The Council of the European Union (EU) is one of the main decision-making bodies of the EU. A number of decisions require a qualified majority, the support of 55% of the member states (currently 15) that represent at least 65% of the total population. We investigate how the power distribution based on the Shapley--Shubik index and the proportion of winning coalitions change if these criteria are modified within reasonable bounds. The power of the two countries, with approximately 4% of the total population each, is found to be almost flat. The decisiveness index decreases if the population criterion is above 68\% or the states criterion is at least 17. Some quota combinations contradict the principles of double majority. The proportion of winning coalitions can be increased from 13.2% to 20.8% (30.1%) such that the maximal relative change in the Shapley--Shubik indices remains below 3.5% (5.5%). Our results are indispensable for evaluating any proposal to reform the qualified majority voting system.

翻译：欧盟理事会（EU）是欧盟主要的决策机构之一。多项决策需要特定多数支持，即获得55%成员国（目前为15国）的赞成，且这些国家需代表至少65%的总人口。本研究探讨了若在合理范围内调整这些标准，基于沙普利-舒比克指数的权力分布及获胜联盟比例将如何变化。研究发现，两个各占约4%总人口的国家，其权力曲线近乎平缓。当人口标准超过68%或国家数量标准达到17国时，决策力指数会下降。部分配额组合违背了双重多数原则。在保证沙普利-舒比克指数最大相对变化低于3.5%（5.5%）的前提下，获胜联盟比例可从13.2%提升至20.8%（30.1%）。本研究成果对于评估特定多数投票制度的改革提案具有重要参考价值。