Voting Districts

One of the problems with democracy is protecting the rights of minorities while obeying the will of the majority. One means to this end is providing minority representation in repesentative bodies such as town councils or the United States Congress. We shall consider two approaches to this problem and how they can enhance or prevent minority representation. These methods are representation districts and at large representatives. We will mention cumulative voting, which is a variation on at-large voting, and mention proportional voting, but develop tha tin the context of proportional representation as in the U. S. House of Representatives. All apply to elections where multiple candidates are elected. This web page considers only voting districts.

It will be useful to define notation for this problem:
n = the number of voters
k = the number of candidates which will be elected (which is the number of voting districts if voting districts are used)
v = the number of votes which each voter casts (for at-large voting, v=1 for voting districts).
x = the number of voters in your party (hence n-x is the number of candidates in the oppositon party, we are assuming two parties or factions).
w = the number of candidates which your party elects (hence k-w is the number of candidates which the oppositon elects, we are assuming two parties or factions).

Voting districts provide voters better access to their legislators because legislators have fewer constituents (e.g., Senators serve everyone in a state, but Congressmen serve only the persons in their district). However, minorities may or may not be able to elect a representative depending on how the population is distributed across voting districts. Gerrymandering refers to the process of drawing district boundaries in order to enhance the power of a constituency. It usually refers to the party in power enhancing its power, but can also refer to providing power to a minority, perhaps under court order. Even with the constraint that voting districts must be connected, it is essentially possible to draw districts which have any number of voters from each party in a district (subject to the constraints that districts have equal size, and the total number of voters for each party), so we shall assume that the party drawing district lines has complete freedom to specify which voters go in which districts.

Consider a state with 8181 voters in 9 voting districts so that there are 909 voters per district. Assume 5454 of the voters support party A and 2727 support party B. If each district has 606 supporters of party A and 303 supporters of party B, then all 9 districts will elect a representative for party A. If the the populaton is distributed so that 6 districts have 909 voters who support party A and the other 3 districts have 909 voters who support party B, then six representatives for party A and three representatives for party B will be elected. Hence it may be necessary to adjust district boundaries to provide elected representatives for party B.

But district boundaries can be drawn to give not just due, but undue power to minorities, for example if five of the districts had 404 voters for party A and 505 voters for party B, two had 859 for party A and 50 for party B and two had 858 for party A and 51 for party B, then although having only one-third of the voters, party B would have a majority (5 out of 9) of the elected representatives. Thus by adjusting district boundaries, a party in power could remain in power after it no longer had a majority of the voters.

We shall gain insight into this problem asking eight questions:

For example: Homework: With 4800 voters and 12 districts of 400 voters:

March 2016

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