Apportionment (politics)

Seat allocation methods are methods of proportional representation, as required in the proportional representation to convert votes into parliamentary seats.

After the end of the count the number of votes are translated into seats in parliament. The purpose of a seat allocation process. The aim of the method is the proportional representation of the parties. If (as usual) all Members have equal rights, to be given a seat number close to their ideal claim any party. The Ideal claim of one party is their votes multiplied by the total number of seats divided by the total number of votes. This quotient is not always an integer (with large total number of votes even rarely ). Therefore a seat allocation process as a rule, are calculated according to the votes of the shares of the parties seat numbers as integers.

Each rule created a different kind of error minimization. What can be regarded as the best, depends on the specified underlying quality criteria for the allocation of seats. Of course, in the interests of the rule be adopted majority have a greater importance for the determination of the right to vote as a mathematical arguments: " suffrage is also power law ".

2.3.1 Upper allocation
2.3.2 sub- allocation
2.3.3 Pros and Cons

3.1 ratio condition and consistency
3.2 Equality of choice
3.3 Majority and minority condition

Survey

For the allocation of seats, there are many defensible claims (including quality standards, requirements, criteria, quality criteria, conditions ). - Methods which do not serve to fulfill a requirement defined, are arbitrary or academic interest.

The following table has claims as columns and seat allocation method as rows. "Fulfilled" An entry, " maximum " or " minimum " applies to any election result. If a field is empty, so the statement is not always valid. - The Divisor in the table ( D' Hondt up of Adams ) are in order of decreasing rounding boundary, thus decreasing favoring large and increasing favoring smaller parties.

* The D' Hondt method always satisfies the weak majority criterion, with an odd total number of seats and the strong.

In the following chapters, the word " parties " for lists, list compounds, states, departments and similar competitors for seats; "Voices " is unavailable for distribution according to population figures for these. - Without loss of generality, the case remains unnoticed that two or more parties claim the same ideal (the same rate ), but this claim is fulfilled only for some of these parties. If this occurs, it may be loose in front of the seat allocation which of these parties which split voice will count towards the number of votes (for example, three parties a party nothing, a third voice and a 2/3 vote). Then all Ideal claims are different.

Types of seat allocation method

There are two sets of seat allocation methods, namely quota sampling and Divisor.

Ratio method

In these methods, so many seats are allocated first to any party as indicating downward rounded quota. The remaining seats are allocated according to a determined rule.

The Hare - Niemeyer method ( in Anglo-Saxon Hamiltonian method ) is the classical rate processes: The remaining seats are distributed according to the size of the fractional shares of quotas to the parties.

Alternatively, the Hare - Niemeyer method, the remaining seats eg Train to be distributed to train for a specific Divisor or all go to the strongest party.

Divisor

Here is a rounding rule is defined and a divisor so sought that the number of votes each party divided by this divisor to a whole number and rounded, the number of seats for this Party results. For this purpose one uses the nested intervals; 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: Calculate the divisor under consideration the sum of the rounded quotient.
Step 2: If this sum is less than or greater than the number of available seats, choose a slightly smaller or larger divisor and leads again from step 1.

(The designation quotient method is also possible, but may lead to confusion with quota sampling. )

Only rounding is neutral to the size of the parties. Method with a rounding boundary below the decimal fraction 0.5 systematically favor smaller parties ( see Panel 1). Method with a rounding boundary above the decimal fraction 0.5 systematically favor larger parties.

In addition to this two-step process, there are other calculation methods which lead to all Divisor at every election result to the same distribution of seats ( see Panel 2).

Divisor with a fixed rounding boundary

The classic Divisor with constant curvature limit are

The D' Hondt method ( in Anglo-Saxon: Jefferson procedure), Divisor with rounding to (especially for D' Hondt ) the Hagenbach -Bischoff method with always the same result as the D' Hondt method, but faster;

At the D' Hondt method of claim seat is always rounded to the nearest whole number, rounded up the Adams methods. At Sainte- Laguë method claims the seat are rounded. Among all the self-imaging process large parties are most favored by the D' Hondt method, small parties most strongly by the Adams method.

Divisor variable rounding boundary

The classic Divisor variable rounding boundary are

The Dean method ( Divisor with a harmonious curve ) and
The Hill - Huntington method ( Divisor with geometric rounding ).

To know whether a non-integral seat at Dean claim process is rounded up or down, is the harmonic mean between the rounded and to calculate the rounded seat claim. This agent is the rounding limit. The harmonic mean is the reciprocal of arithmetic mean of the reciprocals of characteristic values. The harmonic mean of 1 and 2, therefore, results from the reciprocal arithmetic mean of 1 and 1/2. The arithmetic mean of 1 and 1 /2 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 method of a party shall assign a seat even at only a single voice. The harmonic mean of 2 and 3 is 2.4. With increase in number of pairs of the harmonic mean approaches more and more the fractional part 5, but never reached it.

To know whether is rounded up or down the Hill - Huntington method, the geometric mean between the next larger and the next smaller integer seat claim is to be calculated. This forms the rounded border. The geometric mean is the n-th root of the product of the n characteristic values. Therefore, the geometric mean of 1 and 2 is approximately 1.4142. The geometric mean of 0 and 1 is 0, so the Hill - Huntington method of a Party shall, even at only a single voice to a seat. The geometric mean of 2 and 3 is approximately 2.4495. With increase in number of pairs of the geometric mean approaches more and more the fractional part 5, but never reached it.

Alternatively, the Dean and Hill - Huntington method, the rounding limits can also be chosen arbitrarily mixed. If you want the seat allocation, for example, according to Dean or Hill - Huntington calculate, but deviating prevent the fact that one party receives a seat already with a very small share of votes, shall be the rounding boundary between 0 and 1 (no seat or a seat), for example, to 0, 1; 0.5; 0.9 or fixed to 1. The higher the rounding limit, the more votes required for a party its first seat, etc. Similarly, one can define the rounding boundary between 1 and 2, 2 and 3, 3 and 4 seats arbitrary and vary. There is no known mathematically formulated requirement could serve to the fulfillment of such arbitrariness.

Digression 1: Effect of rounding border on favoring by party size

The following table should make favoring small parties significantly by lower rounding limits.

After Dean and Hill - Huntington enough (like Adams ) for the first seat, a citizen's vote. With the increase in the number of seats shrinks, the advantage of these two methods for small parties, because the fractional shares approach the neutral value 0.5 at Sainte- Laguë.

Digression 2: Calculation method

For each Divisor there are at least five different, leading to the same result algorithms:

The two-step process;
The highest average method;
The Rangmaßzahlverfahren;
Iterative selection procedures and the number
The pairwise comparison method.

Highest averages method: The number of votes the parties are divided by a divisor, and from this the seats awarded in the order of the largest resulting maximum numbers. The divisor can be easily derived from the specified rounding rule. When D' Hondt method is the divisor 1; 2; 3; 4; 5, etc., at the Sainte- Laguë method 0.5; 1.5; 2.5; 3.5; 4.5, etc., at the Adams method 0; 1; 2; 3; 4, etc. Taking the rounding limit to 1/3 determines the divisor is 1/3; 1 1/3; 2 1 /3; 3 1 /3; 4 1/ 3, etc. When Dean procedure it is 0; 1 1/3; 2 2 /5; 3 3/ 7; 4 4/ 9, etc. (formation of the harmonic mean ), the Hill - Huntington method 0; Square root of 2; Root 6; Root of 12; Root 20, etc. ( of geometric averaging ). - Taking the rounding threshold for the first seat (arbitrarily ) to 0.8, the second 1.7, the third 2.5, the fourth to 4, for the fifth fixed to 4.9, and are all other seats are allocated according to Sainte- Laguë, the divisor is 0.8; 1.7; 2.5; 4; 4.9; 5.5; 6.5; 7.5, etc.

The divisor can be multiplied by any factor without this makes a mathematical difference. Thus, the Sainte- Laguë method, for example, the divisor 1; 3; 5; 7; 9, etc., or 500; 1500; 2500; 3500; 4500, etc. are used.

Interpretation of quantitative limits: If the divisor is not multiplied by an arbitrary factor, means the maximum number x of the last seat allocated to D' Hondt that each party but gets no rest seat for x votes a seat for less than x remaining votes. The maximum number x of the last seat allocated to Sainte- Laguë means that each party still gets a rest seat for x votes for a seat and at least 0.5 x residual votes. The maximum number x of the last assigned seat to Adams means that each party still gets a rest seat for x votes a seat and rest for at least one vote.

Rangmaßzahlverfahren: The Rangmaßzahlverfahren is a trivial modification of the maximum number procedure. The Rangmaßzahlen are the reciprocals of the maximum numbers. Since these are very small numbers, it makes sense to be multiplied by the total number of votes. The Rangmaßzahlen specify the access level for a seat. The seats are allocated in order of smallest Rangmaßzahlen.

Non- self-imaging process

One method is selbstabbildend ( idempotent in terms of proportionality ) when at an election result, which leads each party to an integer ratio, each party seats in accordance with their ideal claim ( their quota ) allocates. For example, if 100 seats are up for grabs and Party A is a ratio of 50.0, Party B a ratio of 30.0 and Party C has a quota of 20.0, the distribution of seats is at every self-imaging method 50-30 - 20

The self-imaging - in addition to the constancy of the rounding limit - a separate criterion. A Divisor with constant or variable rounding limit may be selbstabbildend or not selbstabbildend.

Non- self-imaging ( disproportionate ) method with the principle of proportionality does not do justice, in violation of the equality of choice and therefore have no practical significance in elections. Also, no mathematically formulated requirement is known to the fulfillment of a non-self -imaging method could be used.

Non- self-imaging process with constant curvature limit

The Imperiali method preferred massively larger parties. Rounding Rule: rounding divisor minus 1 when using the maximum number of process: 2; 3; 4; 5; 6, etc. In the example, 1,000 votes were submitted. Thus, should accrue to Party A 500 , 300 to Party B and Party C 200 votes. The distribution of seats is 51 - 30-19 ( suitable divisor: 9.6). After rounding rounding rule minus 5 is the divisor 6; 7; 8; 9; 10, etc. In the example, the distribution of seats 53 results - 29-18 ( suitable divisor: 8.6 ). After rounding rounding rule minus 145 results in the allocation of seats 99-1 - 0 ( suitable divisor: 2.045 ). From the rounding rule rounding minus 148 Party A receives all 100 seats ( suitable divisor: 2.015 ). Conclusion: For any election result, a seat allocation method can be created, which the strongest party, and it was this allocates only with a single voice projection, the total number of available seats.

The counterpart to the Imperiali method is the method with the rounding rule rounding plus 1 It prefers massively smaller parties. Divisor when using the maximum number method: 0; 0; 1; 2; 3; 4, etc. This example gives the distribution of seats 49-30 - 21 results ( of appropriate divisor: 10.5 ). After rounding rule rounding up plus 5 is the divisor 0; 0; 0; 0; 0; 0; 1; 2; 3; 4, etc. This example gives the distribution of seats 47-31 - 22 results ( of appropriate divisor: 11.95). After rounding rounding rule plus 32 gives the distribution of seats 34 - 33-33 ( proper divisors: All 301-499 ). Conclusion: For any election result, a seat allocation method can be created, which distributes the total number of available seats with maximum uniformity on the parties, and are the differences in party strengths even as large. The only requirement is that each party gets at least one vote.

Other non- self-imaging process with constant curvature limit

Procedure with the rounding rule " rounding minus 1": preference of larger parties, the divisor when using the maximum number procedure: 1.5; 2.5; 3.5; 4.5; 5.5, etc.
Procedure with the rounding rule " rounding minus 2": Even stronger preference for larger parties, the divisor when using the maximum number procedure: 2.5; 3.5; 4.5; 5.5; 6.5, etc.
Procedure with the rounding rule " rounding plus 1": preference for smaller sized parties divisor when using the maximum number method: 0; 0.5; 1.5; 2.5; 3.5, etc.
Procedure with the rounding rule " rounding plus 2": Even stronger preference for smaller parties, the divisor when using the maximum number method: 0; 0; 0.5; 1.5; 2.5, etc.
Procedure with the rounding rule " rounding boundary at 4 decimal place with subtraction of 1": preference for larger parties, the divisor when using the maximum number procedure: 1.4; 2.4; 3.4; 4.4; 5.4, etc.
Procedure with the rounding rule " rounding boundary at fractional part 4 with addition of 1": preference for smaller parties, the divisor when using the maximum number method: 0; 0.4 1.4; 2.4; 3.4, etc.

Non- self-imaging method with variable rounding boundary

Procedure with the rounding rule " harmonious rounding minus 1": preference of larger parties, the divisor when using the maximum number procedure: 1 1/3; 2 2 /5; 3 3/ 7; 4 4/ 9; 5 5/ 11, etc.
Procedure with the rounding rule " harmonious rounding minus 2": Even stronger preference for larger parties, the divisor when using the maximum number procedure: 2 2 /5; 3 3/ 7; 4 4/ 9; 5 5/ 11; 6 6/ 13, etc.
Procedure with the rounding rule " harmonious rounding plus 1": preference for smaller sized parties divisor when using the maximum number method: 0; 0; 1 1/3; 2 2 /5; 3 3/ 7, etc.
Procedure with the rounding rule harmonious roundness plus 2: Even stronger preference for smaller parties, the divisor when using the maximum number method: 0; 0; 0; 1 1/3; 2 2/ 5, etc.
Procedure with the rounding rule " geometric rounding minus 1": preference of larger parties, the divisor when using the maximum number of process: root 2; Root 6; Root of 12; Root 20; Root of 30, etc.
Procedure with the rounding rule " geometric rounding minus 2": Even stronger preference for larger parties, the divisor when using the maximum number procedure: root 6; Root of 12; Root 20; Root of 30; Root of 42, etc.
Procedure with the rounding rule " geometric rounding plus 1": preference for smaller sized parties divisor when using the maximum number method: 0; 0; Square root of 2; Root 6; Root of 12, etc.
Procedure with the rounding rule " geometric rounding plus 2": Even stronger preference for smaller parties, the divisor when using the maximum number method: 0; 0; 0; Square root of 2; Root 6, etc.

Automatic method

In an automatic method, the total number of seats is not set beforehand, but depends on the number of voters or voter turnout. Instead of a fixed total number of seats there is a fixed dial number, and the number of votes the parties divided by this number and rounded choice after a specified rounding rule immediately yields the claim seat.

All Divisor can be designed as an automatic method, you need to define only the number and choice to take over the rounding rule. Their properties have the same ingredients as when used for the allocation of a fixed total number of seats and can be better illustrated here even.

The necessary number of votes for the first seats can be, for example, in an election of 1000 easily the table in Digression 1 below:

After D' Hondt each party for each 1,000 votes a seat, any balance remaining votes but no rest seat. That is, with a number of votes in 1999, only one seat is allocated. That this process systematically disadvantaged small parties and large favors, is easily recognizable.

The counterpart to the D' Hondt method forms the Adams methods. Each party receives 1,000 votes for a seat and with only a single residue voice still had a residual Stitz. That is, in a number of votes of a seat 1 is already allocated, in this case a rest seat. For two seats are at least 1001 votes needed etc.

After Sainte- Laguë a rest seat will be allocated from 500 surplus votes. That is, in the first seat at least 500 parts will be required for the second at least 1500, etc.

Dean only a single voice is needed for the second at least 1334, at least 2400 for the third, the fourth at least 3429, at least 4445 for the fifth to the first seat, etc.

By Hill Huntington only a single voice is needed for the second at least 1415, at least 2450 for the third, the fourth at least 3465, at least 4473 for the fifth to the first seat, etc.

Ratio method can not be described as an automatic method due to their inconsistency: The seat claim of a party depends on the balance of power between others.

Biproportionales method

Starting point for a distribution of seats after the biproportionalen process is a layed out in constituencies electoral area, each constituency (to its population supported for example) is entitled to a certain number of seats. The biproportionale seat allocation method is performed in two steps.

Upper allocation

First, the seats are on the so-called list groups distributed within the whole electoral area (so-called upper -allotment). List groups are the contracted lists all constituencies with the same designation; in fact the list of groups that is, the political parties meet in the constituency. This is done with a Divisor, for example, that according to Sainte- Laguë ( rounding ). Have the constituencies entitled to a different number of seats and has each voter as many votes as there are seats to be awarded in the relevant constituency, so first the voting power must be balanced: the list votes to be divided by the seat of claim of the constituency and yields the so-called voter number. The overhead allocation is a list of all lists group due to the added number of voters. The sum of all electoral numbers, divided by the number of seats is called dial key.

Example: parliament with 15 seats. The Wahlgebebiet is divided into electoral districts I, II and III, where the constituencies are entitled to 4, 5 or 6 seats. It occurred three political parties ( list groups) A, B and C on. In essence, the following table shows the list votes are given, also in italics voters numbers each list, determined by dividing the list votes by the seat of claim of the constituency. The column to the right is from the total of the electoral numbers each list group; whose sum is. With a choice of key result based on the totals of the number of voters who claims 4, 5 and 6 seats for the list of groups A, B and C.

Under allocation

In the second step the other list groups allocated seats are passed on to the individual lists that group. For this purpose the votes of a list are divided by the Listengruppendivisor the relevant list group and by the Wahlkreisdivisor of the relevant constituency. The rounded quotient gives the seat claim this list. The Listengruppendivisoren and Wahlkreisdivisoren be chosen so large that the following conditions are fulfilled if for all lists like The procedure just described:

Example: In the following table, the seat of the claims lists are left registered, how they arise from the upper allotment. Law have added the Listengruppendivisoren and down the Wahlkreisdivisoren. The table lists the core list cast and - sold with a hyphen - the seat of the claim list. Reading example: The List A WW I made 5100 party votes. This value divided by the divisor the list group A (= 0.9) and the divisor of the constituency I (= 4090 ) yields 1.26. Rounded result is a claim from a seat in this list.

It can be shown mathematically that the application of the method results in an unambiguous allocation of seats. This means that there are no two different divisors that meet all the conditions, but lead to different distributions seat.

Pros and Cons

The main advantage of the method is the maximum figure accuracy in the composition of the Parliament on the list of groups ( political parties). For in the upper allotment all seats are distributed in a single step. The disadvantage is that there is no direct, but only a trend towards proportionality between votes and seats claim exists within a list group and within a constituency. For every list of a list group, although there is the same Listengruppendivisor; however, the Wahlkreisdivisoren the lists that list group are different.

The method is based on an idea by Michel Balinski and was made by Friedrich Pukelsheim for the Canton of Zurich operable and is known there under the name Double Pukelsheim. On 12 February 2006, Parliament was first elected to this method - that of the city of Zurich. In 2007, the Parliament of the Canton of Zurich was chosen by this method.

Quality criteria for selection of a seat allocation process

No seat allocation method can meet all the criteria simultaneously. Therefore, there is room for policy prioritization in the selection of the allocation process, as long as no constitutional limitations preclude. Diverting about the constitutional jurisdiction in Germany from the principle of electoral equality in elections, the success ratio value equality of votes on what the preferable use of the large parties and their voters D' Hondt method should actually exclude. This procedure was nevertheless declared constitutional, since - according to the knowledge available to the Federal Constitutional Court of 1963 - a more accurate practically feasible system that would lead to fairer outcomes are not (BVerfGE 16, 130 < 144 > ). The testing of the performance and weightings of the following and even prioritized by the Constitutional Court of quality criteria at the time and in many subsequent process did not take place. A to pending before the Federal Constitutional Court Verification complaint to the federal election of 2002 (May 2008) not yet decided.

Rate condition and consistency

Rate condition ( also: rate criterion, Ideal frame condition, ideal frame criterion): The number of seats a party is allowed to differ by less than 1 of their ideal claim ( their quota ). Only rate method with at most one remaining seat per party always meet the quota condition. All Divisor can hurt them.

House monotony (also sitting 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 a mandate growth paradox. Only Divisor meet the house monotony.

Votes monotony (also: voters growth criterion): An increase in votes of one party must never lead to mandate shifts between two other parties. See also increase voter paradox. Only Divisor meet the votes monotony.

The double demand of domestic monotony and monotony voices called consistency. A seat allocation method can not be both consistent and the rate condition is met (impossibility set of Balinski and Young). All Divisor is consistent with the result that Alabama paradox and voters growth paradox in these processes can not occur.

Electoral equality

The choice should give the same opportunity to influence the composition of the Board to be elected each voter. This requires a proportional possible conversion of votes into political mandates, ie a seat allocation for each party as close to its theoretical ideal claim. A suitable measure of this is the representation of value as well as its inverse, the success value.

The representative value (also: the representative weight) of a party in a particular seat allocation is the number of votes for this party divided by the number of seats to be allocated to that party. The representative value of a party is thus the same for all the seats of that party. He is a pure number, without units of measurement ( as opposed to a value, so it is a representative number). This ( Break ) number indicates very clearly how many voters are at the center behind any deputy of the party. - Equality of choice requires that the representation values are for all parties as close as possible to each other ( and near its mean value). - With which to compare the average representative value of a European election of a municipal election, however, has little meaning.

The success value (also: the success of weight) of voters vote for a party is the quotient of the number of seats of the elected party and the number of their votes, so the reciprocal of the value representation. It is a measure of the weight of a citizen's vote in the composition of the elected body.

Since the ideal seat entitlements ( quotas ) of the parties must be rounded to whole numbers ( the awarding of seat fractions is unlikely to be realized ), arise between the parties inevitably differences in the success value of their votes and, consequently, the representative value of their deputies. There are several dimensions of such differences. From the following here you can always optimize only one, not two at the same time.

Maximizing the smallest value representation: The representative value of the party with the lowest representation of value is to be maximized. This criterion is only the result of the D' Hondt method met ( regardless of the method of calculation ). Given the election result there is no other seat allocation, in which the votes - seats relationship of the party would be higher with the lowest votes - seat ratio than the votes - seats relationship of the party with the lowest votes - seat ratio in the allocation by D' Hondt. The proof is immediately recognizable in the calculation method of D' Hondt: The lowest maximum number for which there is a seat is assigned, is the smallest representative value; any other allocation would result in a smaller minimum value representation. - This maximization is ( according to the above definition of both values ) is equivalent to minimizing the maximum success value.

Minimize the spread of success values: The Sainte- Laguë method minimizes the standard deviation of the performance figures.

Minimize the largest difference in the relative success of values: The Hill - Huntington method minimizes the biggest difference in the relative success values and at the same time maximizes so that the smallest difference in the relative representation of values . Both goals are correlated strictly positive.

Minimize the largest difference of the representative values: The Dean method minimizes the biggest difference between the two ( absolute ) representation values.

Minimize the largest value representation: The representative value of the party with the highest representation of value is to be minimized. This criterion is only the result of the Adams method met ( regardless of the method of calculation ). Given the election result there is no other seat allocation, in which the representative value of the party would be lower with the highest representation value as the representative value of the party with the highest representation of value in the allocation to Adams. - This minimization is ( according to the above definition of both values ) is equivalent to maximizing the minimum value of success.

Majority and minority condition

Majority condition ( also: majority criterion, weak majority condition): A party that unites at least 50 % of the ( allocation legitimate ) votes, should always receive at least 50 % of the seats. Only Divisor with rounding meet the majority condition.

Strong majority Condition: target beyond an absolute majority of ( allocation legitimate ) votes always get a party the absolute majority of seats, with the seat number must be odd. Only then will the D' Hondt method satisfies this condition. Example: There are awarded 10 seats. Party A: 501 votes, Party B 499 votes. Allocation of seats according to D' Hondt: Party A: 5 seats, Party B: 5 seats. Although party A can unite the absolute majority of votes, but does not receive an absolute majority of (at least) 6 seats.

Minority condition ( also: minority criterion): A party that unites more than 50 % of the ( allocation legitimate ) votes is to receive more than 50 % of the seats. Only Divisor with rounding meet the minority condition.

Non-integer voting weights of Elected

The above seat allocation methods are all based on the fundamental property that all parliamentary seats have the same voting weight, ie that every Member has exactly one vote at each voting. Alternatively, a method is also conceivable, are equipped with the parliamentary seats with different voting weights, eg could get a last fractional seat which only has the vocal portion of the rear portion of the comma Ideal claim his party each party. Even with Weighted seats seat numbers but must be calculated as whole numbers to determine how many delegates allocated to each party. Here you need then also need a rule for the different weighting of the seats in parliament.

Voting system Hagenbach-Bischoff system Biproportional apportionment

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