The Adams method (also: divisor method with rounding up ) is a method of proportional representation ( seat allocation method ), as it is e.g. B. in elections with the principle of proportional representation (see proportional representation ) is required to convert votes into members of parliament.

The American politician John Quincy Adams proposed the procedure named after him in 1832 as a method for a population-proportional distribution of the seats in the House of Representatives among the states. However, it was never used for this purpose. In France, the procedure is used to distribute the 577 seats in the National Assembly among the 100 departments , but with the proviso that each department has at least 2 seats.

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The calculation of a distribution of seats according to the Adams method is explained below using the example of a proportional representation.

Seats to be allocated: 50

The votes of the parties are divided by a suitable divisor. The divisor is not necessarily an integer. It must be determined empirically (by trial and error). The quotient of votes cast and seats to be allocated (S / M) can be used as a benchmark for the divisor, in the example 20. The quotients resulting from the division are rounded up to the next whole number. If these integers add up to 50, then the seat allocation calculation is correct. Each party receives seats equal to the whole number calculated for them. If the sum of these whole numbers is more or less than 50, the divisor is unsuitable and must be increased or decreased - until exactly 50 seats are distributed. Because of the rounding up rule, the divisor S / M is usually too small, because it leads to the allocation of more than M seats, but never too large!

Division of the number of votes of the parties by the divisor 20:

Party A: 22.5; Party B: 17.5; Party C: 9.95; Party D: 0.05

The non-whole numbers obtained are rounded up to the next whole number:

Party A: 23; Party B: 18; Party C: 10; Party D: 1

The sum of all mandates is 52. The divisor 20 is too small. Try a larger divisor.

Division of the number of votes of the parties by the divisor 21:

Party A: 21.43; Party B: 16.67; Party C: 9.48; Party D: 0.05

After rounding up, this results in the following allocation of seats:

Party A: 22 seats; Party B: 17 seats; Party C: 10 seats; Party D: 1 seat

The total of all mandates is now 50. The divisor 21 is therefore a suitable divisor.

One can show that suitable divisors for this example are all numbers between . All divisors in this area result in the same distribution of seats. ${\ displaystyle 20.59 \ approx {\ frac {350} {17}} \ leq Divisor <{\ frac {450} {21}} \ approx 21.43}$

Note: The rounding up rule means that each party receives a seat with just one vote, provided that the total number of seats is not smaller than the number of parties with at least one vote.

### Maximum payment method

Alternatively, as with any other divisor method, the Adams seat allocation can also be calculated on the basis of the corresponding maximum number method. The number of votes of the parties is divided by a series of divisors . The resulting quotients are called maximum numbers . The seats are distributed to the parties in the order of the highest maximum number. This algorithm is more complex than the one described above. The advantage is that in the event of an enlargement or reduction of the committee to be elected by z. B. 1 seat can see at first glance which party would receive an additional seat or would have to do without a seat.

The divisor series for the Adams maximum number method is:

0; 1; 2; 3; 4; 5; 6; 7; 8th; 9 etc.

The divisors result from the rounding rule of the seat allocation procedure or the rounding limits established by them between 2 consecutive seat claims. In general, if a number to be rounded is above the rounding limit, it must be rounded up, otherwise it must be rounded down. Because of the rounding up rule, the rounding limit for each seat entitlement is a whole number. From the proportional representation example, it becomes clear that the rounding limit for the first seat is 0, for the second 1, for the third 2, and so on. This results in the above integer series of divisors. The integer of the divisors follows from the integer of the rounding limits. The divisor for the nth seat is also the rounding limit between the nth and (n + 1) th seat according to the algorithm shown in the proportional representation example.

Division by zero: Although division by the number zero is not mathematically possible, the quotient can still be regarded as a number of the size "infinite". The first maximum number of each party with at least one vote is therefore "infinite", so that no party - no matter how large - is assigned a second seat before all others with at least one vote have received their first seat.