Seat Allocation Error and Degree of Negation

The principle of proportional representation asserts that the share of the total seats of a party that is entitled to receive a seat is equal to its share of the total votes of all the parties that are entitled to receive a seat.

The ideal number of seats that a party is entitled to receive is equal to the product of its share of the total votes of all the parties that are entitled to receive a seat and the total number of seats available for the party-list system.

The seat allocation error of a seat allocation formula is equal to the ideal number of seats determined by the principle of proportional representation and the actual number of seats allocated by the seat allocation formula.

For example, if a party obtains 10% of the total votes and the total number of seats available for the party-list system is 55. Then by the principle of proportional representation the ideal number of seats that the party is entitled to receive is 5.5. Suppose that a seat allocation formula allocates 3 seats to the party. Then the seat allocation error is 5.5 - 3 = 2.5.

The degree of negation of a seat allocation formula on a given party is equal to the integer part of the absolute value of the seat allocation error. If the degree of negation is equal to zero then the formula affirms the principle of proportional representation. Otherwise, if the degree of negation is at least one seat, then the formula negates the principle of proportional representation on the given party.

In the example above, the degree of negation of the given formula is 2 seats. This means that the given formula negates the principle of proportional representation on the said party by at least two seats.

Proportional representation tends to elect more representatives from small parties to a legislature than district-based representation would. This is because it enables scattered groups of people to elect at least one representative, even if their numbers in any given electoral district would be insufficient to elect one. This may give small parties power out of proportion to their size if they are needed to form a coalition. (The influence of religious parties in the Israeli legislature, the Knesset, has been used as an example of this effect.)

See ["The Negation of Party-List System Act on the Principle of Proportional Representation"] [http://www.math.admu.edu.ph/~fpmuga/partylist.htm] which appeared as a chapter in Oligarchic Politics: Elections and the Party-List System in the Philippines, CenPEG Books, May 2007.