Parliamentary Allocation

Suppose we run an election. We want to allocate seats in the parliament proportionally to the number of votes each party received.


  • $n$ - number of voters that participate in the elections
  • $N$ - number of classes (parties) that are being elected
  • $P_1, ..., P_k$ - are partitions of the population, \sum_P = N
  • size of $P_i$ is $p_i$
  • goal: to select $S$ representatives, $S$ - the total number of seats that are allocated

A quota $q_i$ of party $i$ is the number of sets the party receives after election:

  • $q_i = S \cdot \cfrac{P_i}{n}$
  • but it must be an integer! Cannot divide one seat between two parties

This is important not only for seats allocation, but for allocating in general.

For example,

  • suppose we have some area that suffers from fire
  • we have only 10 medical units
  • how to allocate them between the regions in that area?
  • area-medicalunits.png
  • can allocate as follows: the number of voters $p_i$ - density of a region, $n$ - all people of the area
  • $S$ is the number of medical units
  • $q_i$ the number of medical units sent to region $i$
  • $q_i = S \cdot \cfrac{P_i}{n}$ - again: it has to be an integer!