A numerical method to determine a degressive proportional distribution of seats in the European Parliament

Distribution of seats in the European Parliament postulated by the Treaty of Lisbon should be degressively proportional. We propose a numerical method which determines uniquely a distribution of seats in the European Parliament which fulfil the requirements of degressive proportionality. More generally, let li, 1≤i≤n, be any non-increasing sequence of real positive numbers. We say that a sequence of positive integers mi is degressively proportional with respect to the sequence li, if mi and limi are non-increasing sequences. For fixed positive integer M we determine inductively a sequence Mi degressively proportional with respect to li which is smaller, then any other such sequence with the first element ≥M. Moreover, for every integer ∑i=1nMi≤Y≤nM we determine a degressively proportional sequence mi with respect to li, which has the first element m1=M and satisfies inequalities Y−(n−2)≤∑i=1nmi≤Y.