Can Likert Scales be Treated as Interval Scales?—A Simulation Study

The Likert scale is widely used in social work research, and is commonly constructed with four to seven points. It is usually treated as an interval scale, but strictly speaking it is an ordinal scale, where arithmetic operations cannot be conducted. There are pros and cons in using the Likert scale as an interval scale, but the controversy can be handled by increasing the number of points. Several researchers have suggested bringing the number up to eleven, on the basis of empirical data. In this article the authors explore this rational and share the same view, but simulate artificial data from both symmetrical normal and skewed distributions where the underlying metric is known in advance. Results show that more Likert scale points will result in a closer approach to the underlying distribution, and hence normality and interval scales. To increase generalizability social work practitioners are encouraged to use 11-point Likert scales from 0 to 10, a natural and easily comprehensible range.     

On the stability of a triplet of scoring rules

When choosing a voting rule to make subsequent decisions, the members of a committee may wish this rule to be self-selected when it is the object of a choice among a menu of different possible voting rules. Such concepts have recently been explored in Social Choice theory, and a menu of voting rule is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this article the menu constituted by the three well-known scoring rules {Borda, Plurality, and Antiplurality}. Under the Impartial Culture assumption, which proposes an a priori model to estimate the likelihood of the profiles, we will derive a probability for the stability of this triplet of voting rules.     

Condorcet vs. Borda in light of a dual majoritarian approach

Many voting rules and, in particular, the plurality rule and Condorcet-consistent voting rules satisfy the simple-majority decisiveness property. The problem implied by such decisiveness, namely, the universal disregard of the preferences of the minority, can be ameliorated by applying unbiased scoring rules such as the classical Borda rule, but such amelioration has a price; it implies erosion in the implementation of the widely accepted “majority principle”. Furthermore, the problems of majority decisiveness and of the erosion in the majority principle are not necessarily severe when one takes into account the likelihood of their occurrence. This paper focuses on the evaluation of the severity of the two problems, comparing simple-majoritarian voting rules that allow the decisiveness of the smallest majority larger than 1/2 and the classical Borda method of counts. Our analysis culminates in the derivation of the conditions that determine, in terms of the number of alternatives k, the number of voters n, and the relative (subjective) weight assigned to the severity of the two problems, which of these rules is superior in light of the dual majoritarian approach.     

Hierarchies achievable in simple games

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games. Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.     

Impression management pressures on racial attitude surveys

For decades, researchers have expressed concern that self-report racial attitude measures are vulnerable to distortion from pressures respondents feel to present themselves as unprejudiced. A common response to this problem is to measure social desirability separately from racial attitudes and control for its variance in statistical analyses. The present study is designed to test whether such controls are sufficient. Participants rated items from both racial attitude and social desirability scales in terms of the amount of pressure they would feel to respond in a particular way regardless of their true attitudes. Participants report significantly greater response pressure on racial attitude items than on social desirability items, and ratings on the two types of items have only moderately shared variance. Implications for controlling social desirability in racial attitude research are discussed.     

Strict proportional power in voting bodies

It is a widely known fact among game theorists as well as political scientists that the distribution of voting weights in a voting body is generally a poor proxy for the distribution of voting power within the body. It has been proposed to equate the distribution of a priori voting power and actual seat distribution by randomizing the decision rule of the voting body over various majority rules. In this paper, some implications of this procedure will be discussed.     

Postulates and paradoxes of relative voting power — A critical re-appraisal

This paper re-evaluates the problem of measuring thea priori relative voting power of a voter in an assembly. We propose several new intuitively compelling postualtes that any reasonable index of voting power ought to satisfy. At the same time we argue that most of the paradoxes of voting power discussed in the literature are paradoxical only in a weak sense, if at all. This leaves three crippling paradoxes — the well-known paradox ofweighted voting, and two new ones presented here: thebloc anddonation paradoxes. We evaluate the four main relative power indices discussed in the literature with respect to these three severe paradoxes. The Shapley-Shubik index is seen to be immune to all three paradoxes, while the Deegan-Packel index is vulnerable to all three. The Banzhaf and the Johnston indices are demonstrably immune to the paradox of weighted voting. However, they are shown to suffer from both the bloc and the donation paradoxes. We argue that this seriously undermines these indices in a hitherto unsuspected way. Several other theoretical issues relating to voting power are discussed.     

Expected frequency and mean size of the paradox of new members

The paradox of new members occurs when the addition of one or more new members to a weighted voting body increases, rather than decreases, the voting power of some of the old members. Extending the computational work of Brams and Affuso (1976), the mean size of the paradox and the relative frequency of its occurrence are presented for small and moderate-size weighted voting bodies. Computational results are presented and conclusions are drawn for the two power indices of Shapley-Shubik and Banzhaf, three different decision rules, and voting bodies with or without dictator. Although the paradox cannot be dismissed as either contrived or improbable, its mean size in moderatesize voting bodies is sufficiently small to question its practical significance.     

A characterization of majority voting rules with quorums

We give a characterization of majority voting rules with quorums in the framework of May (Econometrica 20:680–684, 1952)’s seminal article. According to these voting rules, an alternative is socially chosen if and only if it obtains the relative majority of votes and the total number of voters not abstaining reaches the quorum.     

An application of social choice theory to U.S.D.A. forest service decision making

To meet the requirements of the National Environmental Policy Act of 1969, the U.S.D.A. Forest Service has changed the way forest plans are developed. The focus of this paper is to address the group-decision problem using social choice theory, specifically the voting models of Condorcet and Borda. The elements of a social choice problem are voters, alternatives, preferences, and aggregation. A case study from the Shoshone National Forest is used to demonstrate the use of the voting models from social choice theory. The solutions derived from the analysis are strategy and coalitional strategy proof implying that behaviors intended to influence the outcome, such as vote trading, would be unsuccessful.     

Ordinal preference representations

Ordinal preferences have several advantages over the traditional cardinal expressions of preference. Three different representations of ordinal preferences useful in multi-participant modelling are presented, and their features compared. One approach is thepayoff representation that is based on an ordinal normal form game. A second representation of ordinal preferences is thepreference vector, based on the option form of the game. The option form consists of a list of players, with each player followed by the options under its control. The third representation of ordinal preferences is thepreference tree. A preference tree is an implied binary tree that captures the information of preference vector in a more compact manner by making use of its lexicographic structure. The preference tree offers considerable compactness and computational efficiency over the other two approaches.     

Short Forms for Attitude Scales in Social Sciences: The Case of Attitude Towards Capital Punishment

Attitude scales are common in the social sciences but quite often they are very long and hence short forms are needed. Short forms can reduce the fatigue problem and make it possible for other measurements to be included, but short forms must be produced without much sacrifice to reliability and validity. In this paper, capital punishment attitude scales are investigated as a special case. A total of 27 items whose psychometric properties have been well established are used. A simple and direct method of producing short forms is used to produce half (14 items) and quarter (7 items) versions of the original. Both two- and six-point Likert scales are used to investigate the interaction effects between number of item-response option and scale length. Results show that there is no clear difference between the two- and six-point scales in terms of reliabilities, but the six-point scale is preferred in respect of trace information and percentages of variance in the one-factor model, and is closer to normality. In practice, we suggest short forms should always be considered for attitude scales in social services, and more Likert scale points can produce even better short forms. Limitations and further directions are discussed.     

Electoral Simultaneity: Expressing Equal Respect

The principle of simultaneity—that citizens should vote as far as possible at the same time—is more significant than is usually appreciated. It is based on a fundamental value of democratic theory, and it has substantial implications for electoral practice. It also invites further normative and empirical research.
The most important value that simultaneity expresses is a form of equality—equal respect. First, if citizens have only information they would have had if they were voting at the same time, the value of each citizen's choice is no greater than that of any other citizen. Second, when citizens go to the polls on the same day, publicly participating in a common experience of civic engagement, they demonstrate their willingness to contribute to the democratic process on equal terms. Taking simultaneity seriously has implications for electoral practices. It would limit the practice of media projections of election results, including the reporting of exit polls, and the calling of elections before all the polls close. It also casts doubt on the increasingly widespread use of early voting—absentee ballots and voting by mail. Early voting weakens the value of the experience of participating in a civic activity. Voting alone may be worse than bowling alone.     

Justice,utility, and interpersonal comparisons

One of the most common objections against utilitarianism is that it cannot account for the claims of distributive justice. This objection can be answered by appeal to the principle of Diminishing Marginal Utility (DMU), which says that the value of a given portion of a commodity tends to increase as a person's supply of the commodity decreases. DMU implies that, other things equal, the way to maximize the value of a commodity is to distribute it equally. Some economists have objected to DMU on the ground that it presupposes interpersonal comparisons of utility, but the arguments which they have advanced against the objectivity of such comparisons are unsound. Moreover, the interpersonal comparisons presupposed by DMU are ordinal, rather than cardinal, and there are reasons for believing that ordinal comparisons, at least, do have an objective basis.     

Race and Environmental Voting in the U.S. Congress

Objective. This article examines long-term trends in the environmental voting behavior in Congress and attempts to identify the factors that account for racial differences in voting patterns. Methods. Hypotheses about the various possible influences on environmental voting behavior are tested using longitudinal analyses and path analysis techniques. Results. Findings indicate African American members of Congress have been consistently more supportive of proenvironment legislation than either Republican or Democrat colleagues from 1981 to 1998. The data further suggest that much of these overall differences are explained by an especially large racial gap in proenvironmental voting among members from the South. Although such a gap among members from outside the South is eliminated by controlling for ideology and other variables, it remains substantial among Southern members. Conclusions. Based on the strong and consistent support for environmental legislation by African American members of Congress in the past, we expect significant racial differences in environmental voting to endure. Although results suggest that ideology plays a partial role in explaining such differences, future research will be needed to identify the reasons for the especially large racial gap in proenvironmental voting among members from the South. Given the growth in their numbers, their relatively safe seats, and their potential to chair important committees as they gain seniority, we expect African American legislators to play an increasingly important role in shaping and deciding the fate of national environmental policy in the future.     

Distance,Turnout, and the Convenience of Voting*

Objectives. This research examines how distance factors into the costs associated with political participation. We hypothesize that the political geography of a voter's residence affects not only the likelihood that he or she will vote, but whether the voter will choose between traditional Election Day voting or nontraditional means, such as casting an absentee ballot by mail, or going to an early‐voting site. Methods. Using a geographic information system (GIS), we calculate Manhattan‐block distances between voter residences and their respective precinct and nearest early‐voting sites in Clark County, NV for the 2002 mid‐term election. We then use these calculated distances to predict, with multinomial logistic regression, the likelihood of nonvoting, precinct voting, and nontraditional voting. Results. Our evidence suggests that the cost of traveling to reach a traditional voting site is associated with nonvoting to a point, but the relationship between distance and participation is nonlinear. Distance to traditional voting sites is also highly associated with choosing to vote by mail. Would‐be nonvoters are more inclined to use proximate election‐day sites than proximate early‐voting sites, probably because they decide to vote so late in the campaign. Conclusions. Our findings have important implications for democratic theory, ongoing efforts to reform the electoral process, and the practice of voter mobilization.     

The potential of iterative voting to solve the separability problem in referendum elections

In referendum elections, voters are often required to register simultaneous votes on multiple proposals. The separability problem occurs when a voter’s preferred outcome on one proposal depends on the outcomes of other proposals. This type of interdependence can lead to unsatisfactory or even paradoxical election outcomes, such as a winning outcome that is the last choice of every voter. Here we propose an iterative voting scheme that allows voters to revise their voting strategies based on the outcomes of previous iterations. Using a robust computer simulation, we investigate the potential of this approach to solve the separability problem.     

A subjectivist approach to consecutive conflict

An easily applied approach is developed to provide one participant in a sequence of conflicts with an optimal strategy. A goal of this article is to demonstrate that it is mathematically feasible to incorporate a decision maker's subjective distributions over the effects his actions will have on the outcomes of future conflicts. Unlike many other approaches, the model of this article does not restrict the beliefs that the participant is allowed to express. The participant, not the decision theorist, decides on what is relevent. Model assumptions required for updating rules, such as Bayesian updating, are not required unless they really are appropriate for the situation.Thanks are due to J. B. Kadane for suggesting the problem and for many helpful discussions.     

The Bicameral Postulates and Indices of a Priori Voting Power

If K is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K -power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index – if it is to be used as a tool for analysing abstract, uninhabited decision rules – should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley–Shubik, Deegan–Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x less powerful than y when measured in the simple voting game G₁ , but more powerful than y when G₁ is bicamerally joined with a second chamber G₂ . Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition – the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley–Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power).     

Sincerity and manipulation under approval voting

Under approval voting, each voter can nominate as many candidates as she wishes and the election winners are those candidates that are nominated most often. A voter is said to have voted sincerely if she prefers all those candidates she nominated to all other candidates. As there can be a set of winning candidates rather than just a single winner, a voter’s incentives to vote sincerely will depend on what assumptions we are willing to make regarding the principles by which voters extend their preferences over individual candidates to preferences over sets of candidates. We formulate two such principles, replacement and deletion, and we show that, under approval voting, a voter who accepts those two principles and who knows how the other voters will vote will never have an incentive to vote insincerely. We then discuss the consequences of this result for a number of standard principles of preference extension in view of sincere voting under approval voting.