Ohio Redistricting Contest

[jimrtex]

I had an idea.   99 House seats/16 congressional seats is 6.1875

So if you apportioned 96 house districts among 16 congressional districts you could then distribute the population of the remaining 3 house districts.

If you divided the state into 3 groups of 31 districts, and one groups of 6 districts, then each of the larger groups would have 5 congressional districts, with only a 0.2% error plus any residual variation because of differences among house districts.

(31/6) / (99/16) = 1.002

31 house districts in NE Ohio covers out through Lorain, Medina, Stark, and Mahoning.

31 house districts in western Ohio, mainly in the Cincinnati to Springfield area but extending north to the Michigan line.

6 house districts in the Toledo ara.

31 in the rest of the state with about 1/2 in the Columbus metro area.

So the extra house district in each region could come from Summit, Franklin, Warren as being somewhat central in the region.

[muon2]

I had an idea.   99 House seats/16 congressional seats is 6.1875

So if you apportioned 96 house districts among 16 congressional districts you could then distribute the population of the remaining 3 house districts.

If you divided the state into 3 groups of 31 districts, and one groups of 6 districts, then each of the larger groups would have 5 congressional districts, with only a 0.2% error plus any residual variation because of differences among house districts.

(31/6) / (99/16) = 1.002

31 house districts in NE Ohio covers out through Lorain, Medina, Stark, and Mahoning.

31 house districts in western Ohio, mainly in the Cincinnati to Springfield area but extending north to the Michigan line.

6 house districts in the Toledo ara.

31 in the rest of the state with about 1/2 in the Columbus metro area.

So the extra house district in each region could come from Summit, Franklin, Warren as being somewhat central in the region.

I was thinking a similar idea with senate districts. There are almost two whole districts in each CD.

BTW, my posted minority plan was scored at 149.0. It should have been 149.8 due to the special rule on city integrity preservation. How did yours turn out?

[jimrtex]

I had an idea.   99 House seats/16 congressional seats is 6.1875

So if you apportioned 96 house districts among 16 congressional districts you could then distribute the population of the remaining 3 house districts.

If you divided the state into 3 groups of 31 districts, and one groups of 6 districts, then each of the larger groups would have 5 congressional districts, with only a 0.2% error plus any residual variation because of differences among house districts.

(31/6) / (99/16) = 1.002

31 house districts in NE Ohio covers out through Lorain, Medina, Stark, and Mahoning.

31 house districts in western Ohio, mainly in the Cincinnati to Springfield area but extending north to the Michigan line.

6 house districts in the Toledo ara.

31 in the rest of the state with about 1/2 in the Columbus metro area.

So the extra house district in each region could come from Summit, Franklin, Warren as being somewhat central in the region.

I was thinking a similar idea with senate districts. There are almost two whole districts in each CD.

BTW, my posted minority plan was scored at 149.0. It should have been 149.8 due to the special rule on city integrity preservation. How did yours turn out?

How did you find out a score?  Is the city integrity score for cities crossing county boundaries?  Does that apply for legislative districts?

Using the 99 house districts has the advantage of distributing the error, instead of having one senate district.  After splitting the state up into groups of 31 house districts which is not too hard to do, since it fits real well in the NE, and in the SE it is obvious that Cincinnati to Springfield is not enough and you don't want to intrude on Columbus, is to create 5 groups of 6 house districts and one with 7.  I then eliminated county cuts before starting to balance populations.  It did a pretty good job of distributing the districts.  Not very good on competitiveness.  I've done a lot of refining but mainly to move some seats into the barely competitive.

I have 3 county fragments in Cuyahoga County, and 1 each in Franklin and Hamilton (or zero depending on how you are counting).  Cleveland and Cincinnati are each in a single district, with the VRA district at 48.5% BVAP.

Using cities to cross county boundaries is really pretty sleazy.  The cities that are kept together are pretty much chosen arbitrarily, and it is as likely to be a toe over the line that is used to grab the whole body.  If city councils could say in advance whether they wanted to be kept together and with which county, I could see that it really serves some policy objective.  In Ohio, I don't think it does anything for elections, since they are county-administered.

If they would simply open the deviation limit for whole counties to +/- 1% it would be more useful.  In a real application, I'd bet on them having to use exact equal population which means that they're going to have be splitting counties anyhow.  And if they try to make that part of the rules, they're going to have people concentrating the splits in a single county.

They could simply break the contest into multiple parts, where the first would divide the population among groups of counties, with the plans that required the least additional adjustment to be considered the best plan.

BTW, did you know the district contiguity theme catches touch point contiguity?

[jimrtex]

BTW, my posted minority plan was scored at 149.0. It should have been 149.8 due to the special rule on city integrity preservation. How did yours turn out?

Total Score:         142.85               

                Splits (50 max):                                           35.75     (17.75H + 18S)                   

                Compactness (100 max):                            36.0        (ave of 36.06H & 36.03S)                               

                Competitiveness (49.5 max):                     13.7       [(77H/3 + 29 = 54.7)/4 = 13.7]                     

                Representational Fairness (100 max):       57.4       (26.8H + 30.6S)

 
For the house plans it wasn't worth worrying about anything other than splits,  Arguably to do so at the suffering of population equality in unconstitutional.

For the congressional plans it could be interesting whether it worth minimizing splits, if one considers the relative weighting of the scores.

[jimrtex]

I was explaining why you can't create two majority BVAP senate districts, and I thought of trying something.  And this was the best I could do: 49.70% overall, and 49.79 and 49.62% for the two districts.

Cuyahoga 2 VRA Senate Attempt (House districts)

Cuyahoga 2 VRA Senate Attempt (Senate districts)

The two districts include 6.0007 / 11 of the county population.

[muon2]

I was explaining why you can't create two majority BVAP senate districts, and I thought of trying something.  And this was the best I could do: 49.70% overall, and 49.79 and 49.62% for the two districts.

Cuyahoga 2 VRA Senate Attempt (House districts)

Cuyahoga 2 VRA Senate Attempt (Senate districts)

The two districts include 6.0007 / 11 of the county population.

I concur, and that's why I believe that an unconstitutional grouping of districts in Cuyahoga makes the most sense. At least one county in NE Ohio must violate the section on senate districts, so why not choose the one that gets out of the problem of the VRA at the same time? Cuyahoga has 3.66 senate districts and if the remaining 0.66 is split between two different senate districts one can use Oakwood and Glenwillow in the VRA districts.

[muon2]

My final competition plans for the legislature are up. The shared links are for the house and senate.

Per competition rules, I did not try to use ward boundaries in the cities.

[jimrtex]

I was explaining why you can't create two majority BVAP senate districts, and I thought of trying something.  And this was the best I could do: 49.70% overall, and 49.79 and 49.62% for the two districts.

Cuyahoga 2 VRA Senate Attempt (House districts)

Cuyahoga 2 VRA Senate Attempt (Senate districts)

The two districts include 6.0007 / 11 of the county population.

I concur, and that's why I believe that an unconstitutional grouping of districts in Cuyahoga makes the most sense. At least one county in NE Ohio must violate the section on senate districts, so why not choose the one that gets out of the problem of the VRA at the same time? Cuyahoga has 3.66 senate districts and if the remaining 0.66 is split between two different senate districts one can use Oakwood and Glenwillow in the VRA districts.

If you include the areas with the highest percentage of BVAP that is equivalent to 6/11 of the county, you can get 50.22%

But that leaves Mayfield Heights isolated.  If you add Pepper Pike for connectivity (it is preferred to Lyndhurst because it is smaller), and drop Walton Hills and Orange you are at 50.17%.

If you abandon Mayfield Heights, and replace it with Cleveland Ward 13, and drop Walton Heights and Newburgh Heights for population equality, you get 50.18%.

And you still have to divide these into 6 house districts, that can be formed into 2 senate districts with almost identical BVAP%

When you create a house district in west Cleveland, you are going to end up around 20% BVAP, so you have to pair it with a couple of districts that overwhelm it with BVAP in the mid 60s.

I figured out a grouping that would give me 49.8%, but I can't get the senate districts to balance, one of the house districts is outside of population range, and I only have 3 majority-minority house seats.

[muon2]

I was explaining why you can't create two majority BVAP senate districts, and I thought of trying something.  And this was the best I could do: 49.70% overall, and 49.79 and 49.62% for the two districts.

Cuyahoga 2 VRA Senate Attempt (House districts)

Cuyahoga 2 VRA Senate Attempt (Senate districts)

The two districts include 6.0007 / 11 of the county population.

I concur, and that's why I believe that an unconstitutional grouping of districts in Cuyahoga makes the most sense. At least one county in NE Ohio must violate the section on senate districts, so why not choose the one that gets out of the problem of the VRA at the same time? Cuyahoga has 3.66 senate districts and if the remaining 0.66 is split between two different senate districts one can use Oakwood and Glenwillow in the VRA districts.

If you include the areas with the highest percentage of BVAP that is equivalent to 6/11 of the county, you can get 50.22%

But that leaves Mayfield Heights isolated.  If you add Pepper Pike for connectivity (it is preferred to Lyndhurst because it is smaller), and drop Walton Hills and Orange you are at 50.17%.

If you abandon Mayfield Heights, and replace it with Cleveland Ward 13, and drop Walton Heights and Newburgh Heights for population equality, you get 50.18%.

And you still have to divide these into 6 house districts, that can be formed into 2 senate districts with almost identical BVAP%

When you create a house district in west Cleveland, you are going to end up around 20% BVAP, so you have to pair it with a couple of districts that overwhelm it with BVAP in the mid 60s.

I figured out a grouping that would give me 49.8%, but I can't get the senate districts to balance, one of the house districts is outside of population range, and I only have 3 majority-minority house seats.

The plan I linked above has senate districts of BVAP 50.65% (east) and 50.43% (west), but as I mentioned it doesn't consider wards.

I didn't use Lyndhurst, Pepper Pike or Mayfield Heights in the east side district but I did use Solon. The southern HD is purely suburban with BVAP of 44.53%. The remaining HD BVAPs on the east are 52.78% and 55.10%.

The west side is as you suggested with the western HD at 18.34% BVAP. The other two come in at 72.76% and 61.16%.

As I look at your ward map overlaid on mine, it looks like it would be easily possible to adjust the boundaries so there was not more than one ward split between any two HDs and still have approximately the same BVAPs.

[jimrtex]

As I look at your ward map overlaid on mine, it looks like it would be easily possible to adjust the boundaries so there was not more than one ward split between any two HDs and still have approximately the same BVAPs.

Easier said than done.

Senate Plan

House Plan

I conformed your plan to the Cleveland ward boundaries.  This resulted in the west Cleveland district having too much population.  So I swapped Ward 18 out by the airport with Ward 13 in south Cleveland.  This is an improvement in BVAP (17% vs 7%) but also dropped about 2000 people.

It also meant I had to redo the remainder of western house district.  The Cleveland ward is too big to go with Parma.   And I didn't want a Cleveland ward in the house district that goes into Medina County.  It still ends up in the senate district.

That was the easy part.

The eastern area ended up underpopulated.  You were depending on getting enough population by taking a bit off the edges of the wards.

It violates the constitution to split city wards unless it is not-feasible to draw equal population districts otherwise.  It is demonstrably feasible to create house districts without splitting any wards.

So I added in Pepper Pike and Brooklyn.  To get 6/11 of the population from units that have the highest BVAP%, you should include Mayfield Heights, Solon, and Walton Hills, and leave out Wards 7, 13, and Brooklyn.  So I ended up with Brooklyn and Pepper Pike in place of Mayfield Heights.  This actually improves the maximum percentage from 50.22% to 50.28% because you are short 2000 people.   If you drop Walton Hills it goes to 50.44%.

If you have barely enough 95% for a senate district, the house districts have to be almost perfectly split.  So I had to keep swapping Cleveland wards between the east and the west.  There is enough variation in size that you can pick up a couple of 1000 here and there.  But they weren't in the right place for the eastern house districts, so you would have to adjust those as well.  And it gets messy because the suburban cities are either larger or smaller than the wards.  I eventually got enough population the the east, but the house districts were close enough in balance that I only had one majority BVAP.

I eventually got the eastern districts too work out, but the senate district was up to 51.6%, so I had to swap some Cleveland wards and got the west back up to 49.6%.  I realized if I dropped Newburgh Heights and Cuyahoga Heights I might get close.  So I did, and calculated that it would be 50.01%.  But then I actually put the district together and it was 49.93%.  I had transcribed a number wrong.

That is why Bratenahl is now in the eastern senate district.

[jimrtex]

As I look at your ward map overlaid on mine, it looks like it would be easily possible to adjust the boundaries so there was not more than one ward split between any two HDs and still have approximately the same BVAPs.

Have you tried to enter your map on SOS web site?

Their application has ward boundaries.  The boundaries in Toledo appear to have changed some from what I had, but those in Cleveland, Cincinnati, Columbus, Dayton, and Akron are the same.

[muon2]

As I look at your ward map overlaid on mine, it looks like it would be easily possible to adjust the boundaries so there was not more than one ward split between any two HDs and still have approximately the same BVAPs.

Have you tried to enter your map on SOS web site?

Their application has ward boundaries.  The boundaries in Toledo appear to have changed some from what I had, but those in Cleveland, Cincinnati, Columbus, Dayton, and Akron are the same.

Since the competition says that my map will be presented by contest organizers, I haven't tried to convert it into a form through the SOS website. I assume (perhaps incorrectly) that the contest group will do that to make an official submission.

I've started looking at the congressional map instead.

[muon2]

I've shared a couple of congressional plans to see what some possibilities might be. One thing I learned in the legislative phase was that fairness scores are very important. The electoral disproportionality has a weight of 4 per point of percentage difference. A two percent shift in electoral result between weak and strong in one district changes the score by about 4 or 5 scoring points and from even to weak by 9 points. An equivalent shift in the electoral result in a district would at most change the score by 1 point for competitiveness.

The geographic factors are also weaker than the fairness factor. An additional county split creates two fragments which is a 2 point penalty, but that same split could easily shift the fairness score by 4 or 9 points by shifting the category of a single district. Similarly, the compactness is averaged so it takes a net gain of 16% improvement in individual districts to gain 1% on the average which results in 1 point on the score.

With that in mind two shared plans represent two implementations of maximized fairness. Both plans score over 99 out of 100 points on that index. One plan minimizes county splits, and gets the number of splits down to 9. The other plan recognizes the need to have exact equality if no compelling state interest exists for the population deviation. Many more counties have to be split to get equality and this is used to improve the compactness quite a bit, but it still lands about 4 points lower overall than the plan that minimizes splits.

Minimum split plan

Exact equality plan

[jimrtex]

I've shared a couple of congressional plans to see what some possibilities might be. One thing I learned in the legislative phase was that fairness scores are very important. The electoral disproportionality has a weight of 4 per point of percentage difference. A two percent shift in electoral result between weak and strong in one district changes the score by about 4 or 5 scoring points and from even to weak by 9 points. An equivalent shift in the electoral result in a district would at most change the score by 1 point for competitiveness.

The geographic factors are also weaker than the fairness factor. An additional county split creates two fragments which is a 2 point penalty, but that same split could easily shift the fairness score by 4 or 9 points by shifting the category of a single district. Similarly, the compactness is averaged so it takes a net gain of 16% improvement in individual districts to gain 1% on the average which results in 1 point on the score.

With that in mind two shared plans represent two implementations of maximized fairness. Both plans score over 99 out of 100 points on that index. One plan minimizes county splits, and gets the number of splits down to 9. The other plan recognizes the need to have exact equality if no compelling state interest exists for the population deviation. Many more counties have to be split to get equality and this is used to improve the compactness quite a bit, but it still lands about 4 points lower overall than the plan that minimizes splits.

Minimum split plan

Exact equality plan

I was wondering how the weighting of the factors would play out.

So a 2 SR/6 LR/0/7 LD/1 SD plan

would give (3+6)/(3+6+7+1.5) = 51.4% partisan index.

If the 13 lean seats were all within 2.5%, that would be 39 competitiveness.  To balance the Cleveland seat, the two strong Republican seats would probably have to both be non-competitive.

Such a plan is probably feasible but would start straining the compactness and county split scores.   So you give up some competitiveness, while maintaining fairness, and reduce the splits and keep the compactness up?

BTW, there appears to be an error in the compactness scoring, which has a bias against more irregular districts.  I had a 19% district that appears to be a 29% district, while the Wayne County district is 63% rather than 62%.

[muon2]

I've shared a couple of congressional plans to see what some possibilities might be. One thing I learned in the legislative phase was that fairness scores are very important. The electoral disproportionality has a weight of 4 per point of percentage difference. A two percent shift in electoral result between weak and strong in one district changes the score by about 4 or 5 scoring points and from even to weak by 9 points. An equivalent shift in the electoral result in a district would at most change the score by 1 point for competitiveness.

The geographic factors are also weaker than the fairness factor. An additional county split creates two fragments which is a 2 point penalty, but that same split could easily shift the fairness score by 4 or 9 points by shifting the category of a single district. Similarly, the compactness is averaged so it takes a net gain of 16% improvement in individual districts to gain 1% on the average which results in 1 point on the score.

With that in mind two shared plans represent two implementations of maximized fairness. Both plans score over 99 out of 100 points on that index. One plan minimizes county splits, and gets the number of splits down to 9. The other plan recognizes the need to have exact equality if no compelling state interest exists for the population deviation. Many more counties have to be split to get equality and this is used to improve the compactness quite a bit, but it still lands about 4 points lower overall than the plan that minimizes splits.

Minimum split plan

Exact equality plan

I was wondering how the weighting of the factors would play out.

So a 2 SR/6 LR/0/7 LD/1 SD plan

would give (3+6)/(3+6+7+1.5) = 51.4% partisan index.

If the 13 lean seats were all within 2.5%, that would be 39 competitiveness.  To balance the Cleveland seat, the two strong Republican seats would probably have to both be non-competitive.

Such a plan is probably feasible but would start straining the compactness and county split scores.   So you give up some competitiveness, while maintaining fairness, and reduce the splits and keep the compactness up?

BTW, there appears to be an error in the compactness scoring, which has a bias against more irregular districts.  I had a 19% district that appears to be a 29% district, while the Wayne County district is 63% rather than 62%.

I used a 4 SR, 3 LR, 1E, 7 LD, 1 SD mix for (6+3+1)/(6+3+2+7+1.5) = 51.3%.

Because the SD is 48% black, I found it uses too much D population to make 6 LR districts. The first plan above shows that it's possible to keep splits down with that mix, and the second plan has high compactness with the same mix.

Your comment about the compactness calculation is interesting, as I have had some suspicion, but no definite case to point at. If you haven't already, you should send your findings to the contest, and they'll forward them to the software developers.

[jimrtex]

BTW, there appears to be an error in the compactness scoring, which has a bias against more irregular districts.  I had a 19% district that appears to be a 29% district, while the Wayne County district is 63% rather than 62%.

Your comment about the compactness calculation is interesting, as I have had some suspicion, but no definite case to point at. If you haven't already, you should send your findings to the contest, and they'll forward them to the software developers.

I have done so, and they pointed me to the source code.  It appears that when they are calculating the diameter of the minimum spanning circle, they are assuming that the centroid of the convex hull around a district is the center of the minimum spanning circle.

I'm pretty sure that the minimum spanning circle for a district and the minimum spanning circle for a district's convex hull are the same.  And I conjecture that the point furthest from the centroid is on the circumference of the minimum spanning circle, but the center of the minimum spanning circle is not the centroid of the convex hull.

[muon2]

I have done so, and they pointed me to the source code.  It appears that when they are calculating the diameter of the minimum spanning circle, they are assuming that the centroid of the convex hull around a district is the center of the minimum spanning circle.

I'm pretty sure that the minimum spanning circle for a district and the minimum spanning circle for a district's convex hull are the same.  And I conjecture that the point furthest from the centroid is on the circumference of the minimum spanning circle, but the center of the minimum spanning circle is not the centroid of the convex hull.

That's an interesting shortcut if true. The center of the convex hull is not generally the same as center of the minimum bounding circle. As an extreme example, consider a half circle. Clearly the minimum bounding circle is the completed whole circle and the compactness is 50%. That bounding circle has a center in the middle of the straight edge of the half circle it bounds. The half circle is a convex shape and its centroid is 2/3pi (about 2/10) of the way in from the edge. That makes for a larger bounding circle.

There is no single solution to the problem, but there are many algorithms that iterate and converge to the solution. I'm surprised that they don't at least do some convergence to a proper minimum.

[jimrtex]

I have done so, and they pointed me to the source code.  It appears that when they are calculating the diameter of the minimum spanning circle, they are assuming that the centroid of the convex hull around a district is the center of the minimum spanning circle.

I'm pretty sure that the minimum spanning circle for a district and the minimum spanning circle for a district's convex hull are the same.  And I conjecture that the point furthest from the centroid is on the circumference of the minimum spanning circle, but the center of the minimum spanning circle is not the centroid of the convex hull.

That's an interesting shortcut if true. The center of the convex hull is not generally the same as center of the minimum bounding circle. As an extreme example, consider a half circle. Clearly the minimum bounding circle is the completed whole circle and the compactness is 50%. That bounding circle has a center in the middle of the straight edge of the half circle it bounds. The half circle is a convex shape and its centroid is 2/3pi (about 2/10) of the way in from the edge. That makes for a larger bounding circle.

There is no single solution to the problem, but there are many algorithms that iterate and converge to the solution. I'm surprised that they don't at least do some convergence to a proper minimum.

I think they had the convex hull for other reasons (it is a quick way to check visibility of a district as you pan, and it probably has utility as a first iteration of checking what district is being pointed at, or etc.   District polygons probably have 100s of vertices.

The code had the correct comments.  But they took the centroid, iterated through the points on the convex hull to find the largest distance from the centroid, and said that is the radius of the smallest spanning circle, squared it and multiplied by pi.

Does the Ohio Constitution require that larger cities be placed in as fewest districts as possible?

11.07(B) refers to "such district" in the singular, so each district that can not be formed from whole counties should preferentially be composed of whole townships and cities rather than city wards.   Which would mean that larger cities should be placed in as few districts as possible as long as it is not necessary to split wards.

[muon2]

I have done so, and they pointed me to the source code.  It appears that when they are calculating the diameter of the minimum spanning circle, they are assuming that the centroid of the convex hull around a district is the center of the minimum spanning circle.

I'm pretty sure that the minimum spanning circle for a district and the minimum spanning circle for a district's convex hull are the same.  And I conjecture that the point furthest from the centroid is on the circumference of the minimum spanning circle, but the center of the minimum spanning circle is not the centroid of the convex hull.

That's an interesting shortcut if true. The center of the convex hull is not generally the same as center of the minimum bounding circle. As an extreme example, consider a half circle. Clearly the minimum bounding circle is the completed whole circle and the compactness is 50%. That bounding circle has a center in the middle of the straight edge of the half circle it bounds. The half circle is a convex shape and its centroid is 2/3pi (about 2/10) of the way in from the edge. That makes for a larger bounding circle.

There is no single solution to the problem, but there are many algorithms that iterate and converge to the solution. I'm surprised that they don't at least do some convergence to a proper minimum.

I think they had the convex hull for other reasons (it is a quick way to check visibility of a district as you pan, and it probably has utility as a first iteration of checking what district is being pointed at, or etc.   District polygons probably have 100s of vertices.

The code had the correct comments.  But they took the centroid, iterated through the points on the convex hull to find the largest distance from the centroid, and said that is the radius of the smallest spanning circle, squared it and multiplied by pi.

If they wanted to do a quick approximation, a better technique would have been to take the three points on the convex hull furthest from the centroid, then circumscribe the circle around that triangle. That's a simple geometric construction.

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I don't think so, at least that's not the way I interpret it. In the current plan the contiguous part of Dayton is split between four HDs. Most of it is in HD 39 and 40, but small parts are in HD 37 and 39. I assume those small fragments are whole wards. Given the 5% tolerance, it shouldn't have been difficult to put Dayton in fewer districts, particularly since Kettering and Huber Heights are split as well.

I would interpret the current map to suggest the following: Once a city must be divided, then any need to keep to its boundaries disappears. Instead the city acts as a collection of wards that remain unbroken.

Using that interpretation I can see HD 39 and 40 splitting Dayton since it is too large for one HD. Then, HD 40 was too small it split Huber Heights with HD 36. HD 37 and 38 split Kettering, but are not counted as splitting Dayton between themselves since Dayton has been effectively reduced to a collection of wards by the original split between HD 39 and 40. That's an important sequence, because otherwise it would appear that HD 37 and 38 share two split units of city government - Kettering and Dayton.

Then again, it could be that the current plan is unconstitutional in Montgomery, and simply went unchallenged.

[jimrtex]

I think they had the convex hull for other reasons (it is a quick way to check visibility of a district as you pan, and it probably has utility as a first iteration of checking what district is being pointed at, or etc.   District polygons probably have 100s of vertices.

The code had the correct comments.  But they took the centroid, iterated through the points on the convex hull to find the largest distance from the centroid, and said that is the radius of the smallest spanning circle, squared it and multiplied by pi.

If they wanted to do a quick approximation, a better technique would have been to take the three points on the convex hull furthest from the centroid, then circumscribe the circle around that triangle. That's a simple geometric construction.

The source code has been updated, but the plans still score the same way.

There are a couple of other compactness measures, so I'm guessing that the people in Ohio asked for Roeck's measure and that was what was entered.

I don't really see any value for a compactness measure for legislative districts.  The most countes possible in a House district is 6.  And use of whole townships and wards limit options within counties.  As a contest measure, it encourages increased population deviation, and splitting of cities.

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I don't think so, at least that's not the way I interpret it. In the current plan the contiguous part of Dayton is split between four HDs. Most of it is in HD 39 and 40, but small parts are in HD 37 and 39. I assume those small fragments are whole wards. Given the 5% tolerance, it shouldn't have been difficult to put Dayton in fewer districts, particularly since Kettering and Huber Heights are split as well.

I would interpret the current map to suggest the following: Once a city must be divided, then any need to keep to its boundaries disappears. Instead the city acts as a collection of wards that remain unbroken.

Using that interpretation I can see HD 39 and 40 splitting Dayton since it is too large for one HD. Then, HD 40 was too small it split Huber Heights with HD 36. HD 37 and 38 split Kettering, but are not counted as splitting Dayton between themselves since Dayton has been effectively reduced to a collection of wards by the original split between HD 39 and 40. That's an important sequence, because otherwise it would appear that HD 37 and 38 share two split units of city government - Kettering and Dayton.

Then again, it could be that the current plan is unconstitutional in Montgomery, and simply went unchallenged.
[/quote]
You left out Riverside as well.  The wards in Dayton are pretty small (Dayton, Columbus, and Cincinnati all have at-large elections) so the wards have nothing to do with city elections.  Toledo has 24 wards and 7 city council districts.   Only Akron and Columbus use the wards as city council districts).

The townships in the western tier are small, so it would have been easy eliminate the split of Huber City and put all of Dayton into two districts.  I bet that you could get rid of the Kettering Split as well.  The only challenge in 2000 would have been that the average for 5 districts would have been 97.5%, so you could only have a variation of a little less than 3000 either side of the ideal.

I suppose I was reading "preference" as "feasible" to each level in the list of units.

It really isn't possible to create more than one house district in Cleveland without splitting wards, because of the size of the wards.  You can create the district in west Cleveland because the wards are a bit larger and five of them create a district.

In Franklin County, you can just barely create a contiguous district outside Columbus, stretching from the southeast corner to Hilliard City, for some reason Columbus has not annexed to the Pickaway County line.  But that would require splitting 4 townships in order to avoid a 10th "split" of Columbus.

It's quite feasible to place Cincinnati into 3 house districts.  With the enclaves (Norwood, St. Bernard, etc.) it is about 20,000 short of 116,530x3 which can easily be made up.  All of Springfield township fits in the NW Hamilton district.  The north central suburban district is displaced to the east, and then there is a district the full extent of the Clermont line.

[muon2]

I think they had the convex hull for other reasons (it is a quick way to check visibility of a district as you pan, and it probably has utility as a first iteration of checking what district is being pointed at, or etc.   District polygons probably have 100s of vertices.

The code had the correct comments.  But they took the centroid, iterated through the points on the convex hull to find the largest distance from the centroid, and said that is the radius of the smallest spanning circle, squared it and multiplied by pi.

If they wanted to do a quick approximation, a better technique would have been to take the three points on the convex hull furthest from the centroid, then circumscribe the circle around that triangle. That's a simple geometric construction.

The source code has been updated, but the plans still score the same way.

There are a couple of other compactness measures, so I'm guessing that the people in Ohio asked for Roeck's measure and that was what was entered.

I don't really see any value for a compactness measure for legislative districts.  The most countes possible in a House district is 6.  And use of whole townships and wards limit options within counties.  As a contest measure, it encourages increased population deviation, and splitting of cities.

I agree that the use of compactness is limited due to constitutional constraints, but I found that it did play more of a role for senate districts. In that case It could guide their grouping when not otherwise constrained.

Personally I don't think an average or median score is the best approach for applying compactness. I think a better technique is to count the number of districts that are less compact than some benchmark value. I would use the existing plan median value as that benchmark as it directly encourages improvement on the current plan.

Alternatively one can set two thresholds. One threshold would be like describe above for districts less compact than some median benchmark. A second threshold would define egregious districts that could have a larger penalty.

[jimrtex]

I agree that the use of compactness is limited due to constitutional constraints, but I found that it did play more of a role for senate districts. In that case It could guide their grouping when not otherwise constrained.

Personally I don't think an average or median score is the best approach for applying compactness. I think a better technique is to count the number of districts that are less compact than some benchmark value. I would use the existing plan median value as that benchmark as it directly encourages improvement on the current plan.

Alternatively one can set two thresholds. One threshold would be like describe above for districts less compact than some median benchmark. A second threshold would define egregious districts that could have a larger penalty.

The problem is that the most "egregious" districts are often due to following political boundaries.  And by scoring them, it leads to a temptation to increase population variance or split city wards.  Once a city ward is divided, does the 5% safe harbor disappear at least for those districts that split the ward, and perhaps for all districts in the city?

These are the districts which are more than 1 SD below my mean of 41%.

Cuyahoga: 4 districts, two drawn in an attempt to use whole wards in Cleveland; one that includes Bratenahl to create better population balance; and one drawn with the objective of excluding Blacks.

Hamilton: 2 districts.  One includes the rat tail of Cincinnati.  If I exclude that ward, my compactness goes from 19% to 37%, but my district is split because that ward continues up into the middle of Cincinnati, and a district that is the full length of the Clermont border.  This is in my plan to place Cincinnati in 3 districts, which means the 4 suburban districts have to be drawn around the outside.

A Medina-Summit district using the northern tier of Cleveland suburbs, and which is placed in a senate district with two Cuyahoga districts.

A Montgomery-Greene district that is comprised of mainly Kettering and Beavercreek, but to which Moraine was added to get closer to the ideal population.  Moraine has annexed to the west, which increases the size of the smallest spanning circle, even though the bulk of its population is close to Kettering.  I can improve compactness by dropping Moraine, and being barely within 5%; or swapping West Carrollton for Oakwood (this boosts compactness because West Carrollton has greater area); or splitting Dayton.

A Lucas-Wood district that gets hit because it includes an unpopulated island in Lake Erie.

A Lorain district, because I drew the Lorain district along the shore line which includes Avon Lake.

A Mahoning district because I provide better population equality.

The two Lake districts because I switched a township to balance the population.

A Licking-Fairfield-Hocking district that is drawn in the way that it is because the constitution won't permit a logical split of Fairfield.

A Portage-Geauga-Stark district that includes Stark for better population balance.

A Mercer-Auglaize-Hardin district which is one of two whole county district that gets tagged, and that is largely because of the narrowness of Auglaize County.

A Preble-Darke-Shelby district (which is my county split in western Ohio) which is drawn in part to wrap around a Clark-Miami pairing.  The district on the other side of the Shelby split has 65% compactness, and I could get the district within 1 SD by switching townships or increasing population inequality.

The Erie-Ottawa district which gets dinged because the districts follow shoreline, and don't include Lake Erie in the calculation.  40% of the population of the district is in townships on the boundary which is Sandusky harbor.

An Ashtabula-Geauga district that includes the entire lakeshore of Ashtabula.

I think that all 17 districts represent reasonable and preferable policy choices (with the possible exception of the district in Cuyahoga which was drawn to exclude Blacks).

[muon2]

I agree that the use of compactness is limited due to constitutional constraints, but I found that it did play more of a role for senate districts. In that case It could guide their grouping when not otherwise constrained.

Personally I don't think an average or median score is the best approach for applying compactness. I think a better technique is to count the number of districts that are less compact than some benchmark value. I would use the existing plan median value as that benchmark as it directly encourages improvement on the current plan.

Alternatively one can set two thresholds. One threshold would be like describe above for districts less compact than some median benchmark. A second threshold would define egregious districts that could have a larger penalty.

The problem is that the most "egregious" districts are often due to following political boundaries.  And by scoring them, it leads to a temptation to increase population variance or split city wards.  Once a city ward is divided, does the 5% safe harbor disappear at least for those districts that split the ward, and perhaps for all districts in the city?

I agree that the distinction between districts that avoid splits is important for determining whether a district is egregious. As I suggested, I would use the current plan as the benchmark, and it has a 35.31% compactness. I would start with a point assessed for every district below that value. That naturally builds in a points tradeoff between county splits and compactness.

One standard deviation below the mean on the current plan is 23.99%. I would consider a district to be egregious if the compactness is below that value, and is assessed two points instead of one. If I consider your distinction, then the assessment remains at one point if the district divides no unit of government smaller than a county.

Using those guidelines, here are the districts in my competition plan that would incur points.

SD 2 (Wood-Ottawa-Erie + Toledo; 19.1%; 2 points) This district divides Toledo and was drawn to gain 2 competitiveness points. That's still political gerrymandering, if if for a good policy reason. It shouldn't get any exemption from the "egregious" penalty.

SD 6 (Green- SE Montgomery; 34.27%; 1 point) This is a pretty good looking district, and only by splitting something in Montgomery does it become more compact.

SD 9 (SE Hamilton; 18.70%; 2 points) This district divides Cincinnati, so it could have been drawn to chop off the western tail. Like SD 2 it was drawn to improve competitiveness points between it and SD 8, and doesn't deserve any credit for the ugly shape.

SD 12 (W Ohio; 29.58%; 1 point) This district already benefits by having no split counties, and is not a very compact-looking district so it seems like it deserves its point.

SD 18 (Lake-Geauga-E Cuyahoga; 27.36%; 1 point) This district also was drawn to gain competitiveness and deserves its point.

SD 19 (Delaware-Licking; 29.56%; 1 point) It looks OK, and has no county splits. This is a district that is hurt more by the choice of Roeck as opposed to some other compactness measure like convex hull or Polsby-Popper.

SD 24 (Medina-S Cuyahoga; 32.73%; 1 point) The constitutional rule forces Medina in one SD, and this district keeps the county splits to one and gives one competitiveness point. The point compactness penalty is reasonable in exchange.

SD 25 (E Cuyahoga; 26.13%; 1 point) A black VAP majority district that splits Cleveland. Since it stays over the "egregious" cut, 1 point is fine.

SD 27 (Summit; 34.69%; 1 point) This district splits Akron, and is designed for competitiveness considerations. It's near enough to the threshold that it might have been drawn above 35.3% if this were the actual rule.

Total points assessed: 11

[jimrtex]

The contest site now has a corrected Roerk's measurement.  My mean score improved from 36% to 41%.

In comparing the individual districts on the contest map to those in the state map, there were 5 districts with significant differences.  The state software only shows two significant digits, vs. 4 for the contest.  Most districts either had the same value after rounding or were off by 1%.

Though they are the same plan, they were independently drawn.

The differences:

HD 16 in Franklin County.   Contest 51, State 48.  The state map appears to have messed up one of the wards (Ward 85), adding a 1/2 mile long strip of roadway, at least according to the Franklin County Board of Elections maps.  Because this happens to be at the extreme SE corner of a fairly compact district it extends the radius of the circle, while adding negligible district area.

HD 33 in Summit County.  Contest 38, State 41.  There is an industrial area of Akron on the eastern edge that includes a few enclaves of Springfield township (12 persons).  The area in Akron is uninhabited.  On the contest map I moved the enclaves into the Akron district to eliminate the discontinuity.  On the state map, I moved the industrial area into the suburban district because this would misclassify 12 fewer persons.  Because the industrial area sticks out to the east, and the Akron district has a slight east-west orientation it extended the circle while not adding that much area.

HD 40 in Montgomery County.  Contest 52, State 56.  This is a fairly compact district in the southern part of the county, south of Kettering.   Carlisle village is mostly in Warren County, but about 250 persons live in Montgomery County (maybe 5%).  These areas are part of Miami and German townships.  Only a tiny bit is in German Township, but it sticks out.

Under my understanding of the Ohio constitution, the district boundary should be between Miami and German townships, since the district is the composed of whole townships.  But the contest software treats Carlisle as if it were a township (many municipalities are legally townships).  And if you include all of the Montgomery part of Carlisle, the circle is extended.

HD 44 in Lucas County.  Contest 26, State 32.  The contest map includes an area of water with what appears to be two small islands.   These "islands" are actually dredging spoils and on Google maps, one is underwater, and the other looks to appear be covered with ice (as if the ice had lodged on the spoils and not yet melted.  The census shows these as water, so they are either shoals or intermittent dry land depending on lake levels, tides, and wave height.  The state does not include this area.

HD 98 in Erie and Ottawa counties.  Contest 23, State 18.  The difference here is that the contest geography includes water around the Lake Erie islands so that they are contiguous.

The state mapping software has a circular selection.  I selected census blocks centered on Ohio Stadium in Columbus.  To get enough population for a congressional district you need to get out to about the interstate loop (Jack Nicklaus Freeway).  At that point you pick up undeveloped areas with a few census blocks that are perhaps a 1/2 mile or mile across (ie a section).  The raw score for this circular district was around 80%.  By chopping off some of the more extreme areas, I can push it up to close to 90%.  If there were a circular selection that would only include areas wholly within a circle, I could probably get within a few percentage points of 100%.

Under the existing plan, HD 13, which is Lakewood and three Cleveland wards to the immediate east has a lower score than HD 8, 11, and 17 which are arguably racial gerrymanders, and is one SD below the statewide average.  HD 11 connects two areas of Cleveland via Cuyahoga Heights and Brooklyn Heights.

If the original editorial cartoon had been about HD 13, we would be talking about gerrybricking.

Cartoon shows district that looks like brick, with perhaps some mortar dripping off.  Elbridge Gerry is depicted as a mason finishing a wall.  Caption reads, "That will do for a brick. Better make it a 'gerrybrick'"

The Roeck measure is hypersensitive to the wrong things, and perhaps not sensitive at all to others.  Districts such as 23, 50, and 91 score reasonably well.

[jimrtex]

My Congressional Plan

I was mainly drawing for compactness, though the final map really doesn't show that.

I started by drawing the Cleveland, Columbus, and Cincinnati districts.

In Cleveland and Cincinnati, I started with those cities and then added on cities incrementally that would produce the best Roerk score.  An interesting phenomena is that adding a larger city that increased the circle some would produce a better improvement than adding a smaller city that was already within the circle.  Basically it has a scale problem where it is oversensitive with smaller districts.

The Cleveland district ended up too far west to be a VRA district.  Cleveland itself has a Roerk score of 21.5% and so it was mainly trying to fatten up to the south.  It also split the remainder of Cuyahoga County between an area in the east and one to the south.

In Cincinnati the initial expansion was to the west because of the Cincinnati rat tail, and then to the north, as the circle extended into Butler County and Kentucky and became more like a pair of parentheses.  Since Hamilton County only has an excess of 80,000 over an ideal seat, the residual area were two areas on the east and west end of the county.

In Columbus, I started with the entire city, added in the enclaves of Whitehall, Bexley, Upper Arlington, Grandview Heights, and Marble Cliff, and the fractured townships (Miflin, Clinton, Franklin, and Norwich).  I then started lopping off the Columbus tentacles and cleaned up the boundary.   The area is fairly compact (56.2%) and uses whole wards, but prevents my plan from being competitive.  The remainder of Franklin County almost formed a doughnut, but I needed 8,000 more people and the ward that reaches Delaware County had the right number and didn't eviscerate any townships.

I then started with one district per county, with the remnants of Cuyahoga, Franklin, and Hamilton treated as a district.  I eliminated the districts one by one based on inverse population.  The eliminated district was joined to adjoining districts based on which union would produce the greatest mean compactness.  This tended somewhat to encourage growth of larger (more county districts) and non-compact districts.

When you join two counties together, the mean usually drops.  You might be eliminating two districts in the mid-50s and replace it with a single district of perhaps 45 (a unit square is 64%, a 2x1 rectangle 51%).   If a district has multiple counties, additions don't drop its score very much and may even improve it.  If a district has a low score because of some extensions, then it might be improved by almost any addition.  The original Hamilton remnant had a low score because it was comprised of two disjoint parts.  It also has a fairly small population (34th overall).  So it was joined to Clermont and then other districts added to it which didn't drop the score too much as the inclusion of the western part of Hamilton had less and less effect.

Whenever a district had 721,000 persons I stopped adding districts to it.  Eventually, I ended up with 13 not so equal districts, plus the original 3 city districts with a compactness of 52%.

There was the natural Canton-Youngstown district.  And the Cleveland area districts had a total population equal to almost exactly 5 districts.  I adjusted these a bit, and they now have a population equivalent to 5.0016 districts (0.16% excess).

The district to the east (Lake, Trumbull, Ashtabula, Geauga was quite a bit short so I started adding areas from eastern Cuyahoga County.  After this was done, I was able to slide the Cleveland district east, and it became 48.16% BVAP.  The Summit-Portage district needed just a bit, as did the the Lorain, etc. district.  I did end up having to split a sliver off of Westlake, but otherwise there are no city splits in this area.  Statewide, this split and the split of Columbus are the only city splits not associated with a county boundary.

I then did the Toledo district.  I think that was the only district I could create without splitting counties.  If a 1% deviation were allowed, I think a lot better plan could be drawn.  I had to add quite a bit to the south to get the northwestern rural district.  Using Roerk, a district with a 50-mile indentation is considered the 4th most compact.

The Dayton district is the only one that comes close to being created with whole counties, and fortunately includes Greene.  I had to shift a bit of Carlisle and Springboro to the south.  I was going to shift one of the western townships when I noticed those.  I really don't like the rule about cities crossing boundaries.  I think that it is too arbitrarily applied to merit raising the allowed deviation from +/- 1 to +/- 0.5%.   And once you get above exact equality you are going to be making N-1 splits.

If I were going to create a whole county rule, I would permit multi-district areas that were within 1% deviation (ie, 4.99 to 5.01 rather than 4.95 to 5.05).  The only splits would be internal to these areas.  I would evaluate the splits on a relative basis.   A split where the smaller fragment is 3% of a county is better than one where the split is 40%.  This would encourage an attempt to follow county boundaries and splitting of larger rather than smaller counties.  If anything smaller than a township or city or city ward (large cities only) was split, the districts on either side of the split would have to have identical as possible populations.  In a state like Ohio, I could see letting cities choose to be considered in the county which they have the largest population, but that decision would be made prior to the census.

The Butler district was drawn around the Dayton district.  If you were creating logically compact metropolitan Cincinnati districts, the 4-county area has enough for 2.193 districts.  You could take the excess from Butler (including Hamilton city) and with Montgomery and Preble be right at 3 districts.  This would permit a more rural southern Ohio district, though there may not be enough people for that district and the eastern district.  You probably are going to be pushing into the Columbus suburbs.  You could also have a district that was based in Green-Clark-Miami which would give it more of a SW Ohio flavor.   Instead of 3 Columbus districts you end up with two more tightly drawn around Columbus.

I then adjusted the boundary for the southern district so that along with Hamilton it included exactly 2 districts.  The Cincinnati district was shifted to the west end of Hamilton, and the boundary in Hamilton is drawn to avoid splitting cities and townships.  At this point the eastern district was just a bit short, and there was no place to make an adjustment.  So Hocking got split.  I didn't really like splitting a small county, but I figured the people would prefer an eastern district over a Columbus district (which might not be the case if I trimmed some townships off eastern Fairfield or Licking counties).  The split in Hocking is 3 discontiguous townships - connected through other counties - and avoids the city of Logan.

This left the 2 Columbus suburban districts.  Originally, the doughnut was in the south/western district, and the north/western district was somewhat short.  I added the NE corner - Westerville to Reynoldsburg but it was still about 20,000 off.  I didn't feel like shifting the Columbus district, since I would have to do that on the SOS software (because it has wards and precincts).  So I moved Union to the other district and took in the NW corner of Franklin (Dublin and Worthington).   I was still just a few 1000 short, and I was at the Hilliard City limits, so would have had to go back to the SOS computer.

So I took the pieces of Dublin in Union County, and Reynoldsburg in Fairfield County.  Both cities are in 3 counties, and I already had the other two parts, so these don't bother me as much.  The Dublin city limits had an enclave, and I straightened out the boundary to follow roads, so technically I have split Union County under the rules.

By my count, I have 9 county fragments: Cuyahoga 4, Franklin 2, Hamilton 1, and Hocking 2, my compactness score is 44.6% (5 districts in the low 30s pulling this down).

Competitiveness is 21 (3 highly competitive, and 6 moderately competitive).

My partisan index is 59.0% (5 strong Republican, 2 strong Democrat, 4 lean Republican, and 5 lean Democrat and no even).  This gives me a fairness score pf 69.6 for an overall score of 176.2, which is pretty good for a plan with few county splits, pretty good community of interest, reasonably compact, and only two city splits (though Columbus is is triple split).

78.4% of Columbus is in the Columbus district.  If I wanted a better score, the clear solution is to split Columbus, which would give me 3 more competitive districts.   This wouldn't do much for the fairness score which would require pushing one or two of the leaning Republican districts to even.