Consider a situation with @math agents or players where some of the players form a coalition with a certain collective objective. Simple games are used to model systems that can decide whether coalitions are successful (winning) or not (losing). A simple game can be viewed as a monotone boolean function. The dimension of a simple game is the smallest positive integer @math such that the simple game can be expressed as the intersection of @math threshold functions where each threshold function uses a threshold and @math weights. Taylor and Zwicker have shown that @math is bounded from above by the number of maximal losing coalitions. We present two new upper bounds both containing the Taylor Zwicker-bound as a special case. The Taylor Zwicker-bound imply an upper bound of @math . We improve this upper bound significantly by showing constructively that @math is bounded from above by the cardinality of any binary covering code with length @math and covering radius @math . This result supplements a recent result where showed how to construct simple games with dimension @math for any binary constant weight SECDED code @math with length @math . Our result represents a major step in the attempt to close the dimensionality gap for simple games.
As mentioned earlier, the dimension of real world voting systems has been the focus for several studies. The Amendment of the Canadian constitution @cite_5 and the US federal legislative system @cite_6 have dimension @math . The voting systems of the Legislative Council of Hong Kong @cite_0 and the Council of the European Union under its Treaty of Nice rules @cite_7 have a dimension of exactly @math . Kurz and Napel @cite_14 have established that the dimension of the voting system of the Council of the European Union under its Treaty of Lisbon rules is between @math and @math .
@cite_14 @cite_7 @cite_6 @cite_0 @cite_5
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The new voting system of the Council of the European Union cannot be represented as the intersection of six or fewer weighted games, i.e., its dimension is at least 7. This sets a new record for real-world voting bodies. A heuristic combination of different discrete optimization methods yields a representation as the intersection of 13368 weighted games. Determination of the exact dimension is posed as a challenge to the community. The system's Boolean dimension is proven to be 3. Abstract In this note it is proved that the European Union enlargement voting system under two different decision rules (simple majority of countries or qualified majority of 2 3) adopted in the Nice European summit, held in December 2000, has exactly dimension 3. This result clarifies that these voting systems cannot realize either as a weighted voting system or as the intersection of two weighted voting systems and, therefore, it is useful to prove the existence of well-known real voting systems of dimension 3. This paper examines the system of Qualified Majority Voting, used by the Council of the European Union, from the perspective of enlargement of the Union. It uses an approach based on power indices due to Penrose (1946), Banzhaf (1965) and Coleman (1971) to make two analyses: (1) the question of the voting power of member countries from the point of view of fairness, and (2) the question of how the threshold number of votes required for QMV should be determined. It studies two scenarios for change from 2005 onwards envisaged by the Nice Treaty: (1) no enlargement, the EU comprising 15 member countries, and (2) full enlargement to 27 members by the accession of all the present twelve candidates. The proposal is made that fair weights be determined algorithmically as a technical or routine matter as the membership changes. The analysis of how the threshold affects power shows the trade-offs that countries face between their blocking power and the power of the Council to act. The main findings are: (1) that the weights laid down in the Nice Treaty are close to being fair, the only significant discrepancies being the under-representation of Germany and Romania, and the over-representation of Spain and Poland; (2) the threshold required for a decision is set too high for the Council to be an effective decision making body. In the following paper we offer a method for the a priori evaluation of the division of power among the various bodies and members of a legislature or committee system. The method is based on a technique of the mathematical theory of games, applied to what are known there as “simple games” and “weighted majority games.” We apply it here to a number of illustrative cases, including the United States Congress, and discuss some of its formal properties. The designing of the size and type of a legislative body is a process that may continue for many years, with frequent revisions and modifications aimed at reflecting changes in the social structure of the country; we may cite the role of the House of Lords in England as an example. The effect of a revision usually cannot be gauged in advance except in the roughest terms; it can easily happen that the mathematical structure of a voting system conceals a bias in power distribution unsuspected and unintended by the authors of the revision. How, for example, is one to predict the degree of protection which a proposed system affords to minority interests? Can a consistent criterion for “fair representation” be found? It is difficult even to describe the net effect of a double representation system such as is found in the U. S. Congress (i.e., by states and by population), without attempting to deduce it a priori . The method of measuring “power” which we present in this paper is intended as a first step in the attack on these problems.