On influence and compromise in two-tier voting systems

We examine two aspects of the mathematical basis for two-tier voting systems, such as that of the Council of the European Union. These aspects concern the use of square-root weights and the choice of quota. Square-root weights originate in the Penrose square-root system, which assumes that votes are cast independently and uniformly at random, and is based around the concept of equality of influence of the voters across the Union. There are (at least) two distinct definitions of influence in current use in probability theory, namely, absolute and conditional influence. These are in agreement when the underlying random variables are independent, but not generally. We review their possible implications for two-tier voting systems, especially in the context of the so-called collective bias model. We show that the two square-root laws invoked by Penrose are unified through the use of conditional influence. In an elaboration of the square-root system, Slomczynski and Zyczkowski have proposed an exact value for the quota $q=q^*$ to be achieved in a successful vote of a two-tier system, and they have presented numerical and theoretical evidence in its support. We indicate some numerical and mathematical issues arising in the use of a Gaussian (or normal) approximation in this context, and we propose that other values of $q$ may be as good if not better than $q^*$. We discuss certain aspects of the relationship between theoreticians and politicians in the design of a two-tier voting system, and we reach the conclusion that the choice of quota in the square-root system is an issue for politicians informed by theory

Square root voting system, optimal threshold and \pi

The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favour of the square root voting system, where the voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic 'union' of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: q \simeq 1/2+1/\sqrt{{\pi} M}. The prefactor 1/\sqrt{{\pi}} appears here as a result of averaging over the ensemble of unions with random populations.Comment: revised version, 21 pages in late

Penrose voting system and optimal quota

Systems of indirect voting based on the principle of qualified majority can be analysed using the methods of game theory. In particular, this applies to the voting system in the Council of the European Union, which was recently a subject of a vivid political discussion. The a priori voting power of a voter measures his potential influence over the decisions of the voting body under a given decision rule. We investigate a system based on the law of Penrose, in which each representative in the voting body receives the number of votes (the voting weight) proportional to the square root of the population he or she represents. Here we demonstrate that for a generic distribution of the population there exists an optimal quota for which the voting power of any state is proportional to its weight. The optimal quota is shown to decrease with the number of voting countries.Comment: 12 pages, 2 figure

The Art of Compromise

Policy is modeled as the outcome of negotiations between two three-party parliamentary states. An election in jurisdiction A determines the composition of the legislature that selects a representative to negotiate an intergovernmental policy agreement with the representative from the legislature of jurisdiction B. Negotiations are modeled using Nash’s (1950) bargaining framework, modified to account for a simultaneous legislative ratification vote. Though agreements favor the legislative representative least willing to compromise, agreements between the bargainers may not follow the ordering of the parties’ ideal policies. An electoral outcome where support for the center party comes from extreme voters may emerge.Vote balancing; intergovernmental bargaining; legislative ratification; willingness to compromise

Globalisation, Competitive Governments, and Constitutional Choice in Europe

The traditional school of economic policy analysis predicts that globalisation will give rise to predatory competition between the governments of the European nation states. The consequence is anticipated to be a marked reduction in, if not the destruction of, the benevolent Welfare State. The objective of this contribution is to present the main arguments that have led us to believe that, this traditional literature notwithstanding, a European constitution should not restrict but rather should encourage horizontal and vertical governmental competition. In our view the European political order, in defining the relationship among member states and also the relationship between the member states and the EU, ought to be inspired by what we know about competition in the commercial sphere.globalisation, globalization

Electoral reform in Asia: institutional engineering against "money politics"

This paper argues that the concept of intraparty competition – as opposed to interparty competition – provides the most useful lens through which to understand recent cases of electoral reform in East Asia. Various democracies in the region have over the past two decades replaced ‘extreme’ systems on the intraparty dimension with more moderate types. Pressure for reform built up as these systems were increasingly blamed for a number of social ills, such as "money politics" and economic mismanagement. The paper will conclude by arguing that the effect of electoral reform has been rather limited. In particular, particularistic strategies of voter mobilization – such as clientelism and vote buying – remain an important electoral tool for many politicians