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.
@cite_14: 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.
@cite_7: 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.
@cite_6: Abstract 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.
@cite_0: The voting system of the Legislative Council of Hong Kong (Legco) is sometimes unicameral and sometimes bicameral, depending on whether the bill is proposed by the Hong Kong government. Therefore, although without any representative within Legco, the Hong Kong government has certain degree of legislative power – as if there is a virtual representative of the Hong Kong government within the Legco. By introducing such a virtual representative of the Hong Kong government, we show that Legco is a three-dimensional voting system. We also calculate two power indices of the Hong Kong government through this virtual representative and consider the C-dimension and the W-dimension of Legco. Finally, some implications of this Legco model to the current constitutional reform in Hong Kong will be given.
@cite_5: A formal analysis of the amending formula of Canada's Constitution Act, 1982 is presented. It is shown that larger provinces have greater voices in determining constitutional amendments, except that there are two groupings of provinces within which power with respect to constitutional amendments is exactly equal. Further, it is argued that differences in power among the provinces are disproportionately small in comparison to differences in population, so that the amending formula systematically directs greater influence to the individual citizens of smaller provinces.
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 .