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.     

Simple Majority Achievable Hierarchies

We completely characterize the simple majority weighted voting game achievable hierarchies, and, in doing so, show that a problem about representative government, noted by J. Banzhaf [Rutgers Law Review 58, 317–343 (1965)] cannot be resolved using the simple majority quota. We also demonstrate that all hierarchies achievable by any quota can be achieved if the simple majority quota is simply incremented by one.     

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.     

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.     

Resolute majority rules

We investigate resolute voting rules that always rank two alternatives strictly and avoid social indecision. Resolute majority rules differ from the standard majority rule in that whenever both alternatives win the same number of votes, a tie-breaking function is used to determine the outcome. We provide axiomatic characterizations of resolute majority rules or resolute majority rules with a quorum. Resoluteness axiom is used in all these results. The other axioms are weaker than those considered in the characterization of the majority rule by May (1952 Econometrica, 20:680–684). In particular, instead of May’s positive responsiveness, we consider a much weaker monotonicity axiom.     

Majority voting on orders

We characterize two lexicographic-type preference extension rules from a set X to the set of all orders on this set. Elements of X are interpreted as basic economic policy decisions, whereas elements of are conceived as political programs among which a collectivity has to choose through majority voting. The main axiom is called tournament-consistency, and states that whenever majority pairwise comparisons based on initial preferences on X define an order on X, then this order is also chosen by a majority among all other orders in . Tournament-consistency thus allows to predict the outcome of majority voting upon orders from the knowledge of majority preferences on their components.     

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.     

A characterization of the maximin rule in the context of voting

In a voting context, when the preferences of voters are described by linear orderings over a finite set of alternatives, the Maximin rule orders the alternatives according to their minimal rank in the voters’ preferences. It is equivalent to the Fallback bargaining process described by Brams and Kilgour (Group Decision and Negotiation 10:287–316, 2001). This article proposes a characterization of the Maximin rule as a social welfare function (SWF) based upon five conditions: Neutrality, Duplication, Unanimity, Top Invariance, and Weak Separability. In a similar way, we obtain a characterization for the Maximax SWF by using Bottom Invariance instead of Top Invariance. Then, these results are compared to the axiomatic characterizations of two famous scoring rules, the Plurality rule and the Antiplurality rule.     

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.     

Intensity of preference and related uncertainty in non-compensatory aggregation rules

Non-compensatory aggregation rules are applied in a variety of problems such as voting theory, multi-criteria analysis, composite indicators, web ranking algorithms and so on. A major open problem is the fact that non-compensability implies the analytical cost of loosing all available information about intensity of preference, i.e. if some variables are measured on interval or ratio scales, they have to be treated as measured on an ordinal scale. Here this problem has been tackled in its most general formulation, that is when mixed measurement scales (interval, ratio and ordinal) are used and both stochastic and fuzzy uncertainties are present. Objectives of this article are first to present a comprehensive review of useful solutions already proposed in the literature and second to advance the state of the art mainly in the theoretical guarantee that weights have the meaning of importance coefficients and they can be summarized in a voting matrix. This is a key result for using non-compensatory Condorcet consistent rules. A proof on the probability of existence of ties in the voting matrix is also developed.     

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.     

Sophisticated voting under the plurality procedure: A test of a new definition

The Niemi-Frank definition of sophisticated voting can now be evaluated on two grounds. First, we can compare our definition to Farquharson's. For the most part, the two definitions yield identical outcomes. Both pickCondorcet winners a very high proportion of the time and prevent the selection of Condorcet losers. The major differences are in the logic underlying the two definitions and in the rate of determinacy of outcomes. Here there is a tradeoff. The logic underlying the Farquharson model is especially persuasive, although it is our feeling that the Niemi-Frank definition comes closer to mirroring the way in which voters might actually analyze a plurality situation. In any case, the price paid by the Farquharson definition for its ironclad logic is a much higher rate of indeterminacy. In over half of the cases, the Farquharson logic fails to lead to any conclusion whatsoever. The Niemi-Frank definition yields many more determinate situations, with mostly Condorcet winners and with strategies that make good, if not completely unassailable sense. A second way of evaluating the Niemi-Frank definition is in comparison with sincere voting. A commonly-cited shortcoming of plurality voting is that often fails to choose a Condorcet winner. As we notedearlier, sophisticated plurality voting, unlike binary voting, is imperfect in this respect. Nonetheless, even taking account of the indeterminacy thatremains in the Niemi-Frank definition, sophisticated voting picked a Condorcet winner about 10 percent more frequently than did sincere voting as well as eliminating the possibility of a Condorcet loser being chosen. By this measure, the Niemi-Frank definition is not only acceptable but suggests that this form of strategic behavior actually leads tobetter outcomes. By proposing and now by testing a new definition of sophisticated voting under plurality rule, we have begun to make some headway on understanding strategic behavior and its effects in an outwardly simple yet deceptively complex voting system. We are, of course, far from finished. Most significantly, our definition applies to only three alternatives, and Farquharson's (even if one is willing to live with its high indeterminacy) becomes extraordinarily cumbersome with more than three alternatives? In any event, the results of this foray into sophisticated nonbinary voting suggests once again that strategic behavior, rather than making things worse, improves the chances that the outcome will be the one most favored by the majority criterion.     

Re-examining the law of iterated expectations for Choquet decision makers

Yoo (Economic Letters 37:145–149, 1991) argues that the law of iterated expectations must be violated if the probability measure of a Choquet decision maker is non-additive. In this article, we prove the positive result that the law of iterated expectations is satisfied for Choquet decision makers whenever they update their non-additive beliefs in accordance with the Sarin and Wakker (Journal of Risk and Uncertainty 16:223–250, 1998) update rule. The formal key to this result is the act-dependence of the Sarin–Wakker update rule, which does not hold for the update rules considered by Yoo (1991).     

Pareto improvements by Pareto strategic voting under majority voting with risk loving and risk avoiding voters — A note

Voters satisfy maximin or maximax in their choices between sets of alternatives and secure a Pareto improvement by all voting strategically under simple majority voting for particular sincere preferences. Thus the assumption that strategic voting is a bad thing is challenged and the idea that we should reject voting because of the possibility of misrepresentation dismissed.     

Environmental disasters as risk regulation catalysts? The role of Bhopal,Chernobyl, Exxon Valdez,Love Canal,and Three Mile Island in shaping U.S. environmental law

Unexpected events such as environmental catastrophes capture wide public attention. Soon after five major shocks—Three Mile Island, Love Canal, Bhopal, Chernobyl, and the Exxon Valdez oil spill—Congress voted on new risk regulation. This paper conducts an event study to test whether individual congressional representatives were “shocked” by these environmental disasters into increasing their probability of voting in favor of risk legislation. On average, representatives were less likely to vote in favor of bills tied to these five events. Significant heterogeneity in representatives’ responses to these shocks is documented. Liberal Northeast representatives were most likely to increase their pro-environment voting in the aftermath of these shocks.

Potential and ‘Power of a Collectivity to Act’<Superscript>*</Superscript>

This paper connects two notions: Hart and Mas-Colell’s ‘potential’, related to the value of coalitional games, and Coleman’s earlier notion of ‘power of a collectivity to act’, related to the easiness to make decisions by means of a voting rule. ^*This research has been supported by the Spanish Ministerio de Ciencia y Tecnología under project BEC2000-0875, and by the Universidad del País Vasco under project UPV/EHU000031.321-HA-7918/2000. The first author acknowledges financial support from the Spanish Ministerio de Ciencia y Tecnología under the Ramón y Cajal programme.     

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.     

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.     

Nader voters in the 2000 Presidential Election: what would they have done without him?

This research estimates the likely voting behavior of Nader voters if he had not been a candidate in the presidential race. Bivariate analysis of ANES data suggests that Nader voters fit the profile of likely voters and have a distinct preference for Democratic candidates. We utilize multinomial logit analysis to include the possibility of abstention as well as the option of voting for Gore, Bush, or another third-party candidate. The results suggest that Nader voters closely resembled the typical voter in educational achievement, and therefore it is likely that a majority of these individuals would have participated in the 2000 election if Nader had not been a candidate. In addition, it is likely that these individuals would have voted for Al Gore over George Bush. However, these Nader voters were younger, less partisan, and were more likely to express feelings of political alienation, so it is quite possible that the absence of the Nader candidacy would have kept a sizeable minority of them at home on election day.     

A Method for Eliciting Utilities and its Application to Collective Choice

Designing a mechanism that provides a direct incentive for an individual to report her utility function over several alternatives is a difficult task. A framework for such mechanism design is the following: an individual (a decision maker) is faced with an optimization problem (e.g., maximization of expected utility), and a mechanism designer observes the decision maker’s action. The mechanism does reveal the individual’s utility truthfully if the mechanism designer, having observed the decision maker’s action, infers the decision maker’s utilities over several alternatives. This paper studies an example of such a mechanism and discusses its application to the problem of optimal social choice. Under certain simplifying assumptions about individuals’ utility functions and about how voters choose their voting strategies, this mechanism selects the alternative that maximizes Harsanyi’s social utility function and is Pareto-efficient.