Consistent allocation of cabinet seats: the Swiss Magic Formula

In 1959, the four largest political parties in Switzerland’s Federal Assembly established a broad coalition government with a specific allocation of the seven cabinet seats among them. This assignment continued unaltered for 44 years up to 1999. It is, therefore, phrased the ‘Magic Formula’. The underlying political paradigm appears to be essentially egalitarian, with a Rawlsian concern for minorities. We thus ask whether the Magic Formula can be implemented by the nucleolus of a weighted voting game for the Swiss government. We provide results over real-valued and integer-valued imputation sets, and for all regular election periods from 1959 to date. It turns out that the Magic Formula is contained in eight out of 11 nucleoli over integer domains between 1959 and 1999, and that it comes very close to the nucleolus in two of the three remaining cases. The game solution likewise predicts in part a major modification of the Magic Formula adopted in 2003.