When the whole electoral region is subdivided into multiple electoral districts, Bazi activates three options (with Option 1 the default) for the user to select:
Denoting districts by i = 1, ..., k and parties by j = 1, ..., l , the three options may differ in their resulting apportionments aij , the number of seats apportioned in district i to party j . We assume that, prior to the election, each district i is allocated mi seats to fill; these numbers are called district magnitudes. The house size h then amounts to h = m1 + ··· + mk . Let vij be the number of votes cast in district i for party j , called the district party votes.
Option 1 solves the apportionment problem for each district separately, without consideration of any interactions with other districts. That is, within district i, its mi seats are apportioned among parties proportionally to the pertinent district party votes vi1, ..., vil .
The other two options have in common that they first calculate regionwide global party seats sj ; this process is called superapportionment. Option 2 is appropriate when all voters in the whole electoral region have the same number of votes, and finds its superapportionment as follows. All district votes for party j are summed across the whole electoral region, giving the party vote total vj = v1j + ··· + vkj . The house size h is then apportioned among parties proportionally to these party vote totals v1, ..., vl , resulting in global party seats sj .
Option 2 then proceeds to calculate its subapportionment, that is, the apportionment matrix aij , in such a way that its rows form a breakdown of the prespecified district magnitudes, mi = ai1 + ··· + ail , while its columns are a breakdown of the global party seats, sj = a1j + ··· + akj . To this end Bazi uses a discrete variant of an iterative algorithm known as alternating scaling or iterative proportional fitting.
Option 3 responds to the specific needs in the 2004 amendment of the Electoral Law of the Canton Zurich (Gesetz über die politischen Rechte). In Zurich, the number of votes to which a voter is entitled depends on the district. Voters in district i have mi votes, that is, as many votes as there are seats to fill. In order to put the weights of each voter on a common scale, district party votes vij are converted into district voter counts wij = < vij / mi > . The angle brackets < > indicate standard rounding, demanding that the quotient vij / mi is rounded to the nearest integer.
Option 3 now finds its superapportionment as follows. All district voter counts for party j are summed across the whole electoral region to obtain the Canton voter count wj = w1j + ··· + wkj that indicates the number of people behind party j . The house size h is then apportioned among parties proportionally to these Canton voter counts w1, ..., wl , yielding the global party seats sj . Given this superapportionment, the subapportionment is calculated in the same fashion as with Option 2.
All apportionment results obtained with the Bazi program can be checked at any time and without a machine, simply using paper and pencil (or a pocket calculator). As for Option 1, Bazi displays as the last entry of row i a divisor Ci . With it, the results in row i are reproduced according to
aij = [ vij / Ci ]
That is, district party votes are subdivided solely by the divisor associated with that district. The square brackets [ ] indicate that the resulting quotients are rounded as demanded by the selected divisor method.
As for Options 2 and 3, Bazi outputs two sets of divisors, the district divisors Ci as the last entry of any district row i , and the party divisors Dj as the last entry of any party column j . The number of seats apportioned in district i to party j is then reproduced via
aij = [ vij / (Ci · Dj ) ]
That is, district party votes are subdivided by both, the corresponding district divisor as well as the corresponding party divisor, and the resulting quotient is rounded as is peculiar to the selected divisor method.
fp/22MAR2004