Power and size: A new paradox

An analysis of several different indices of voting power reveals that the voting power of a member of a weighted voting body may increase, rather than decrease, when new members are added to the original body. Real instances of this phenomenon, called the paradox of new members, are shown to have occurred when new states were added to the U.S. Electoral College and new countries to the European Community Council of Ministers. Conditions for the existence of the paradox, and probabilities of its occurrence in small and moderate-size voting bodies, are given. Efficient algorithms for the calculation of the voting power indices, based on generating functions, are also outlined.     

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.     

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.     

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.     

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).     

Achievable Hierarchies In Voting Games

Previous work by Diffo Lambo and Moulen [Theory and Decision 53, 313–325 (2002)] and Felsenthal and Machover [The Measurement of Voting Power, Edward Elgar Publishing Limited (1998)], shows that all swap preserving measures of voting power are ordinally equivalent on any swap robust simple voting game. Swap preserving measures include the Banzhaf, the Shapley–Shubik and other commonly used measures of a priori voting power. In this paper, we completely characterize the achievable hierarchies for any such measure on a swap robust simple voting game. Each possible hierarchy can be induced by a weighted voting game and we provide a constructive proof of this result. In particular, the strict hierarchy is always achievable as long as there are at least five players.     

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.     

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.     

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.     

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.     

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.     

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.      

Subjective expected utility theory revisited: A reductio ad absurdum paradox

A paradox is posed and analyzed in which people reverse their preferences for information on probabilities versus prizes once the range of the unknown probabilities is sufficiently narrowed. This reversal is shown to be incompatible with both objective expected utility (EU) as well as subjective versions in which the same probability transformation applies to all random variables. Experimental data are presented showing that the reversals occur with small, medium and large payoffs.The present paradox is compared with those of Allais and Ellsberg, and found to differ in substantive ways. It raises further questions about the normative status of expected utility theory, especially its treatment of probability and value. The paradox specifically calls into question EU's substitution and compound probability axioms.     

从两个公式看累积投票制的适用

2006年1月1日开始实施的《公司法》明确规定股份公司在选举董事、监事时可实行累积投票制,原意是更好地维护中小股东的权益,然而从两个简单的公式即可看出累积投票制的实施必然会遇到一些困难。     

Subjectively weighted linear utility

An axiomatized theory of nonlinear utility and subjective probability is presented in which assessed probabilities are allowed to depend on the consequences associated with events. The representation includes the expected utility model as a special case, but can accommodate the Ellsberg paradox and other types of ambiguity sensitive behavior, while retaining familiar properties of subjective probability, such as additivity for disjoint events and multiplication of conditional probabilities. It is an extension, to the states model of decision making under uncertainty, of Chew's weighted linear utility representation for decision making under risk.     

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.     

Economic Determinants of Voting in an Era of Union Decline*

Objective. The objective of this article is to test competing hypotheses regarding union vote effects by economic sector. Overlooked in existing research on political participation and the labor movement is de‐unionization's sectoral dimension: declining union rolls is a private‐sector phenomenon. The sectoral dimension of union decline carries important political consequences if the influence of unions on voter turnout varies by sector. Method. Using Current Population Survey (CPS) November Voting and Registration Supplements for all national elections between 1984 and 2006, I estimate union vote effects for public‐ and private‐sector employees. Results. The results of the analyses reveal that while union members continue to vote at higher rates than otherwise similar nonmembers, the union effect is nearly three times as large for private‐sector members: private‐sector unionists have a predicted probability of voting 6.7 points higher than nonmembers, while public‐sector members have a predicted probability of voting only 2.4 points higher than nonmembers. Conclusions. Given the small fraction of private‐sector workers now in labor unions, recent fluctuations in the unionization rate have little aggregate affect on turnout. Given that private‐sector union members tend to be less educated and earn less than their public‐sector counterparts, the near disappearance of private‐sector unions from the economic landscape removes an important institutional buffer against political inequality in the United States.     

Measuring Legislator Ideology*

Objective. Previous models of roll‐call voting have either ignored the role of legislator ideology or have employed questionable measures of it. A defensible measure of legislator ideology is needed. Methods. This study surveys former members of Congress (MCs) to obtain a new measure of legislator preferences. The measure has several attractive properties, appears to be valid, and is unique compared to a preference measure others have used. Results. Validating the measure in a model of roll‐call voting reveals that the importance of legislators' ideologies is similar to that of their party affiliations. Conclusions. A useful measure of legislators' ideologies can be generated using a postretirement instrument.     

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.