Jagiellonian compromise

The Jagiellonian compromise is an electoral system for two-tier voting bodies originally proposed in 2004 for the Council of the European Union as a way of achieving "one person, one vote" within the union. The compromise was analysed by various authors and received attention in the popular press. The system is based on the square root law of Penrose, which implies that a priori voting power defined by the Penrose–Banzhaf index of a member of a voting body is inversely proportional to the square root of its size. Hence the number of votes obtained by a representative of a state $$ j $$ with population $$ N_j $$ is proportional to  $$ \sqrt{N_j} $$. Jagiellonian Compromise is based on a single criterion only. Decision of the Council of the union of $$ M $$ member states is taken if the sum of the weights of states voting in favour of a given proposal exceeds the qualified majority quota $$ q $$  equal to
 * $$q = \frac12 \cdot\left(1 + \frac{\sqrt{\sum_{i=1}^M N_i}}{\sum_{i=1}^M \sqrt{N_i}}\right)$$

For a generic distribution of population among $$ M $$ states of the union, the optimal threshold $$ q_* $$ decreases with $$ M $$ as $$q_* \approx 1/2 +1/\sqrt{\pi M} $$.