Math for the People

6.

When using grid examples, it is often easy for populations to be equal in every district, but on real maps, some deviation is allowed. The population deviation of a district plan is the percentage difference between the most populous and least populous districts, relative to the ideal population (the population if all districts were equal).

For example, suppose a state with 1,000,000 people was divided into 10 districts. The most populous district contains 102,300 people and the least populous district contains 98,400 people. This means that the ideal population is 100,000 and the population deviation is 3.9%.

Courts have dictated that the maximum allowed population deviation is 0.75% for congressional plans and 10% for state legislative plans.

1. Suppose a state is planning to create new congressional districts and proposes a plan in which their 3 districts have the following populations:

   1. District #1: 232,934

   2. District #2: 231,956

   3. District #3: 231,158

   Calculate the ideal population and the population deviation for this plan. Would this proposal be allowed to move forward based on population alone?

2. Suppose a state with a population of 2,300,000 is set to create new district plans for 8 congressional districts and 40 state legislative districts

   1. The congressional proposal's least populous district has a population of 286,790. What is the maximum population allowed for another district?

   2. The state legislative proposal's most populous district has a population of 59,440. What is the minimum population allowed for another district?

10.

For each statement, explain why the statement is not an appropriate interpretation of compactness scores.

1. The average Polsby-Popper score of districts in Maryland is 0.115, thus the districts are not compact.

2. The Reock score of Massachusetts' 6th district is 0.45 while the Reock score of its 5th district is 0.23, thus the 6th district is more compact.

3. The average Polsby-Popper score of New Jersey's congressional districts is 0.20 while the average Polsby-Popper score of Pennsylvania's congressional districts is 0.33, thus Pennsylvania is more compact.

11. Efficiency Gap.

1. Calculate vote totals for each district.

2. Calculate wasted vote totals for each district.

3. Compute the efficiency gap.

4. Interpret the efficiency gap answer. Which party holds the advantage in this plan? How significant is their advantage?

12.

There is an alternative “shortcut” formula to calculate the efficiency gap in a districting plan:

Efficiency Gap = (% of Districts won \(-\) 50%) \(-\) 2 * (% of Votes \(-\) 50%)

For example, in the district plan in Example 1.6.1 (given in Table 4.6.2), Heart wins 3/5 = 60% of districts and has 26/45 = 57.8% of voters.

Using the formula: Efficiency Gap = (60% \(-\) 50%) \(-\) 2 * (57.8% \(-\) 50%) = \(-5.6\)%. Since we used values for Heart, the negative number here represents that the Club party holds the advantage in this map.

Note: This answer is slightly different from the original calculation due to rounding error and the fact that our grid districts needed at least 5/9 = 55.6% of voters to win (rather than 50% +1 vote).

1. Use this alternative formula to calculate and interpret the efficiency gap in Exercise 4.11.11. Are your answers closeWindow()?

2. Suppose a party has 60% of votes on an entire map. The “most fair” outcome according to the efficiency gap formula is that this party wins what percentage of districts? What does your answer tell you about the efficiency gap formula and proportionality?

13. Efficiency Gap and Mean-Median.

14.

15.

16.

Suppose we collected data from all precincts from the scenarios in the previous exercise, then created a scatterplot and found the line of best fit.

17.

Use the MGGG 5x5 grid interactive tool (opens new tab)² to answer the following questions.

1. Place 9 Hearts in the center of the grid and 16 Clubs around the outside. Given that there are 5 districts of 5 voters each, do you think it is more likely that Hearts wins 0 districts or 3 districts? Test this by creating an example of each scenario and comparing the bar graphs of results.

2. Now place 9 Hearts on the border of the grid and use Clubs for the remaining 16 positions. Does this position of Hearts make it easier or harder for them to win districts compared to clustering them in the center (in the previous question)? Use the bar graphs to support your answer.

18.

Use the MGGG 7x7 grid interactive tool (opens new tab)³ to answer the following questions.

Even though the 7x7 grid is only roughly double the amount of spaces as a 5x5 grid, the number of possible district plans vastly increases. While it might not be practical to consider all possibilities, we can use the grid simulation to take a sample of possible district plans and obtain a distribution of results (similar to the previous exercise) among the sample.

1. Create a valid district plan (7 districts of 7 voters each) with exactly 22 Hearts in which Hearts wins 4 out of 7 districts. Is your plan an outlier for the distribution of voters that you chose? Use the “sample with MCMC” button to create a bar graph of results. What happens when you change nothing on your district plan but click the sample button again?

2. Change your district plan by changing 4 Hearts to Clubs which do not change the outcome of any district. You should now have exactly 18 Hearts and a plan in which Hearts wins 4 out of 7 districts. Is your plan an outlier for the distribution of voters that you chose?

3. Create multiple grids of exactly 18 Hearts but arrange the Hearts in different ways. For example, you might arrange them together in one large group, in two groups in opposite corners, or randomly scattered in the grid. For each arrangement, create the distribution of districts won by Hearts. Are there certain arrangements of the 18 Hearts that tend to result in winning more districts?