Belgium has repeatedly tinkered with its electoral system over the past several decades.
BELGIUM 1 (1991)
This system may have also been used in previous elections.
Here is the apportionment of mandates within arrondissements (first tier) nested wtihin provinces (second tier) for 1978-91:
First Allocation to Parties in Arrondissements
Seats were allocated first in arrondissements to parties for multiples of full Hare quotas (valid votes divided by the number of seats). Parties had to surpass a legal threshold of 5% of the arrondissement vote to qualify to receive any seats. Here is the 1991 allocation of seats in Dinant-Phillipeville and Namur Arrondissements of Namur Province:
In Dinant-Phillippeville, only the PS earned more votes than a full Hare quota (95,609 valid votes/3 seats equals a Hare quota of 31,690) and received a seat. In Namur Arrondissement, the PS gained more than 2 full Hare quotas (165,441 valid votes/6 seats equals a Hare quota of 27,574) and thus 2 seats. The PSC won a seat for exceeding more than 1 full Hare quota.
Second Allocation to Parties in Provinces
After allocation at the arrondissement level, any remaining seats were allocated at the provincial level based on the collective votes received by parties in all arrondissements according to the d’Hondt highest average method of proportional representation. Parties had to have received at least one-third of a Hare quota in at least one arrondissement to participate in the provincial level allocation. Here is the distribution of the remaining five seats in Namur Province in 1991:
Only four parties (PS, PSC, PRL, and ECOLO) received more than one-third of a Hare quota in at least one arrondissement (see previous table). The first d’Hondt quota is equal to the total number of votes received by each party. The quotas highlighted in yellow indicate the four seats already won by the PS and the PSC in the first level of allocation in the two arrondissements. The remaining seats were allocated in descending order of d’Hondt quotas as indicated by the numbers in boldface to the left of each quota.
Allocation of Seats Won at the Provincial Level Back to Arrondissements
Note: I am uncertain regarding the accuracy of this part of the process. The next step is the allocation of the seats won at the provincial level back to each arrondissement. The calculation of local fractions for each party in each arrondissement that won seats at the provincial level facilitated this process. The local fraction equaled the number of seats won by the party in the arrondissement during the first allocation divided by the total number of Hare quotas that the party received in the arrondissement. The local fraction equaled 0 if the party won no seats in the arrondissement.
The rules for the allocation of seats won at the provincial level back to arrondissements appear to have been as follows. First, seats were allocated back to arrondissements in the order that they were won by the parties. Second, seats went to the arrondissement where the party had received a higher local fraction (or portion of a Hare quota if it had won no seats and the local fraction equaled zero). An exception is that if all of the seats in an arrondissement have already been filled, then the seat goes to the arrondissement with the next highest local fraction. Similarly, if a party has already received a provincially-won seat in an arrondissement, then its next provincially-won seat goes to the the arrondissement with the next highest local fraction.
The above table shows the order in which mandates were gained in Namur Province in 1991 and thus were allocated back to arrondissements. The previous table presents the local fractions greater than zero for all parties in the two arrondiseements in the province. The following table shows how the seats were allocated in this province back to arrondissements and the controlling rule. The fifth seat went to the PRL and to Namur Arrondissement as the PRL had won a higher remainder (i.e. portion of Hare quota) in that arrondissement. The sixth sent went to ECOLO in Namur following the same rule. The PSC had a higher local fraction in Dinant-Phillipeville, so it gained its seventh seat there. The PRL had already won a seat in Namur so it gained the eighth seat in Dinant-Phillipeville–the last seat in that arrondissement. The PS received the ninth and final seat in Namur where it had a higher local fraction but also the only seat left available.
Allocation of Seats to Candidates
Belgian parties have to file ordered lists in each electoral constituency with no more names on the list than seats available. Voters may cast a vote for the list as ordered–called header votes–or preference votes for as many candidates as desired for their chosen party.
Candidates need to have won preferences equal to a Droop quota–all valid ballots cast for the list divided by the number seats it won plus one (rounded up to the next whole number)–to gain a seat. Any candidates that exceed a Droop quota are elected (in descending order of the total number of preference votes received if there is more than one). If any seats remain, sufficient header votes are added to those preference votes received by other candidates so that they too can reach a Droop quota until all of the header votes have been devolved to candidates. Any seats still remaining are distributed based on the order of preference votes.
The following is a fictional example of the allocation process to candidates using these rules. (The names of candidates and preferences are for the New Flemish Alliance in Antwerp Province in 2014 but the allocation process was changed starting with the 2003 election.) In this case, the party won 11 of the 24 mandates.
The Droop quota equals 37,461 (rounded up result for 449,531 votes cast for the list divided by 12, which is one greater than number of seats won by the party). Only the leading candidate, Bart De Wever, exceeds the Droop quota and immediately wins a seat.
Afterwards, one must begin the devolution of header votes to remaining candidates in list order. The Header Votes Devolution column shows how many are given to each candidate. The Header Votes Left column shows how many header votes are left as the devolution proceeds. The Preferences after Devolution column reveals the number of preferences for each candidate after the distribution of header votes is complete.
There were enough header votes for the second through fifth candidates on the list to reach full Droop quotas. Since all seats are not yet awarded, remaining seats go to candidates in order of preferences after the devolution of header votes. Rob van de Helde, the sixth on the list, wins election next as the devolution pushed his number of preference votes higher than Jan Jambon, who is last on the list. However, Jambon wins the seventh seat as he has the next highest number of preferences. The last four seats go in descending order of preferences–not in list order–as shown in the last column of the table.
BELGIUM 2 (1995-99)
The number of seats in the Chamber declined form 212 to 150 with the 1995 elections. Belgium also aggregated arrondissements together in several places to create larger first-tier electoral constituencies. Here was the apportionment for the 1995 and 1999 elections:
The final change was parties now needed to receive at least 2/3 of a Hare quota in one arrondissement to participate in the allocation of seats at the provincial level in that province.
BELGIUM 3 (2003-2010)
Belgium simplified its electoral system for the 2003 elections. Leaving aside for the moment the now former province of Brabant, seats were allocated in a single allocation at the provincial level by the d’Hondt method.
Brabant had been divided into three parts by the Belgian state reform: Flemish Brabant (part of Flanders), Walloon Barbant (part of Wallonia), and Brussels-Capital which was its own region separate from Flanders or Wallonia. However, the former constituencies and electoral system were maintained. So seats were first allocated for full Hare quotas in the three electoral districts–Leuven, Brussels-Halle-Vilvoorde, and Walloon Brabant (Nivelles)–with seats then allocated by d’Hondt across former Brabant.
Here is the apportionment for the 2003-2014 elections with the three linked constituencies in Brabant highlighted in yellow.
Additionally, Belgium altered the allocation of preference votes so that they became more likely to influence the order of the list. Specifically, the share of header votes devolved to candidates in list order was reduced from 100 to 50 percent.
The following shows the distribution of seats to candidates of the New Flemish Alliance in Antwerp Province in 2014 where the party won 11 of the 24 mandates. (Compare to the distribution under the old rules above.)
The Droop quota equaled 37,461 (rounded up result for 449,531 votes cast for the list divided by 12, which was one greater than number of seats won by the party). Only the leading candidate, Bart De Wever, exceeded the Droop quota and immediately won a seat.
Afterwards, the devolution of header votes to remaining candidates began in list order.
There were enough header votes for the second and third candidates on the list to reach full Droop quotas. Since all seats were not yet awarded, remaining seats went to candidates in order of preferences after the devolution of header votes. Jan Jambon gained the fourth seat as he the most preference votes of any unreturned candidate, despite being placed last on the list. remaining seats go in descending order of preferences–not in list order–as shown in the last column of the table.
BELGIUM 4 (2014)
Belgium altered the method for allocating seats in the former Province of Brabant starting with the 2014 election. There is no longer a second allocation tier. The Brussels-Halle-Vilvoorde district was also split. Brussels-Capital now serves as an electoral constituency with seats allocated as in any province. Halle-Vilvoorde was combined with Leuven into the electoral constituency of Flemish Brabant. Walloon Brabant (or Nivelles) remains an electoral constituency. As Flemish Brabant and Walloon Brabant are provinces, all seats in Belgium are now allocated within provinces or the Brussels-Capital Region.
Voters in six Flemish Brabant municipalities near Brussels with language facilities for Francophones may choose to cast their ballots in Brussels instead of Flemish Brabant.