obtained for the Condorcet efficiency of each for large electorates under impartial culture. The Condorcet efficiencies of both rules are equal and are shown to be substantially greater than the Condorcet efficiency of plurality and negative plurality.

There are two different fundamental conceptions of the task of social aggregation: interest aggregation and judgment aggregation. While the former aims at the fair and efficient trade-off of personal interests, the latter aims the best resolution of disagreement. Both are possible interpretations of the standard preference aggregation and preference-based voting problems. This note makes two broad points. First, the judgment aggregation conception is not yet well understood and remains undertheorized. Second, the difference between the two conceptions matters; it is a difference that makes a difference. To substantiate the first point, we argue that judgment aggregation as resolution of disagreement cannot be subsumed under statistical aggregation in the manner of Condorcet’s jury theorem, and sketch a more specific articulation as “reflexive consensus under arguable disagreement”. To substantiate the second point, we illustrate the difference in the context of two important voting rules, the Kemeny rule and Majority Judgment.

The Borda Compromise states that, if one has to choose among five popular voting rules that are not Condorcet consistent, one should always give preference to the Borda rule over the four other rules. We assess the theoretical as well as the empirical support for the Borda Compromise. We find that, despite considerable differences between the properties of the theoretical framework and the characteristics of two sets of observed ranking data, all three analyses provide considerable support for the Borda Compromise.

Vulnerability of Voting Paradoxes As a Criteria For Voting Procedure Selection

Voting procedures are known to be plagued with a variety of difficulties such as strategic voting, or where a voter is rewarded with a better election outcome by not voting, or where a winning candidate can lose by receiving more support. Once we know that these problems can occur, the next objective should be to completely understand how, why, and where they arise. Namely, for each election procedure, the new goal is to determine when such problems can occur, all voter types who can cause these difficulties, and the actions they must take. This paper develops an easily used approach to handle all of these issues for standard voting methods. New intuitive explanations for these various oddities follow from this approach.

We investigate the effect of beam coherence on four-wave mixing via reflection gratings in photorefractive media. For the case of phase conjugation, the results of our theoretical analysis indicate that partial coherence always leads to a drop of signal gain and phase conjugate reflectivity in non-depleted cases. In general, the mutual coherence of the signal beam and the pump beam can be enhanced due to the process of wave mixing. The mutual coherence of the phase conjugate beam and one of the pump beams depends on the beam intensity ratio as well as the optical path difference. This is distinctly different from the four-wave mixing case with a transmission grating.

By means of compressive sensing (CS) technique, this paper considers the collection of sensor data with spatiotemporal correlations in wireless sensor networks (WSNs). In energy-constrained WSNs, one-dimensional CS methods need a lot of data transmissions since they are less applicable in fully exploiting the spatiotemporal correlations, while the Kronecker CS (KCS) methods suffer performance degradations when the signal dimension increases. In this paper, an appropriate sensing matrix as well as an efficient sensing method is proposed to further reduce the data transmissions without the loss of the recovery performance. Different matrices for the temporal signal of each sensor node are separately designed. The corresponding energy-efficient data gathering method is presented, which only transmitting a subset of sensor readings to recover data of the entire WSN. Theoretical analysis indicates that the sensing structure could have the relatively small mutual coherence according to the selection of matrix. Compared with the existing spatiotemporal CS (CS-ST) method, the simulation results show that the proposed efficient measurement method could reduce data transmissions by about 25% with the similar recovery performance. In addition, compared with the conventional KCS method, for 95% successful recovery, the proposed sensing structure could improve the recovery performance by about 20%.