# Seat allocation process

Seat allocation process are computational methods for proportional representation, as in the proportional representation needed for votes in parliamentary seats to be converted.

Each party's share of MPs should be equal to its share of the vote. The ideal requirement of each party is its number of votes multiplied by the total number of seats divided by the total number of votes. For the party it is ${\ displaystyle i}$

${\ displaystyle {\ text {seats}} S_ {i} = {\ frac {({\ text {votes}} v_ {i})} {({\ text {all votes}} v)}} \ times ( {\ text {Number of members of parliament}} N)}$

This quotient is not always an integer (if there is a large number of voters, it is rarely). Therefore, a rounding rule is required, according to which the number of seats is calculated as whole numbers from the voting shares of the parties ( total-preserving rounding ).

Each rule produces a different way of minimizing errors. Which one can be considered the best depends on the underlying quality criteria for the allocation of seats. Of course, as a rule, the interests of the decision- making majority are more important for the determination of the right to vote than mathematical arguments: “The right to vote is also the right to power”.

## application

There are three main allocation procedures used worldwide:

There are also variations on these procedures, such as the balanced method .

The Hill-Huntington system is used for the distribution of seats in the United States House of Representatives among states according to their population.

## properties

There are many justifiable requirements for the allocation of seats (including quality requirements, requirements, criteria, quality criteria , conditions ).

The following table has requirements as columns and seat allocation procedures as rows. An entry “fulfilled”, “maximum” or “minimum” applies to each election result. If a field is empty, the statement does not always apply. - The divisor methods in the table (from D'Hondt to Adams) are in the order of decreasing rounding limit, i.e. decreasing favoring large and increasing favoring small parties.

Comparison of seat allocation procedures
Quota criterion House and voice monotony neutral regarding party size smallest representation value Spread of the success values greatest difference in relative success values greatest difference in representation values greatest representation value Majority and minority criterion
Hare-Niemeyer
= Hamilton
Fulfills Fulfills
D'Hondt
= Jefferson
Fulfills maximum Majority criterion *
Sainte-Laguë
= Webster
Fulfills Fulfills minimal
Hill Huntington Fulfills minimal
dean Fulfills minimal
Adams Fulfills minimal Minority criterion met

* The D'Hondt procedure always fulfills the weak majority criterion, if the total number of seats is uneven, also the strong one.

In the following chapters, the word “parties” also stands for lists, list connections, states, departments and similar competitors for seats; “Votes” stands for these when distributed according to population numbers. - Without loss of generality, the case is not taken into account that two or more parties have the same ideal claim (the same quota), but this claim can only be fulfilled for some of these parties. If this occurs, you can draw a lot before the allocation of seats , which of these parties gets which fraction of a vote added to their number of votes (for example, if there are three parties, one party has nothing, a 1/3 vote and a 2/3 vote ). Then all ideal claims are different.

## Types of seat allocation procedures

A distinction is made between two groups of seat allocation procedures, namely quota procedures and divisor procedures.

### Quota procedure

In these procedures, as many seats are allocated to each party as the quota rounded down . The remaining seats are allocated according to a rule to be determined.

The Hare-Niemeyer method (in the Anglo-Saxon region, the Hamilton method ) is the classic quota method: The remaining seats are distributed to the parties according to the size of the fractional shares of the quotas.

As an alternative to the Hare-Niemeyer process, the remaining seats can e.g. B. be distributed step by step according to a certain divisor procedure or (as with Thomas Hare ) all go to the strongest party.

### Divisor method

A rounding rule is specified here and a divisor is sought in such a way that the number of votes for each party, divided by this divisor and rounded to a whole number, results in the number of seats for this party. To do this, you use the interval nesting ; it always leads to the result. A good estimate for the divisor is the quotient of the total number of votes and the total number of seats.

Step 1
The divisor just considered is used to calculate the sum of the rounded quotients.
step 2
If this total is smaller or larger than the number of seats to be allocated, choose a slightly smaller or larger divisor and carry out step 1 again.

(The term quotient method is also possible, but can lead to confusion with quota method.)

Only commercial or symmetrical rounding is neutral to the size of the parties. Procedures with a rounding limit below the fractional part 0.5 systematically favor smaller parties (see excursus 1 ). Procedures with a rounding limit above the fractional part 0.5 systematically favor larger parties.

In addition to this two-step procedure, there are other calculation procedures that lead to the same distribution of seats for all divisor procedures for every election result (see excursus 2 ).

#### Divisor method with a fixed rounding limit

The classic divisor methods with a constant rounding limit are

• the D'Hondt method (in the Anglo-Saxon area: Jefferson method), divisor method with rounding, plus (especially for D'Hondt)
• the Sainte-Laguë method (in the Anglo-Saxon region: Webster method), divisor method with commercial rounding;
• the Adams method , divisor method with rounding up.

With the D'Hondt procedure, the seat entitlement is always rounded down to the next whole number, with the Adams procedure rounded up. In the Sainte-Laguë procedure, the seat claims are rounded commercially. Of all the self-mapping processes, large parties are most favored by the D'Hondt process, and small parties are most favored by the Adams process.

#### Divisor method with variable rounding limit

The classic divisor methods with a variable rounding limit are

You can also arbitrarily specify fixed rounding limits individually. For example, if you want to calculate the seat allocation according to Dean or Hill-Huntington, but deviating from this request more votes for the first seat, you set the rounding limit between 0 and 1 (no seat or one seat), for example, to 0.6 or to 0, 9 or fixed to 1. The higher the rounding limit, the more votes a party needs for its first seat. In the same way, the rounding limit between other neighboring numbers of seats can be set arbitrarily. There is no known mathematically formulated requirement which such arbitrariness could serve to fulfill.

Rounding rule in the Dean method

In order to know whether a non-integer seat claim is rounded up or down in the Dean procedure, the harmonic mean between the rounded up and the rounded seat claim must be calculated. This mean forms the rounding limit. The harmonic mean is the reciprocal arithmetic mean of the reciprocal feature values. The harmonic mean of 1 and 2 results from the reciprocal arithmetic mean of 1 and 1/2. The arithmetic mean of 1 and 1/2 is 3/4. The reciprocal arithmetic mean of 1 and 1/2 and thus the harmonic mean of 1 and 2 is 1 1/3. The harmonic mean of 0 and 1 is defined as 0 (popular: 1 / [infinity + 1] = 0). Therefore, the Dean procedure gives a party a seat with just one vote. The harmonic mean of 2 and 3 is 2.4. When the number pairs increase, the harmonic mean approaches the decimal value 5, but never reaches it.

Hill-Huntington's rounding rule

In order to know whether the Hill-Huntington method is rounded up or down, the geometric mean between the next larger and next smaller integer seat claim must be calculated. This forms the rounding limit. The geometric mean is the -th root of the product of the feature values. The geometric mean from 1 and 2 is therefore around 1.4142. The geometric mean of 0 and 1 is 0. This is why the Hill-Huntington method gives a party a seat with just one vote. The geometric mean from 2 and 3 is around 2.4495. As the number pairs increase, the geometric mean approaches the decimal value 5, but never reaches it. ${\ displaystyle n}$${\ displaystyle n}$

#### Excursus 1: Influence of the rounding limit on beneficiary according to party size

The following table is intended to make it clear that smaller parties are favored by lower rounding limits.

1 seat 2 seats 3 seats 4 seats 5 seats ${\ displaystyle n}$ Seats 1 2 3 4th 5 ${\ displaystyle n}$ 0.5 1.5 2.5 3.5 4.5 ${\ displaystyle {\ tfrac {2n-1} {2}}}$ > 0 1.4142 ... 2.4495 ... 3.4641 ... 4.4721 ... ${\ displaystyle {\ sqrt {n \ cdot (n-1)}}}$ > 0 1.3333 ... 2.4 3.4286 ... 4.4444 ... ${\ displaystyle {\ tfrac {2 \ cdot n \ cdot (n-1)} {n + n-1}}}$ > 0 > 1 > 2 > 3 > 4 > ${\ displaystyle n-1}$

According to Dean and Hill-Huntington (as with Adams), one vote is sufficient for the first seat. As the number of seats increases, the advantage of these two procedures for small parties diminishes, because the fractional shares approach the neutral value of 0.5 at Sainte-Laguë.

#### Excursus 2: Calculation method

For each divisor method there are at least five different algorithms that lead to the same result :

• the two-step process ;
• the maximum payment method ;
• the rank metric method ;
• the iterative voting method and
• the pairwise comparison procedure .

Maximum number procedure: The number of votes of the parties is divided by a series of divisors and the seats are assigned in the order of the highest resulting maximum number. The divisor series can easily be derived from the specified rounding rule. In the D'Hondt method, the divisor series is 1; 2; 3; 4; 5 etc., with the Sainte-Laguë method 0.5; 1.5; 2.5; 3.5; 4.5 etc., in the Adams method 0; 1; 2; 3; 4 etc. If the rounding limit is set to 1/3, the divisor series is 1/3; 1 1/3; 2 1/3; 3 1/3; 4 1/3, etc. With the Dean method, it is 0; 1 1/3; 2 2/5; 3 3/7; 4 4/9 etc. (formation of the harmonic mean), in the Hill-Huntington method 0; Root 2; Root 6; Root 12; Root 20 etc. (formation of the geometric mean). - If you set the rounding limit for the first seat (arbitrarily) to 0.8, for the second to 1.7, for the third to 2.5, for the fourth to 4, for the fifth to 4.9, and should all other seats are allocated to Sainte-Laguë, the divisor series is 0.8; 1.7; 2.5; 4; 4.9; 5.5; 6.5; 7.5 etc.

The divisor series can be multiplied by any factor without making a mathematical difference. Thus, in the Sainte-Laguë method, for. B. also the divisor series 1; 3; 5; 7; 9 etc. or 500; 1500; 2500; 3500; 4500 etc. can be used.

Interpretation of the maximum numbers: If the divisor series is not multiplied by any factor, the maximum number of the last seat awarded according to D'Hondt means that each party receives one seat for votes, but no remaining seat for fewer than the remaining votes. The maximum number of the last seat awarded after Sainte-Laguë means that each party receives one seat for votes and one remaining seat for at least the remaining votes. The maximum number of the last seat awarded according to Adams means that each party receives one seat for votes and one remaining seat for at least one remaining vote. ${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle 0 {,} 5x}$${\ displaystyle x}$${\ displaystyle x}$

Rank measure method: The ranking method is a trivial modification of the maximum number method. The rank measures are the reciprocal of the maximum numbers. Since these are very small numbers, it makes sense to multiply with the total number of votes. The rank metrics indicate the access rank for a seat. The seats are assigned in the order of the lowest ranking numbers.

#### Not self-imaging procedures

A procedure is self-mapping ( idempotent in terms of proportionality ) if, in the case of an election result which leads to an integer quota for each party, it allocates seats to each party according to its ideal claim (its quota). Example: If there are 100 seats to be allocated and party A has a quota of 50.0, party B a quota of 30.0 and party C a quota of 20.0, the distribution of seats in each self-mapping procedure is 50-30-20.

The self-mapping is - in addition to the constancy of the rounding limit - an independent criterion. A divisor method with constant or variable rounding limit can be self-mapping or not.

Non-self-mapping seat allocation procedures deviate more than inevitably from proportionality (are disproportionate ) and thus violate the demand for equality of choice. Nevertheless they do occur. There is no known mathematically formulated requirement that a non-self-mapping method could serve to fulfill.

#### Automatic procedures

In an automatic procedure, the total number of seats is not determined in advance, but depends on the number of voters or the turnout. Instead of a fixed total number of seats, there is a fixed number of votes, and the number of votes of the parties divided by this number of votes and rounded according to a fixed rounding rule results in the seat entitlement.

All divisor methods can be designed as an automatic method, you only need to specify the number to choose and apply the rounding rule. Their properties are the same as when used for the allocation of a fixed total number of seats and can even be illustrated better here.

The number of votes required for the first seats can easily be found in the table in excursus 1 , for example with a number of 1000 votes :

• According to D'Hondt, each party receives one seat for every 1000 votes, but no remaining seat for any remaining votes. That is, if the number of votes was 1999, only one seat will be allocated. It is easy to see that this procedure systematically disadvantages small parties and favors large parties.
• The counterpart to the D'Hondt process is the Adams process. Each party receives one seat for 1000 votes and one remaining seat if there is only one remaining vote. This means that if the number of votes is 1, one seat is already allocated, in this case one remaining seat. For 2 seats at least 1001 votes are required, etc.
• According to Sainte-Laguë, a remaining seat is allocated from 500 remaining votes. This means that at least 500 votes are required for the first seat, at least 1500 for the second, etc.
• According to Dean, only a single vote is needed for the first seat, at least 1334 for the second, at least 2400 for the third, at least 3429 for the fourth, at least 4445 for the fifth, and so on.
• According to Hill-Huntington, only one vote is required for the first seat, at least 1415 for the second, at least 2450 for the third, at least 3465 for the fourth, at least 4473 for the fifth, and so on.

Due to their inconsistency, quota procedures cannot be described as an automatic method: a party's entitlement to a seat depends on the balance of power between others.

### Biproportional procedure

The starting point for a distribution of seats according to the biproportional procedure is an electoral area divided into constituencies, with each constituency (e.g. based on its population) being entitled to a certain number of seats. The bi-proportional seat allocation process takes place in two steps.

#### Allotment

First of all, the seats are allocated to the so-called list groups within the entire electoral area (so-called upper allocation ). List groups are the merged lists of all constituencies with the same name; in fact, the list groups correspond to the political parties in the constituency. This is done with a divisor method, for example that according to Sainte-Laguë (commercial rounding). If the constituencies are entitled to a different number of seats and if each voter has as many votes as there are seats in the constituency concerned, the voting power must first be balanced: The number of votes in the list must be divided by the seat entitlement of the constituency and this results in the so-called number of voters. The upper allocation is based on the total number of voters of all lists in a list group. The sum of all voter numbers divided by the number of seats is called the voting key .

Example: Parliament with 15 seats. The electoral area is divided into constituencies I, II and III, whereby the constituencies are entitled to 4, 5 or 6 seats. There are three political parties (list groups) A, B and C. In the core of the following table, the list votes are given, also in italics the number of voters of each list, determined by dividing the list votes by the seat entitlement of the constituency. The column on the right shows the total number of voters for each list group; the sum of which is . With a voting key of, based on the totals of the voter numbers, the entitlements of 4, 5 and 6 seats for list groups A, B and C. ${\ displaystyle 3985 + 5500 + 6015 = 15500}$${\ displaystyle 15500/15 \ approx 1033}$

 WK I WW II WW III Total voter numbers Dialing key Seat claim (rounded) (4 seats) (5 seats) (6 seats) List group A 5100 9800 4500 1275 1960 750 3985 ./. 1033 4 seats List group B 6000 10,000 12000 1500 2000 2000 5500 ./. 1033 5 seats List group C 6300 10200 14400 1575 2040 2400 6015 ./. 1033 6 seats

#### Sub-allocation

In the second step, the seats assigned to the list groups are passed on to the individual lists in this group. For this purpose, the number of votes in a list is divided by the list group divisor of the relevant list group and the constituency divisor of the relevant constituency. The rounded quotient gives the seat entitlement of this list. The list group divisors and the constituency divisors are chosen so large that the following conditions are met if the procedure for all lists is as just described:

1. Each list group (political party) receives as many seats as were allocated to it in the overall allocation.
2. Each constituency receives as many seats as were previously allocated (for example based on the size of the population).

Example: In the table below, the seat entitlements of the lists are entered on the left as they result from the upper allocation. On the right are the list group divisors and below the constituency divisors. The table core names the list votes and - separated by a hyphen - the seat entitlement of the list. Reading example: List A in WK I made 5100 party votes. This value divided by the divisor of list group A (= 0.9) and the divisor of constituency I (= 4090) results in 1.26. Rounded off, this results in a claim of one seat for this list.

 WK I WW II WW III List group divisor (4 seats) (5 seats) (6 seats) List group A 5100 -1 9800 -2 4500 -1 0.9 (4 seats) List group B 6000 -1 10,000 -2 12000 -2 1 (5 seats) List group C 6300 -2 10200 -1 14400 -3 1.025 (6 seats) Constituency divisor 4090 6635 5150

It can be proven mathematically that the application of the procedure results in a clear distribution of seats. This means that there are no two different divisors that meet all the conditions but lead to different seat distributions.

The main advantage of the procedure is the maximum accuracy in mapping the composition of the parliament with regard to the list groups (political parties). Because with the upper allocation, all seats are allocated in a single step. The disadvantage is that within a list group and within a constituency there is no direct, but rather only a tendency towards proportionality between the number of votes and the right to a seat. This is because the same list group divisor exists for every list in a list group; however, the constituency divisors of the lists of this list group are different.

The process is based on an idea by Michel Balinski and was made operable by Friedrich Pukelsheim for the Canton of Zurich and is known there under the name Doppelter Pukelsheim . On February 12, 2006, a parliament was elected using this procedure for the first time - that of the City of Zurich. In 2007, the Parliament of the Canton of Zurich was elected using this procedure.

## Quality criteria for the selection of a seat allocation process

No seat allocation process can meet all of the criteria at the same time. There is therefore room for the political setting of priorities in the selection of the allocation procedure, as long as there are no constitutional restrictions. For example, the constitutional jurisdiction in Germany derives from the principle of equal electoral equality in proportional elections the equality of success of the votes, which should actually exclude the use of the large parties or their voters preferring D'Hondt procedure. This procedure was nevertheless declared constitutional because - according to the knowledge of the Federal Constitutional Court of 1963 - there is no more precise, practically feasible system that would lead to fairer results (BVerfGE 16, 130 <144>). The examination of the fulfillment and weighting of the following quality criteria prioritized by the Constitutional Court itself did not take place at that time and in many subsequent proceedings.

### Quota condition and consistency

Quota condition (also: quota criterion, ideal framework condition, ideal framework criterion)
A party's number of seats may only deviate by less than 1 from its ideal (its quota). Only quota procedures with a maximum of one remaining seat per party always meet the quota condition. It can violate all divisor methods.
House monotony (also: seat or mandate growth criterion)
An increase in the total number of seats to be distributed must never reduce the number of seats for a party and vice versa. See also Alabama Paradox as the Mandate Growth Paradox . Only divisor methods satisfy house monotony.
Vote monotony (also: voter growth criterion)

The dual requirement of house monotony and voice monotony is called consistency . A seat allocation process cannot be consistent and meet the quota condition at the same time ( Balinski and Young's rule of impossibility ). All divisor methods are consistent with the consequence that the Alabama paradox and the voter growth paradox cannot occur with these methods.

### Equality of choice

The election should give every voter the same opportunity to influence the composition of the body to be elected. This requires a proportional conversion of the votes into political mandates, i.e. an allocation of seats for each party as close as possible to their arithmetical ideal. A suitable measure for this is the representation value and also its reciprocal value, the success value.

The representation value (also: the representation weight) of a party for a certain seat allocation is the number of votes for this party divided by the number of seats that are allocated to this party. The representation value of a party is therefore the same for all seats of this party. It is a pure number , without a unit of measurement (in contrast to a value , so it is a representative number ). This ( fraction ) number shows very clearly how many voters stand behind each member of the party on average. - Equality of choice requires that the representation values ​​for all parties are as close as possible to each other (and close to their mean). - Comparing the mean representation value of a European election with that of a local election, on the other hand, makes little sense.

The success value (also: the success weight) of a vote for a party is the quotient of the number of seats of the elected party and the number of their votes, i.e. the reciprocal of the representation value . It is a measure of the weight of one vote in the composition of the body to be elected.

Since the ideal seat claims (quotas) of the parties have to be rounded to whole numbers (the allocation of fractions of seats is unlikely to be possible), differences inevitably arise between the parties in terms of the success value of their votes and consequently also the representation value of their representatives. There are several degrees of such differences. Of the following, you can only optimize one, not two at the same time.

Maximizing the smallest representation value
The representation value of the party with the lowest representation value should be maximized. This quality criterion is only fulfilled by the result of the D'Hondt method (regardless of the calculation method ). Given the election result, there is no other allocation of seats in which the vote / seat ratio of the party with the lowest vote / seat ratio would be higher than the vote / seat ratio of the party with the lowest vote / seat ratio at the time of allocation D'Hondt. The proof is immediately recognizable in D'Hondt's calculation method: The lowest maximum number for which a seat is allocated there is the lowest representative value; any other allocation would result in a smaller minimum substitute value. - This maximization (as defined above for both values) is synonymous with minimizing the maximum success value.
Minimizing the spread of the success values
The Sainte-Laguë method minimizes the standard deviation of the success values .
Minimizing the largest difference in relative success scores
The Hill-Huntington's method minimizes the greatest difference in the relative success values ​​and thereby maximizes at the same time the smallest difference in the relative representation values. Both goals are strictly positively correlated.
Minimizing the greatest difference in the representation values
The Dean method minimizes the greatest difference between two (absolute) representative values.
Minimizing the greatest representation value
The representation value of the party with the highest representation value should be minimized. This quality criterion is only fulfilled by the result of the Adams method (regardless of the calculation method ). Given the election result, there is no other allocation of seats in which the representation value of the party with the highest representation value would be lower than the representation value of the party with the highest representation value in the allocation to Adams. - This minimization (according to the above definition of both values) is synonymous with maximizing the minimum success value.

### Majority and minority condition

Majority condition (also: majority criterion, weak majority condition)
A party that collects at least 50% of the (eligible) votes should always receive at least 50% of the seats. Only divisor methods with rounding off meet the majority condition.
Strong majority condition
If, in addition, a party with an absolute majority of the (eligible) votes always receives the absolute majority of the seats, the total number of seats must be odd . Only then does the D'Hondt process meet this requirement. Example: There are 10 seats to be allocated. Party A: 501 votes, Party B 499 votes. Allocation of seats according to D'Hondt: Party A: 5 seats, Party B: 5 seats. Party A can gain an absolute majority of the votes, but does not receive an absolute majority of (at least) 6 seats.
Minority condition (also: minority criterion)
A party that collects a maximum of 50% of the (eligible) votes should receive a maximum of 50% of the seats. Only divisor methods with rounding up meet the minority condition.

## Not integer voting weights of the elected

The above seat allocation procedures are all based on the basic property that all MPs have the same voting weight; H. that each MP has exactly one vote in each vote. Alternatively, a method is also conceivable in which seats are equipped with different voting weights, e.g. B. each party could get a last fractional seat, which only has the voting share of the fractional part of the ideal claim of his party. Even with weighted seats, however, the number of seats must be calculated as whole numbers in order to determine how many MPs there are for each party. Here you also need a rule for the different weighting of the seats in parliament.

## Individual evidence

1. Ernst Gottfried Mahrenholz: All voters are the same, some remain the same. In: FAZ.net . May 18, 2011, accessed October 13, 2018 .
2. See Thomas Hare , in English
3. See balanced method and electoral law (Swedish) , accessed November 18, 2015
4. ^ The new Zurich allocation procedure for parliamentary elections ( memento from September 26, 2017 in the Internet Archive ), on math.uni-augsburg.de
5. Sainte-Laguë , on wahlrecht.de
6. Huntington / Hill - Example , on wahlrecht.de
7. Dean , on wahlrecht.de