Gerrymandering and the Political Process week 5

Squaretopia: As compact as possible

Mattie Di Giovanni

 

Redistricting Commission Mission. Squaretopia’s latest census results have been tallied, and now it’s time to redistrict the state. You’ve been selected to be on a commission to propose a district map to the state legislature for the next elections. The members of the commission are given a very detailed map of the state geography, the exact location of its residents, and prediction of each resident’s preferred party candidate (with 100% certainty). Squaretopia has been apportioned 10 federal legislators for its 90 electorates. The Squaretopia Constitution requires that districts are single-member, contain exactly the same number of individuals (i.e., squares), are contiguous, and should be as compact as possible.

PART I

  1.  First, create a district map according to the state constitution, prioritizing compactness. Record the number of districts that the Gray Party will win in the next election in the column labeled Compact in the table below.  
  2.  Now, your instructor will give you an identity (Gray Party, Not Gray Party, an unaffiliated individual that strongly believes in free and fair elections). Discuss with the other like-minded individuals in the class what your strategy might be in creating an optimal district map for your group (that still adheres to the state constitution). On your own, create a district map of Squaretopia that tries to achieve your goal. (Use a second Squaretopia gird; don’t erase what you did in Step 1.) Record the number of districts predicted to be won by the Gray Party in the appropriate column in the table below.
  3.  Return to your group and determine who made the best map. Discuss what elements your group considered and prioritized when deciding on the best map. I think the proportional map is the best because it best carries out what the purpose of proportional representation. Also it’s very accurate in still being compact. It just allows for the most representation on both sides.

Continue to the next portion of this Investigation before filling in the rest of the table.

Compact Proportional Gerrymandered for Gray Gerrymandered against Gray
Number of districts Gray wins 7 6 9

 

PART II   Measuring Fairness with Compactness Metrics.

  1.  In this question, you will measure the compactness of the districts in your COMPACT district map using each of the four measures: Skew, Isoperimetric, Square Reock, and Convex Hull.  
  2. a) For the Skew measure:

  (i) Calculate the measure for each of the districts. Record your answer to 2 decimal places.

Districts: 1  2 3    4 5 6 7  8 9 10

Skew:      1  1 1  .6 1 1  .6 .33 .75 .33

 

 (ii) Calculate the average of the measures over all the 10 districts to 2 decimal places.  .761

 (iii) Determine the measure’s worst score (i.e., the minimum value of the 10 measures) .33

 (iv) Fill in the table below with your calculated averages and worst scores to 2 decimal places

 (v) Transfer the information above to this shared Google Sheet to compile and examine the results from  

    the entire class.

  1. b) Repeat (a) for the Isoperimetric measure, the Square Reock measure, and the Convex Hull measures. (Note that there are separate tabs in the Google Sheet for each measure.)
  2. c) Calculate the total perimeter of your COMPACT district map. Record in the table below and transfer the information to the shared Google Sheet.

 

  1. Repeat question 1 for the other district map you created in class  (Proportional, Gerrymandered for Gray, or Gerrymandered against Gray).

 

FORMULAS: Recall the calculations for the measures discussed in the lecture:

  • Skew measure: WL, where W is the district’s shorter dimension (length or width) and L is its longer one.
  • Isoperimetric measure: 16AP2, where A is the district’s area and P is its perimeter.
  • Square Reock measure: AS, where A is the district’s area and S is the area of the smallest square containing the district.
  • Convex Hull measure: AH, where A is the district’s area and H is the area of its convex hull.
  • Total perimeter: P1+P2+ +P10, where P1is the perimeter of the first district, P2is the perimeter of the second district, and so on. (Note, no average or worst, just the sum!)

 

District Maps
Compact Proportional Gerrymander for Gray Gerrymander against Gray
avg worst avg worst avg worst avg worst
Skew .761 .33 .59 .28 .58 .33
Isoperimetric .741 .56 .682 .44 .68 .44
Square Reock .68 .25 .46 .18 .45 .25
Convex Hull .97 .9 .95 .8 ..93 .8
Total Perimeter 138 148 144

 

PART III

  1.  Once you’ve filled in the compactness measures for the two district maps you created in class, return to the district map-making portion of this Investigation and choose one more strategy for creating a district map (Proportional, Gerrymandered for Gray or Gerrymandered against Gray).  
  2. a) On a separate Squaretopia grid (don’t erase your previous district maps), draw a district map that best meets that strategy. Fill in the number of districts Gray wins in the first table.
  3. b) Calculate all the compactness measures and fill in your answers in the second table.
  4. c) Transfer the information to the table to this shared Google Sheet to compile and examine the results from the entire class.

 

After you’ve completed this, both tables should be filled in for the COMPACT district map and 2 out of the 3 others strategies and transferred to the shared Google Sheet (linked above).

  1. Take photos of the three maps you’ve made. Insert them into this document with labels indicating which strategy was used in each.

  1.  Follow-up questions about the stipulation from the Squaretopia Constitution that districts “… should be as compact as possible.”
  2. a)  Which measure of compactness seems to best reflect the fairness of a district map? Why?- I think the best way that reflects compactness is the skew because I had a lot of perfect compactness but then I had a couple that were not very compact and I think the average reflects that. The average is fairly high like most of the districts were in compactness. Plus there’s not room for large amounts of error. When I did the convex hull it showed I had little to no error which didn’t seem accurate at all.
  3. b) If you were a Squaretopia Supreme Court Justice, how would you use measures of compactness to determine the constitutionality of a proposed district map? Why?

I would come up with a certain number that the compactness had to be as a minimum that way I would know that no weird or odd shapes are being made as a result of gerrymandering. I think that having to be at least .66 compact would beat out a lot of odd shapes and it would help maintain proportional representation the way it’s supposed to be. It would help both parties have the right number of representatives and therefore be more fair.

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