Weighted Voting System Research Articles

Complete simple games

Completeness is a necessary condition for a simple game to be representable as a weighted voting system. This paper deals with the class of complete simple games and centers on their structure. Using an extension of Isbell's desirability relation to coalitions, different from the extension normally used, we associate with any complete simple game a lattice of coalition models based upon the types of indifferent players. We establish the basic properties of a vector with natural components and a matrix with non-negative integer entries, both closely related to the lattice, which are also shown to be characteristic invariants of the game, in the sense that they determine it uniquely up to isomorphisms.

Weighted Voting, Multicameral Representation, and Power

A simple game ( P, W) can serve as a model of a voting system in which an alternative is pitted against the status quo. In what follows, we investigate the following three aspects of such games as they apply to four real-world examples of voting systems: a characterization of weighted voting systems in terms of the ways in which coalitions can gain or lose by trading players; the application of a graph-theoretic notion of dimension to simple games and voting systems; and the consideration of a way to measure the power of a player as an interval of real numbers. American Mathematical Society Classification Numbers: 90A28, 90D80, 05C65. Journal of Economic Literature Classification Number: C71.

The Borda method is most likely to respect the Condorcet principle

We prove that in the class of weighted voting systems the Borda Count maximizes the probability that a Condorcet candidate is ranked first in a group election. A direct result is that the Borda Count maximizes the probability that a transitive, binary ranking of the candidates is preserved in a group election. A preliminary result, but one of independent interest, is that the Borda Count maximizes the probability that a majority outcome betweenany two candidates is reflected by the group election. All theorems are valid when there is a uniform probability distribution on the voter profiles and can be generalized to other “uniform-like” probability distributions. This work extends previous results of Fishburn and Gehrlein from three candidates to any number of candidates.

A characterization of weighted voting

A simple game is a structure G = ( N , W ) G = (N,W) where N = { 1 , … , n } N = \{ 1, \ldots ,n\} and W W is an arbitrary collection of subsets of N N . Sets in W W are called winning coalitions and sets not in W W are called losing coalitions. G G is said to be a weighted voting system if there is a function w : N → R w:N \to \mathbb {R} and a "quota" q ∈ R q \in \mathbb {R} so that X ∈ W X \in W iff ∑ { w ( x ) : x ∈ X } ≥ q \sum {\{ w(x):x \in X\} \geq q} . Weighted voting systems are the hypergraph analogue of threshold graphs. We show here that a simple game is a weighted voting system iff it never turns out that a series of trades among (fewer than 2 2 n {2^{{2^n}}} not necessarily distinct) winning coalitions can simultaneously render all of them losing. The proof is a self-contained combinatorial argument that makes no appeal to the separating of convex sets in R n {\mathbb {R}^n} or its algebraic analogue known as the Theorem of the Alternative.

Effects of Two-Member Formal Coalitions on the Power Distribution in a Weighted Voting System

A “formal coalition” in a voting body is defined as a subgroup of members in which their vote is completely dependent, voting as a bloc. The formation of such a formal coalition sometimes yields the unexpected effects on the power of members in the voting body. They are the non-increase of power of coalitional members and the increase of power of non-coalitional members. This paper shows that the unexpected effects above will occur with respect to the Banzhaf power index and explores that the formal coalition of what members will yield what effects on the Banzhaf power of members. Throughout the paper, it is assumed that there is only one two-member formal coalition in the voting body and that there are no other formal coalitions in it. Using the well-known concepts of veto and dummy, we then obtained the necessary and sufficient condition that the unexpected effects on coalitional member’s power will occur. In addition, introducing the three new concepts about non-coalitional members, we obtained several sufficient conditions that the unexpected effects on non-coalitional member’s power will occur. Finally, by use of the obtained results we analysed the effects of all possible two-party formal coalitions on the power distribution in the House of Representatives of Japan.

The ultimate of chaos resulting from weighted voting systems

The ultimate of chaos resulting from weighted voting systems

Evaluation of the Distribution of Power in a Weighted Voting System

A distribution of weights in a weighted voting system sometimes yields quite undesirable effects on the outcome of voting, which have not been foreseen until such outcomes actually occur. Examples are the existence of so-called “dummies,” unexpected power of veto, the failure of reflecting the dominance order of weights proportionally on the order of the effective power. The present paper introduce a number of criteria to distinguish such undesirable aspects which potentially exist in some particular configurations of the weight distribution. The criteria were stated in simple relations among weights which can be checked quite easily. Mathematical relations among the criteria were also presented in theorems with their empirical interpretations. Using the proposed criteria, the recent results of the general election for the House of Representatives in Japan was analyzed.

Developments in the Law and Institutions of International Economic Relations

[Ed. Note: Since World War II the administration of international monetary, financial, and commodity agreements has been performed by institutions operating on weighted voting principles, the weight dependent largely on the stake of various countries in the assetsdisposed of through the agreement. There is much discussion now concerning the adoption of such a system for the trade field, perhaps through amendment of the GATT, or in codesof trade liberalization among developed countries. Mr. Gold's article, relating the limitations and problems attendant upon the operation of the weighted voting system in the International Monetary Ftmd, is therefore particularly timely. Stanley D. Metzger.]