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 a given total population and number of districts, what is the smallest number of voters of your party that could be provided a representative? (i.e., if you drew district lies, how may voters would you need to win a seat?).
For a given total population and number of districts, how many voters must a party have to be assured of at least one representative? (i.e., if the opposition drew district lines, how many voters would you need to win a seat?)
For a given total population and number of districts, what is the smallest number of voters of a party that could be provided a majority of the rerpresentatives? (i.e., if your party drew district lines, how many voters would you need to win a majority of the representatives?)
For a given total population and number of districts, how many voters must a party have to be assured of a majority of the representatives? (i.e., if the opposition drew the district lines, how many voters would you need to win a majority of the seats?)
For a given total population and number of districts, what is the smallest number of voters of a party that could win all of the rerpresentatives? (i.e., if your party drew district lines, how many voters would you need to win all of the representatives?)
For a given total population and number of districts, how many voters must a party have to be assured of winning all of the representatives? (i.e., if the opposition drew the district lines, how many voters would you need to win all of the seats?)
For a given total population size, number of districts, and number of party supporters, what is the largest number of representatives a party could win? (i.e., if you drew the district lines.)
For a given total population size, number of districts, and number of party supporters, what is the smallest number of representatives a party could have? (i.e., if they drew the district lines.)

For example:

For 10,000 voters in 10 districts of 1000 each, what is the smallest number of voters in a party that could be provided a representative?
If you put all your voters i one district, 501 voters would provide a majority for that district and you would have one representative.
For 10,000 voters in 10 districts of 1000 each, how many voters must a party have to be assured of at least one representative?
If the opposition placed 500 of your voters in each district (and 500 of their voters in each district), then each district would be tied. There are many ways of dealing with ties, one of which is to flip a coin. If coin flips were employed none of the districts with 500 for each party would be guaranteed to go for a specified party. (Different interpretations of how to deal with ties and rounding off numbers to integers may change answers by one or two individuals.) Therefore 5001 voters are necessary to win at least one seat, because one district would need to have at least 501 voters of your party.
For 10,000 voters in 10 districts of 1000 each, what is the smallest number of voters in a party that could be provided a majority (6) of the representatives?
A majority is six out of 10; if you filled the districts, you could put 501 of your voters in each of six districts, hence a total of 6 × 501 = 3006 voters can win 6 districts.
For 10,000 voters in 10 districts of 1000 each, how many voters must a party have to be assured of a majority (6) of the representatives?
The strategy of the opposition is to underfill districts with your voters, i.e., they would put up to 500 voters in a district before they had to concede a district. Thus the opposition would place 5000 of your voters in in 10 districts with 500 each and not lose a district. If you had 5001 voters, you would need to win one district as noted above, but the opposition would fill that district before they conceded another district, hence 5500 voters would still only win one district. In summary, 5000 votes would win no districts for you, an additional 2500 voters paced in fve districts would win you 5 districts with 1000 of your voters and leave 5 districts in undetermined ties. You would ned one more voter for a total of 7501 in order to win 6 districts.
For 10,000 voters in 10 districts of 1000 each, what is the smallest number of voters in your party that could win all of the representatives?
If you put 501 of your voters in each district, you would win all, hence 501 × 10 = 501 could win all.
For 10,000 voters in 10 districts of 1000 each, how many voters must a party have to be assured of winning all of the representatives?
If the opposition were filling the districts, they would put 500 of your voter in each district before conceding a district, then fill each district before conceditin another, hence 5000 (win none) + 4500 (fill 9) + (1 lose the last district) would win all 10 districts.
The stategy is to minimally win districts if you are drawing district lines. For the opposition, they will maximally underfill (500 voters in the above case) all districts, then completely fill a district before conceding another.
For 10,000 voters in 10 districts of 1000 each, what is the fewest number of representatives a party with 7500 supporters could have?
Put 500 members of that party in as many districts as possible. If there were 5000 or fewer party supporters, they could have no representatives as noted above. Then fill the districts to 1000 if they must have a majority in order to have as few with a majority as possible. Giving five districts an additional 500 will use up 5000 voters, so that party may have only 5 representatives.
For 10,000 voters in 10 districts of 1000 each, what is the largest number of representatives a party with 2500 supporters could have?
If they had 5010 voters, they coud be located with 501 in each district, and they would have all 10 representatives. With 2505, they could control 5 districts with 501 voters each. But because we are not counting ties, they need 2004 voters to win 4 districts, and 496 will not win another district, so they can only win 4 districts.

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

For 4800 voters in 12 districts of 400 each, what is the smallest population of a party that could be provided a representative? (i.e., if they drew district lines)
For 4800 voters in 12 districts of 400 each, how large must a party be to be assured of at least one representative? (i.e., if the opposition drew district lines)
For 4800 voters in 12 districts of 400 each, what is the smallest number of voters a party could have and be provided a majority of the representatives? (i.e., if they drew district lines)
For 4800 voters in 12 districts of 400 each, how many voters must a party have to be assured of a majority of the representatives? (i.e., if the opposition drew district lines)
For 4800 voters in 12 districts of 400 each, what is the smallest number of voters a party could have and win all of the representatives? (i.e., if they drew district lines)
For 4800 voters in 12 districts of 400 each, how many voters must a party have to be assured of winning all of the representatives? (i.e., if the opposition drew district lines)
For 4800 voters in 12 districts of 400 each, what is the fewest number of representatives a party with 3000 supporters could have?
For 4800 voters in 12 districts of 400 each, what is the largest number of representatives a party with 2000 supporters could have?

March 2016

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