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.     

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.     

Monotonicity of power in games with a priori unions

Power indices are commonly required to assign at least as much power to a player endowed with some given voting weight as to any player of the same game with smaller weight. This local monotonicity and a related global property however are frequently and for good reasons violated when indices take account of a priori unions amongst subsets of players (reflecting, e.g., ideological proximity). This paper introduces adaptations of the conventional monotonicity notions that are suitable for voting games with an exogenous coalition structure. A taxonomy of old and new monotonicity concepts is provided, and different coalitional versions of the Banzhaf and Shapley–Shubik power indices are compared accordingly.      

Political influence in multi-choice institutions: cyclicity,anonymity, and transitivity

We study political influence in institutions where each member chooses a level of support for a collective goal. These individual choices determine the degree to which the goal is reached. Influence is assessed by newly defined binary relations, each of which ranks members on the basis of their relative performance at a corresponding level of participation. For institutions with three options (e.g., voting games in which each voter may vote “yes”, “abstain”, or vote “no”), we obtain three influence relations, and show that their strict components may be cyclic. This latter property describes a “paradox of power” which contrasts with the transitivity of the unique influence relation of binary voting games. Weak conditions of anonymity suffice for each of these relations to be transitive. We also obtain a necessary and sufficient condition for each of these relations to be complete. Further, we characterize institutions in which the rankings induced by these relations, and the Banzhaf–Coleman and Shapley–Shubik power indices coincide. We argue that extending the influence relations to firms would be useful in efficiently assigning workers to different units of production. Finally, we provide applications to various forms of political and economic organizations.     

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.     

Calculable Power: A Case Study of IMF Quotas and Voting Rights Reform*

Voting procedure is an important mechanism for public choice in collective bodies such as international organizations. This paper measures and compares IMF member countries’ voting power before the 2008 reform and after the 2010 reform on the basis of datasets on IMF quotas and voting rights distribution provided by IMF. Our study verifies that IMF’s quotas and voting rights reforms do help to reduce the voting power gap among member countries. The 2008 and 2010 reforms produce a greater improvement in emerging members’ voting power under the 70 percent majority rule than the 85 percent rule; the 70 percent majority rule means the United States would lose its absolute veto. Moreover, the paper disproves the underlying assumption that regards a member’s voting power as proportional to its voting rights. Countries with different amounts of voting rights can still have the same voting power.     

Committee Decisions with Partisans and Side-Transfers

A dichotomous decision-making context in committees is considered where potential partisan members with predetermined votes can generate inefficient decisions and buy neutral votes. The optimal voting rule minimizing the expected costs of inefficient decisions for the case of a three-member committee is analyzed. It is shown that the optimal voting rule can be non-monotonic with respect to side-transfers: in the symmetric case, majority voting is optimal under either zero, mild or full side-transfer possibilities, whereas unanimity voting may be optimal under an intermediate side-transfer possibility. The side-transfer possibilities depend on the power of partisans (their ability or willingness to pay for neutral votes) relative to the corruptibility of neutral members (personal cost of deliberately casting a `wrong' vote).     

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).     

Analysis of California propositions: An experiment in the empirical revelation of shared mental models

This paper uses factor analysis to analyze Californians’ voting on 12 propositions considered during the 2008 general election. The result is to create quantitative representations of shared mental models. After generating two county-level shared mental model indices, the indices are further examined to explain their variation across counties and to evaluated their performance as substitutes for previous presidential voting as an independent variable in a presidential voting model.     

Groups can make a difference: voting power measures extended

The voting power of a voter—the extent to which she can affect the outcome of a collective decision—is often quantified in terms of the probability that she is critical. This measure is extended to a series of power measures of different ranks. The measures quantify the extent to which a voter can be part of a group that can jointly make a difference as to whether a bill passes or not. It is argued that the series of these measures allow for a more appropriate assessment of voting power, particularly of a posteriori voting power in case the votes are stochastically dependent. Also, the new measures discriminate between voting games that cannot be distinguished in terms of the probability of only criticality.     

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.     

A Special Case of Penrose’s Limit Theorem When Abstention is Allowed

In general, analyses of voting power are performed through the notion of a simple voting game (SVG) in which every voter can choose between two options: ‘yes’ or ‘no’. Felsenthal and Machover [Felsenthal, D.S. and Machover, M. (1997), International Journal of Game Theory 26, 335–351.] introduced the concept of ternary voting games (TVGs) which recognizes abstention alongside. They derive appropriate generalizations of the Shapley–Shubik and Banzhaf indices in TVGs. Braham and Steffen [Braham, M. and Steffen, F. (2002), in Holler, et al. (eds.), Power and Fairness, Jahrbuch für Neue Politische Ökonomie 20, Mohr Siebeck, pp. 333–348.] argued that the decision-making structure of a TVG may not be justified. They propose a sequential structure in which voters first decide between participation and abstention and then between ‘yes’ or ‘no’. The purpose of this paper is two-fold. First, we compare the two approaches and show how the probabilistic interpretation of power provides a unifying characterization of analogues of the Banzhaf (Bz) measure. Second, using the probabilistic approach we shall prove a special case of Penrose’s Limit Theorem (PLT). This theorem deals with an asymptotic property in weighted voting games with an increasing number of voters. It says that under certain conditions the ratio between the voting power of any two voters (according to various measures of voting power) approaches the ratio between their weights. We show that PLT holds in TVGs for analogues of Bz measures, irrespective of the particular nature of abstention.     

A longitudinal test of the gender turnover model among U.S. House and Senate members

This study builds on previous research by examining the impact of gender when predicting roll call voting behavior in the U.S. House and the U.S. Senate over several recent congresses. In order to unearth gender effects, it employs a longitudinal design based on turnover in the membership of both the House and the Senate. Through a comparison of the voting records of members of Congress representing the same geographic territory it holds constituency constantly while allowing for gender and party to vary. It does so with models including dependent variables that measure roll call ideology and support for women's issues exhibited in the voting records of members in both institutions. The results show that male and female members in each chamber representing the same constituency amass virtually indistinguishable voting records on the liberal-conservative policy dimension. However, on votes dealing with issues of concern to women, female senators tend to be more supportive than the male senators they replace and male senators tend to be less supportive than the female senators they replace.     

The joy of ruling: an experimental investigation on collective giving

We analyse team dictator games with different voting mechanisms in the laboratory. Individuals vote to select a donation for all group members. Standard Bayesian analysis makes the same prediction for all three mechanisms: participants should cast the same vote regardless of the voting mechanism used to determine the common donation level. Our experimental results show that subjects fail to choose the same vote. We show that their behaviour is consistent with a joy of ruling: individuals get an extra utility when they determine the voting outcome.     

The inferred referendum — A rule for committee decisions

We present a new method of social choice. The result of our method coincides with that of majority voting when it does not produce an intransitivity among the alternatives under consideration. When majority voting would produce an intransitivity, our method orders the alternatives in the same way as the transitive constituency would whom the committee members are most likely to represent. Analysis of the application of our method to three alternatives shows that (a) the resulting order depends only on the committee members' votes between pairs of alternatives (b) the resulting order is less in conflict with Sen's Property than the orders provided by other schemes (c) when majority voting provides an intransitivity, the hypothesis that, in fact, the committee's constituency is as we assume it to be is almost as likely as the hypothesis that it precisely mirrors the committee.     

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.     

Notes on Ostrogorski's paradox

Conclusion This note has investigated the necessary conditions for the existence of Ostrogorski's Paradox, and it showed that the paradox depends on the sizes of the groups taking each possible set of stands on the salient issues. By revising its positions on relevant issues, the majority party can avert the paradox. With increased emphasis on single-issue voting observed in the United States and other Western democracies, the analysis of the paradox should guide the understanding of parties' responses to coalition formation processes among special interest groups. Indeed, the Apex conditions characterizing the paradox provide clues as to the strategy to be followed by the majority party in selecting its platform: the party should generally follow positions advocated by the largest group while presenting a coalition of all other groups.The helpful comments of Howard Tamashiro on an earliar draft of this paper are greatfully acknowledge.     

A Game-Theoretic View of Voting

Two game-theoretic arguments for the potential rationality of voting are presented. The first argument suggests that people make choices that allow the most favorable forecasts. People choose to vote inasmuch as they project their own choices between voting and abstaining more strongly onto members of their own political group than onto members of other political groups. Relevant evidence is reviewed and extended by new findings in a simulated public-goods dilemma. The second argument suggests that people preview how they will feel about each of the four possible scenarios generated by the conjunctions of their own choices (to vote or to abstain) and the election outcome (victory or defeat). They choose to vote inasmuch as they feel their own vote will not be wasted. The implications of both arguments for efforts to increase turnout are discussed.     

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.