How are votes counted in an election and how are seats allocated afterward in the Knesset? The formula for translating votes into seats has major implications for the outcome of the elections and the map of political parties. This article looks at the various formulas used and their implications.
The division of power among candidates is a result of the electoral preferences expressed by the citizens on Election Day. Alongside the human factor, there are also other significant elements that are sometimes less conspicuous. Although a party’s success is influenced primarily by its percentage support among the voting public, there are other factors at work, such as the electoral system (plurality/majority or proportional), the threshold, and use of districts which influence the composition of the legislature after all the votes have been counted. In proportional systems, it is also matters how votes are converted into seats and how surplus votes are allocated.
This article looks into several formulas used to translate votes into parliamentary seats (the “electoral formula”) and considers the implications of each one.
The Largest Remainder system is implemented in two stages. First a quota (a certain number of votes cast) is set for dividing up the seats in the legislature (120 in Israel); in the second stage, the surplus votes are ranked according to the number of surplus votes received by each party. The more surplus votes a party has, the greater its precedence over other parties for receiving an unallocated seat no matter how many seats it has already received.
The Hare Quota
In this method, the total number of valid votes cast in the election is divided by the number of seats in the legislature to determine the quota, or the “vote price” of each seat. When an electoral threshold is in force, votes for parties that fail to pass the threshold are deducted from the count of valid votes. A party that received a number of votes equal to the quota receives one seat in the legislature. In most cases, a party’s vote total is not evenly divisible by the quota, which means that it has leftover votes equal to some fraction of the quota. If these surplus votes are ignored when parliamentary seats are distributed, the leftovers of all the parties may add up to several quotas/seats; when all seats have been allocated in the first stage, a number of empty seats will remain unfilled.
After the division by integral quotas, the leftover seats must be divided among the parties. It is clear that this process rewards some parties, so the rules for allocating these seats are extremely important. When the Hare quota is used, the allocation of the leftover seats is influenced by each party’s vote total; the party with the largest remainder has priority in receiving unallocated seats. For example, if there is only one unallocated seat left, it will go to a party that received 7.89 quotas rather than a rival party that polled 32.55 quotas, because the first party has a larger surplus than the second (0.89>0.55). The logic here is that the party with the larger surplus was closer to winning another seat and consequently deserves to receive it.
The method can be illustrated for a legislature of 100 seats and a campaign in which six parties run. To determine the quota, the number of valid ballots cast (100,000) is divided by the number of seats, yielding a quota of 1,000 votes. Table 1 lists the number of seats received by each party on the basis of integral quotas and the number of leftover votes that do not add up to a full quota for each. After the seats are distributed to all the parties on the basis of integral quotas (97 seats), three seats remain. These will be awarded to the three parties with the largest surpluses (party F, party E and party B).
Table 1: Allocation of Parliamentary Seats using the Hare Quota
The Largest Remainder system ignores the total number of votes received by a party, which means that, in theory, all parties running have an equal chance of winning an extra seat. This is apt to lead to parties to prefer to split into a number of lists in order to increase their odds for a remainder—although this also carries the danger that a list will not pass the threshold and all its votes will be wasted. The Hare Quota gives the most proportional results and is followed in Costa Rica, Denmark, El Salvador, Hong Kong and Slovakia.
The Droop Quota
The Droop Quota is the quota most commonly used in elections held under the Single Transferable Vote (STV) system (i.e., the minimum number of votes a candidate must receive in order to be elected). The main difference between the two formulas is that in the Hare system the number of valid ballots cast is divided by the number of seats, whereas in the Droop system the number of valid ballots is divided by the number of seats plus one (returning to the previous example, the number of votes is divided by 101 instead of 100). This formula produces a smaller quota (990 instead of 1,000). Its advantage is that, as a matter of arithmetic, it means that each party is likely to receive more integral quotas, leaving fewer leftover seats to be allocated. That is, it is the smallest possible quota that is large enough to avoid the “election” of more candidates than there are seats in the legislature. This system is followed in the Republic of Ireland, Northern Ireland, Australia and Malta, as well as in Greece, South Africa and the Czech Republic.
In the Largest Average system, a party receives seats in the legislature both as a function of the number of votes it receives and of the “price” it can pay for them. In this system, there are no quotas and no surplus votes. To begin with, all the valid ballots are assigned to the parties that receive them, after which the seats are allotted in several stages, as described below, on the basis on the number of votes credited to each party in each stage.
The d’Hondt System
In this system, the total number of votes received by each party is divided by the number of seats it has received thus far, plus one. Consider the example of four parties running for a legislature with six seats (Table 2). In the first stage, we look at the number of votes cast for each party; the party with the most votes is awarded the first seat in the parliament. In our example (Table 2), Party A, with 600 votes, is the frontrunner, so it is the first to receive a seat. After this, the number of votes it received (600) is divided by the number of seats it has been allotted thus far (1) plus one (=2). The party’s actual vote total is then divided by this number, and in the next round of seat allocation it is considered to have received 300 votes. If it is still the largest party in the second round, it is awarded another seat, after which its original vote total (600) is divided by three (2+1, that is, the number of seats it has received plus one). For the allocation of the third seat, it is considered to have received 200 votes—and now it falls short of another seat. So the third seat is awarded to the party with the largest number of votes (Party B), which received 280 votes. After it has received its first seat, its vote total is divided by two, and it is now considered to have received 140 votes. As a result, the fourth seat in the legislature goes to party A, with its 200 votes; its original total is now divided by four. The fifth seat goes to party C, whose votes are now divided by two. Finally, the sixth and last seat goes to Party A. Party D remains without representation in the legislature.
Table 2: Allocation of Legislative Seats using the d’Hondt System
|Votes||Seat 1||Seat 2||Seat 3||Seat 4||Seat 5||Seat 6||Total Seats|
|Party A||600||600 (600/1)||300 (600/2)||200 (600/3)||200||150 (600/4)||150||4|
|Party B||280||280 (280/1)||280||280||140 (280/2)||140||140||1|
|Party C||160||160 (160/1)||160||160||160||160||80 (160/2)||1|
|Party D||60||60 (60/1)||60||60||60||60||60||0|
This system favors large parties, because the allocation of seats is based on their electoral performance. The repeated divisions do not undermine a large party’s ability to be awarded additional seats because the quotient of a large number remains large enough to exceed the votes cast for small parties. This system also deters factionalism within parties, since the size of the party counts. The d’Hondt system is employed in many countries, including Argentina, Brazil, Bulgaria, Switzerland, Finland, Belgium, Austria and Portugal.
The Sainte Laguë System
The Sainte Laguë system is very similar to the d’Hondt system, the main difference being the divisors employed. Whereas the d’Hondt system uses 1, 2, 3, 4, 5, and so on, the Sainte Laguë system divisors are 1, 3, 5, 7, 9, and so on. This means that after a party has won its first seat, its vote total is divided by 3 instead of 2, after it wins a second by 5 instead of 3, and so on.
Table 3: Allocation of Legislative Seats using the Sainte Laguë System
|Votes||Seat 1||Seat 2||Seat 3||Seat 4||Seat 5||Seat 6||Total Seats|
|Party A||600||600 (600/1)||200 (600/3)||200||200||120 (600/5)||150||3|
|Party B||280||280 (280/1)||280||93.3 (280/3)||140 (280/2)||93.3||140||2|
|Party C||160||160 (160/1)||160||160||160||53.3 (160/3)||80 (160/2)||1|
|Party D||60||60 (60/1)||60||60||60||60||60||0|
This system offers an advantage to smaller parties because it reduces the number of times that large parties keep receiving a seat, inasmuch as the larger divisors reduce their vote total more rapidly. A comparison of the examples for the d’Hondt system (Table 2) and the Sainte Laguë system (Table 3) shows that, in the former, the largest party receives four seats, but only three seats under the latter, while Party B receives two seats under the Sainte Laguë system and only one under d’Hondt. The Sainte Laguë system is used in New Zealand, Iraq, Bosnia and Herzegovina, Latvia, Kosovo, Denmark (for the 40 supplementary seats in the national parliament), and in Germany on the federal level and for some state legislatures. It has also spawned the modified Sainte Laguë system, which employs the divisors 1.4, 3, 5, 7, and so on, and is used in Norway and Sweden, and was used in Nepal’s Constituent Assembly poll in 2008.
A larger increment between the successive divisors works to the benefit of small parties. The d’Hondt system uses the smallest possible increment, so a large party can remain the largest even after it has been awarded a seat and its vote total divided. In the Sainte Laguë system, the increment between divisors is larger and this significantly reduces the odds that the largest party will remain the largest after it is awarded a seat. In the modified Sainte Laguë system, the first divisor is the largest of all systems (1.4 instead of 1); this reduces a party’s vote total even more and seriously impacts the smallest parties and reduces their chances of winning any seats at all.
As noted, Israel has changed its electoral formula three times. In the first two Knesset elections, in 1949 and 1951, it employed the Hagenbach-Bischoff Quota, to be explained below. It then switched to the Hare Quota (a member of the Largest Remainder family). Finally, in April 1973, Amendment 5 to the Knesset Elections Law (Consolidated Version) 5729-1969 modified the formula again. The amendment is often known as the Bader-Ofer Law, but in practice represents a return to the Hagenbach-Bischoff system. It combines the Largest Remainder and Largest Average principles, using the Hare Quota in the first stage but the d’Hondt system for allocating surplus votes. This system has the relative simplicity of computing the integral seats using the Hare Quota, but then awards the remainder seats in a way that favors the large parties. It was no coincidence that the sponsors of this change in Israel were representatives of the large parties, which supported the amendment.
To illustrate the importance of which calculation method is used, consider the following example. In the elections for the 18th Knesset, in 2009, Kadima won 29 seats thanks to the Hagenbach-Bischoff system. Had the Largest Remainder system been employed, it would have received only 28 seats. The Likud, too, received one more seat than it would have using the Largest Remainder system. On the other hand, the Pensioners’ Party and Hadash would each have received an additional seat under the Largest Remainder system. In the elections for the 17th Knesset (2006), the large parties—Likud, Labor, and Shas—received extra seats thanks to the Hagenbach-Bischoff system, while Meretz, Hadash, Yisrael B’aliya, and Ra’am (the United Arab List) each received one fewer seat than they would have under the Largest Remainder system.
In order to increase their chances of benefiting from the allocation of surplus votes, lists can sign pre-election agreements. Under these surplus-vote agreements, two parties are considered to be one in the second stage of the allocation of seats, and the total number of votes for both is used for the allocation of surplus seats. To determine which of the two lists receives the extra seat, a new quota is computed for the two lists (using the same system by which the quota is calculated for all the lists) and the votes of the two lists simply are divided by this quota. The results determine which of them wins the seat. If a surplus seat remains after this stage as well, the d’Hondt system is employed, computed for each of these two parties.
The outcome of Knesset elections is influenced by many factors, ranging from the citizens’ decision of whom to vote for to the parties’ decision concerning surplus-vote agreements. Individual voters’ decisions are influenced not only by ideological considerations but also by many other factors, including the electoral system, since the chance that one’s first-choice party will or will not clear the electoral threshold is of significant weight in the decision to vote for it.
The implications of the electoral system are traditionally classified as psychological or mechanical. The psychological implications affect the decisions made by voters and parties and guide their behavior. For voters, it may contribute to their decision as to whether to cast an ideological vote or a strategic vote; for parties, it may be manifested in a decision to split, to invest campaign resources in a district or the entire country, and so on. The mechanical implications are built into the system and affect the results—by filtering out small parties, undermining the principles of proportionality and representation, and so on. The results published in the media the morning after Election Day are produced by several non-random factors. It is true that citizens are supposed to vote according to their conscience, but the electoral laws affect the outcome and mold the legislature in accordance with preset theoretical lines.