Chops and Erosity - Great Lakes Style

[muon2]

Let me pose another question to your erosity eye. How would you rate your CDs 9-14 above on erosity compared to CD 3, 5 and 8?

Excellent, let me play with these and see what a heuristic model can produce.

Edit: I notice that you have two microchopped subunits: Lenox Twp in Macomb and Brandon Twp in Oakland. I assume that is to come within the inequality limit.

[Torie]

Correct.

[muon2]

Correct.

I can add Plainfield Twp in Kent to the subunit chops, but that may be due to thinking the the village line was a city line.

[muon2]

Here's a coarse erosity measure of the list using regional links between counties and local links within counties. Let's see how it might compare.

CD 11: Torie 0, muon 20
CD 09: Torie 3, muon 17
CD 08: Torie 4, muon 21
CD 03: Torie 5, muon 29
CD 05: Torie 6, muon 13
CD 10: Torie 7, muon 30
CD 13: Torie 8, muon 30
CD 12: Torie 9, muon 27
CD 14: Torie 10, muon 37

The two CDs that are most mismatched are CD 11 and CD 05. It's easy to find why CD 11 is high. The straight line through Oakland happens to cut through 9 pairs of contiguous munis and that accounts for half the score in this crude model. If the line were just one township to the west the score would drop by 5 and be below the score for CD 09. The low score for CD 05 is somewhat harder to detect. Part is due to the long rounded boundary with Lake Huron that adds nothing to erosity. It also appears to be aided by the fact that the two chop pieces are relatively small and go through mostly rural areas. If I treat those chops as if they were in a larger urban county the score would rise to 21 which would be more in keeping with Torie's eye.

Thoughts?

[Torie]

We have a problem. Maybe for zoom situations, one needs to use a ratio of length of perimeter to area contained within. Giving MI-11 with its clean straight lines a reasonably close to square rectangular shape, is a dog that won't hunt in my view.

I cleaned up Kent, got rid of the chop for MI-09, and have a micro-chop into Clawson. Avoiding chops within counties is more important to me than erosity, and I think the public square would agree.

[muon2]

We have a problem. Maybe for zoom situations, one needs to use a ratio of length of perimeter to area contained within. Giving MI-11 with its clean straight lines a reasonably close to square rectangular shape, is a dog that won't hunt in my view.

I cleaned up Kent, got rid of the chop for MI-09, and have a micro-chop into Clawson. Avoiding chops within counties is more important to me than erosity, and I think the public square would agree.

As I look at it more, the problem for your CD 11 is not a zoom problem, it's a question of the number of geographic units on the perimeter. Your CD 4 is about as rectangular as your CD 11 by the usual perimeter to area measures if I square off the counties in CD 4 along their lines in Saginaw Bay. But the county erosity for CD 4 is 18 (or 20 depending on the count around the Saginaw chop). Note that's also about the same as CD 11 using the townships in Oakland as quasi-counties.

The problem for CD 5 is worse in CD 2, because they both benefit by sitting at the perimeter where the lake adds no erosity. I could up each score by the number of counties on the state border. But that's really a kluge, since in the end what matters is the score for the whole state. Every plan must have the same base score for the counties at the border, so whether they are included or not has no effect on the plan rankings. It only impacts the score comparing individual districts.

BTW your new CD 14 is what I would draw in Oakland, though I would smooth it in Detroit city so that Highland Park and Hamtramck weren't sticking out, even at the cost of one neighborhood chop. Did you consider a version of CD 9 that puts Shelby twp in 10 and New Baltimore in 9? What about Sylvan Lake and Keego Harbor instead of the Clawson microchop?

[Torie]

Maybe your system works then. The test is whether the golf score goes up (bad) with say MI-11 becoming something other than a squarish rectangle. It might be worth playing with that, since then we correct for non erosity issues from actually drawing lines that arise solely by the hood that the CD is in. Make sense?

As per usual, all your suggestions "worked," (you probably already knew that ), but for the Detroit hood chop, it became an intermediate rather than a micro-chop, so there was a cost.

[muon2]

If those changes work, here's another version. I've eliminated the microchop in Brandon twp by extending the chain to Orchard Lake in Oakland. It has no effect on erosity (nor would it have any discernible effect on other compactness measures) since the area is very small and each of the munis only touch two others - one in each district. You could leave Orchard Lake in CD 11, but the inequality between the districts is a bit higher.

The biggie is in Wayne. There is no reason to treat Dearborn like a sacred cow. Keeping it in CD 12 creates most of the excess erosity. Here local connection erosity is minimized while keeping the only chop in one of the Detroit planning areas. It maintains two VRA CDs (CD 13 @ 50.8% BVAP, CD 14 @ 51.3% BVAP), and I don't think they would be easy to challenge in today's SCOTUS.

[muon2]

I took the Detroit modification above and embedded it into a modification of your state plan. I eliminated the Kent macrochop and replaced it with a chop in Ionia and a microchop in Barry, as well as reducing the Clinton macrochop just under the 5% threshold. I also rearranged CD 7 so that the Jackson chop ceases to be a macrochop. Boundaries were adjusted in the north to keep things in whole counties. This plan has 5 chops and one microchop outside of the big three counties. There are 5 chops in the big three, plus the chop of Detroit, plus the chop of one subunit of Detroit.



The erosity should be quite a bit reduced with this as well. Is it worth calculating, or would you rather see something else?

[muon2]

In this current sequence we haven't addressed the impact of UCCs. I made this variation of the previous plan to do that. The county chop count is the same as before, but by rearranging the location of the chop through Clinton the part in CD 4 is now under 5% of the quota rather than the CD 8 part falling under 5% of the quota. The other chops and microchops were then shifted to balance population.

If the UCC approach is to treat them as super-counties, then the question returns of when and if to double penalize a chop. Under my proposal I use the 5% rule to open the door to a double penalty. The threshold is intentionally the same as the one used to count subunits as quasi-counties. One desirable property is that if one is just over the limit, then it's possible to make an adjustment while shifting population so as to make no substantial changes to any district.

In the previous plan the Clinton chop was under 5%, but it is CD 8 that has the smaller part, putting the CD 4 chop over the limit for the UCC chop. It was just barely over however, so with a few minor shifts the CD 4 chop goes under the limit and avoids the double penalty. I would argue that the adjustment to get this from the other was not a substantial change in the plan.

[Torie]

A macrochop opens more highways to being cut, correct?  If there is another chop penalty, that arises just how?  When you intermediate chop a subunit?  Otherwise, I am still unclear when subunit chops count and when.

Sure calculate the score for your map. I am interested in seeing what the erosity score is for your MI-07 versus what I drew. If it does not have a higher score, something is wrong, because it should given it is much more elongated. Or the erosity score of adjacent CD's should rise, if MI-07 is artificially kept low by virtue of using so much of the state border, displacing MI-08 from the border. 

[muon2]

A macrochop opens more highways to being cut, correct?  If there is another chop penalty, that arises just how?  When you intermediate chop a subunit?  Otherwise, I am still unclear when subunit chops count and when.

Sure calculate the score for your map. I am interested in seeing what the erosity score is for your MI-07 versus what I drew. If it does not have a higher score, something is wrong, because it should given it is much more elongated. Or the erosity score of adjacent CD's should rise, if MI-07 is artificially kept low by virtue of using so much of the state border, displacing MI-08 from the border. 

Separate the problems from the proposed solutions. If we can agree on the problems we can make headway on solutions.

Let's also separate the UCC implementation form the more general rule for counties and their subunits, since some would suggest it not be used at all in scoring.

The question comes down to when do county subunits count as quasi-counties as opposed to counting a whole chop piece as a quasi-county? The high density urban areas need it or they get off with unusually low erosities. But rural areas with small chops get whacked if it applies everywhere because they already tend to have a lot of counties with links that are cut. My 5% rule is just one possible way to address this, there are certainly others.

On CD 7 you are correct that the state border matters, but so does the answer to the question in the above paragraph. Another factor to note is that CD 8 becomes much more square in my revision and that would offset the additional east-west stretch in CD 7 with conventional compactness measures. When I'm comparing two plans, I prefer to focus on the shape and erosity of the border between two districts and not so much on their overall score. In the end the statewide score is just the sum of all the individual segment erosities between each pair of districts. My choice of focus gets to the eventual goal of reducing overall erosity without getting caught up in the shape of an individual district.

[Torie]

Yes, overall is the thing of course. It is just that I think my combined MI-07 and MI-08 collectively is a bit less erose than your two CD's collectively. Thus I am interested in the relative erosity score for these two CD's.

I am still confused by macrochops. I understand the subunit trigger, but I believe you had a macro-chop count as another chop automatically if into (or out of for that matter (thus the Lapeer chop earned a penalty, or would, if it were macrochopped, even though it is not within a UCC), a multi county UCC, while we both tended to not want to do that for a single county UCC. And I think where we still are at odds, is that you have no penalty for microchops, albeit it you sum them when there is more than one per CD.

Have I summarized the above properly?

[muon2]

Yes, overall is the thing of course. It is just that I think my combined MI-07 and MI-08 collectively is a bit less erose than your two CD's collectively. Thus I am interested in the relative erosity score for these two CD's.

I am still confused by macrochops. I understand the subunit trigger, but I believe you had a macro-chop count as another chop automatically if into (or out of for that matter (thus the Lapeer chop earned a penalty, or would, if it were macrochopped, even though it is not within a UCC), a multi county UCC, while we both tended to not want to do that for a single county UCC. And I think where we still are at odds, is that you have no penalty for microchops, albeit it you sum them when there is more than one per CD.

Have I summarized the above properly?

Mostly. I have a proposed subunit trigger that I call a macrochop. When subunits are invoked, they can be chopped and that adds to the score. If a subunit is macrochoped it spawns sub-subunits, eg the Detroit planning areas, and they can be chopped. In MI and other states with statutory townships I would require that a county fragment produced by a chop could not have more than one chopped subunit. Note that units that have less than 10% of the quota cannot be macrochopped.

The confusion is with the UCC. I would treat a UCC like a supercounty and its subunits are the counties within it. Other than that all other rules are the same.

Lapeer county is not in a UCC and its population is large enough to macrochop. Any chop in excess of a microchop counts as a one chop. If the county is macrochopped, and the county is replaced by its subunits as quasi-counties. The chop of the county still counts but there is no excess for the macrochop unless one of the subunits is chopped.

And here I thought my flow chart was so helpful. Sigh. I can try one of my favorite nursery rhymes instead.

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[Torie]

I'm a lawyer. I like text, tightly written and clear, preferably with defined terms, not charts or mathematical formulas, and if such formulas are absolutely necessary, as they sometimes are in contracts with complex financial mechanics, then attended by examples. Your text above was for some reason unusually clear to me. Kudos! We still have the issue of no penalty for microchops.

I am impressed with your intelligent thought on all of these issues. You are now probably the leading expert on the planet on this little niche. That is what high powered brains and obsessive perseverance can accomplish.

"the chop of the county still counts but there is no excess for the macrochop unless one of the subunits is chopped."

Does such a chop of a subunit for a macro-chopped county include a micro-chop, to return to my favorite leitmotif?

[muon2]

Let me return to the problem that microchops address. Two years ago we noticed that plans had districts that otherwise whole counties needed tiny shifts of population to reach equality. These shifts were the same size as the allowable quota, so they were consistent with shifts that would be used to bring a plan with 1% range into exact equality. The agreement then was such pieces should not have a chop penalty, and later discussion took us in the direction that they could have an erosity penalty.

The same concept of small shifts appeared in jimrtex's population flow diagrams (see for example the OP linked MI thread) a year and a half ago and there was general agreement that the microchops would have a special status. I note that some wanted a larger threshold for microchops consistent with a larger allowable deviation from the quota. The value of 0.5% became the middle ground between that and no allowance for small de minimus shifts at all.

Another perspective is to take a plan that has deviations within 0.5% like the ones we have drawn. To bring any of them into exact equality I could pepper them with a bunch of microchops. Once I did that, you can't distinguish between the microchops used in the original plan and the microchops that bring it to equality. If the microchops could be required to make a plan legal (esp before Tennant), then they shouldn't factor into the scoring. Since we can't tell which of those microchops were needed, none factor into chop scoring.

[Torie]

I have no problem with the 0.5% deviation. You are allowed that much population inequality under the law in all probability, so you use that to get rid of chops. But an additional 0.5% on top of that for microchops, that bear no penalty, is where I dissent, and I don't think that will be accepted in the public square - at all. Maybe there was "general agreement" on that, but certainly not by me. So we will just have to agree to disagree on that one. Microchops don't create quasi counties adding roads to be cut for erosity purposes either do they, as opposed to intermediate and macrochops? If so, than there is no penalty at all, and while maybe practically there is not much risk of gaming, and creating a lot of microchops, I don't want to run that risk myself.

[muon2]

I have no problem with the 0.5% deviation. You are allowed that much population inequality under the law in all probability, so you use that to get rid of chops. But an additional 0.5% on top of that for microchops, that bear no penalty, is where I dissent, and I don't think that will be accepted in the public square - at all. Maybe there was "general agreement" on that, but certainly not by me. So we will just have to agree to disagree on that one. Microchops don't create quasi counties adding roads to be cut for erosity purposes either do they, as opposed to intermediate and macrochops? If so, than there is no penalty at all, and while maybe practically there is not much risk of gaming, and creating a lot of microchops, I don't want to run that risk myself.

Microchops do create quasicounties. And I agree that you have been most skeptical on this from the start of those who weighed in over the last couple of years. OTOH, there are others who pushed for a wider variance than 0.5%.

Let me restate what I think is fundamental here. When we draw a plan with 0.5% tolerance I can see what microchops are in the plan. But if I started with a plan that had many microchops and exact equality you could not necessarily say which of those microchops I might remove to allow for greater inequality in a plan. If the microchops used to create equality are free of a chop penalty, then all such must be since one doesn't know a priori which I will choose to leave in my final plan.

If one says that microchops used to move a plan from 0.5% deviation to zero deviation would be scored as full chops, then microchops would not exist in discussion at all. they would just be chops. In that case any zero inequality plan would be full of chops. The inequality score is a tiebreaker in large part because one might otherwise knock out competing plans by use of many zero chop scoring microchops. So if there were no microchops then I think that the inequality score should move to equal Pareto weight with chops and erosity, and not be a mere tiebreaker.

[Torie]

Microchops that are within the 0.5% deviation are penalty free. They just disappear, because the law allows for such population deviations. I would not attempt to distinguish between plans within the 0.5% deviation, as to which have the most perfect populations (way too complicated). The next 0.5% beyond that is what we are calling microchops. They should be penalized, and where they appear would be based on what creates the lowest erosity score (unless there is enough penalty vis a vis the quasi county erosity rule, with more highway cuts put in play - that is an empirical issue). You might balance more having more microchops to get rid of an intermediate or macro-chop elsewhere. I favor giving microchops a half point penalty.

I suspect when others weighed in, they might have been confused or mixed up, the microchops that just disappear (everyone I think agreed on that), from the next trench of microchops.

I fail to see the disadvantage of my approach here. We are having trouble communicating perhaps because of the conflation of the two kinds of microchops.

[muon2]

Microchops that are within the 0.5% deviation are penalty free. They just disappear, because the law allows for such population deviations. I would not attempt to distinguish between plans within the 0.5% deviation, as to which have the most perfect populations (way too complicated). The next 0.5% beyond that is what we are calling microchops. They should be penalized, and where they appear would be based on what creates the lowest erosity score (unless there is enough penalty vis a vis the quasi county erosity rule, with more highway cuts put in play - that is an empirical issue). You might balance more having more microchops to get rid of an intermediate or macro-chop elsewhere. I favor giving microchops a half point penalty.

I suspect when others weighed in, they might have been confused or mixed up, the microchops that just disappear (everyone I think agreed on that), from the next trench of microchops.

I fail to see the disadvantage of my approach here. We are having trouble communicating perhaps because of the conflation of the two kinds of microchops.

It is the sentence I have emphasized that I think brings us to disagreement. For the VA discussion I reposted my analysis of chop-free plans and found that a simple table captures the ranges of inequality with the same granularity as the chop score. If all chops count in the score then I see no argument for treating all plans equally with respect to population inequality. It is only when one could otherwise game the system with microchops, that it makes sense to me to relegate inequality to a tiebreaker role.

In this regard I strongly agree with IA where inequality is minimized while preserving counties and keeping compact districts. All share a role, and more compact shapes are routinely rejected in favor of greater equality, yet extremely erose shapes that are even closer to equality are also rejected. Without codifying a Pareto principle they have effectively sought a balance between compactness and equality. Without the potential of gaming scoreless microchops, then that balance is desirable in my view for any scoring system.

[jimrtex]

So I took some time this morning to put together the complete county connection map for MI. As usual this represents all the regional connections defined by a path between seats of county government that follows state and federal numbered highways. When there is more than one path between two counties, the path is defined by the one that takes the shortest time according to Mapquest.

Technically, I-75 jogs into Emmet county for a brief part just south of the Mackinac bridge, so it can't be on a path between Mackinac and Cheboygan. But I-75 also can't be on a path between Mackinac and Cheboygan since the south end of the bridge is in Cheboygan. There is ferry service across the strait and in mild winters it can run all year, but usually it doesn't. So that leaves no possible link, so the map shows the best alternative which is Mackinac to Cheboygan.

It is possible to travel directly (ie not in a circuitous fashion, nor substantially through any other counties) between Mackinac and either of the two LP counties.

It is a similar situation to Whitman-Asotin, and Snohomish-Chelan, and distinguishable from Franlin-Columbia.

[Torie]

Microchops that are within the 0.5% deviation are penalty free. They just disappear, because the law allows for such population deviations. I would not attempt to distinguish between plans within the 0.5% deviation, as to which have the most perfect populations (way too complicated). The next 0.5% beyond that is what we are calling microchops. They should be penalized, and where they appear would be based on what creates the lowest erosity score (unless there is enough penalty vis a vis the quasi county erosity rule, with more highway cuts put in play - that is an empirical issue). You might balance more having more microchops to get rid of an intermediate or macro-chop elsewhere. I favor giving microchops a half point penalty.

I suspect when others weighed in, they might have been confused or mixed up, the microchops that just disappear (everyone I think agreed on that), from the next trench of microchops.

I fail to see the disadvantage of my approach here. We are having trouble communicating perhaps because of the conflation of the two kinds of microchops.

It is the sentence I have emphasized that I think brings us to disagreement. For the VA discussion I reposted my analysis of chop-free plans and found that a simple table captures the ranges of inequality with the same granularity as the chop score. If all chops count in the score then I see no argument for treating all plans equally with respect to population inequality. It is only when one could otherwise game the system with microchops, that it makes sense to me to relegate inequality to a tiebreaker role.

In this regard I strongly agree with IA where inequality is minimized while preserving counties and keeping compact districts. All share a role, and more compact shapes are routinely rejected in favor of greater equality, yet extremely erose shapes that are even closer to equality are also rejected. Without codifying a Pareto principle they have effectively sought a balance between compactness and equality. Without the potential of gaming scoreless microchops, then that balance is desirable in my view for any scoring system.

I don't understand what I put in red above. Would you elaborate please? Nobody is suggesting that all chops count equally. If I understand your system (yes, I have difficulty there I admit), macrochops activate chops of subunits that are more than microchops counting as a full chop, intemediate chops do not so activate but count as a county chop, and microchops are not chops but along with macrochops, activate more highways for the created quasi counties vis a vis the erosity score, for both subunits and counties. Is that right?

[jimrtex]

Oh, I was mixing up cuts and chops. And you treat a multi-county UCC as one county for this purpose, right?  Anyway, as I think I mentioned before, within UCC's smaller unit chops should count, whether or not you have a macro chop into a UCC. So I was focused on erosity measures within a UCC, which I thought was what we were discussing. And outside UCC's, I am still not persuaded why some state highways should count, and not others, and what to do if there are no state highways between adjacent counties. Is that deemed a chop when two counties with no state highway between them are in the same CD? Or do payed county highways count to avoid a chop?  None of this may obtain to WI or MI perhaps (although some state highways are poorly labeled and hard to find on Dave's matting utility.

There are various mathematical formula that are used to measure compactness of districts. 

One of them compares the perimeter of a district (the length of its outer boundary) with the circumference of a circle with the same area as the district.  It is a measure of the convoluted-ness of the boundary, which may indicate gerrymandering.

But it has certain negative features:

(1) It includes the external boundary of the state since this is part of the boundary of the district.  In Michigan you will have a district that include the upper peninsula.   Even if you have a straight east-west boundary across the lower peninsula, it will score poorly.  And you can make that boundary across the lower peninsula more irregular, with little increased penalty.

(2) It penalizes use of boundaries that follow natural boundaries, such as rivers.  In Ohio, one rules exploit was to create a district along the Ohio River, producing one district with a bad score, but permitting the other districts to have lower score.  It in effect plastered over the rough surface to produce a smooth internal surface.

(3) It has little penalty in districts that mix urban areas with intricate boundaries combined with rural areas.  For example, you could have a quite complex boundary in Cuyahoga County.  Its score would improve if you added Medina County, and improve even more if you added Wayne County.  The same thing can happen in the Detroit area.

You can eliminate the first two problems by ignoring external boundaries and using a simplified measure of the boundary.  Muon prefers a count of highway cuts, while I would use a straight line measurement between boundary nodes (a boundary node is where three (or more) counties meet, as well as the where a boundary between counties meets an external land boundary.

For example, the border between Genesee and Oakland would be measured as a straight line between the Livingston-Genesee-Oakland junction, and the Lapeer-Genessee-Oakland junction.

The above approaches work when you have whole county districts.

Both my method and Muon's method can be extended to chopped counties.  Consider a north-south split of Clinton County.  As long as the boundary followed township boundaries, it does not really matter whether the boundary within Clinton is straight across, or jogs up a township and back, particularly if the jog is for purposes of population equality.

Both Muon and I would treat the two parts of Clinton as the equivalent to counties for measuring erosity.   If Shiawasee were in a 3rd district, then Muon would check for a highway connection between Shiawassee and the two parts of Clinton.  If there is no highway, then Muon would in essence say that the boundary does not disrupt communication and therefore does not disrupt community of interest.

I would simply measure the two segments of the boundary between Shiawasee and Clinton.

In addition, Muon would count a cut between the two parts of Clinton, while I would measure the distance across Clinton.

A third approach would be to ignore the split of Clinton County for purposes of measuring erosity.  Instead Clinton County would be treated as being in the district in which the majority of the population resided.

But the above still does not address the problem within areas of concentrated population such as the Detroit area.  For the most part, it will be impossible to create whole-county districts.  At most, you might control excessive division of counties, and block double (or multiple spanning) where two or more districts include territory from a pair of counties.  It may be impractical to not have a district crossing the Macomb-Oakland boundary, but it is unnecessary to have two or more.

My preferred approach would be to treat the districts in a large UCC as a single unit when measuring erosity.   One could still check for excessive county chopping within the UCC.

After the statewide plan were adopted, then a plan for the 6 Detroit districts could be created, using townships and cities as the building blocks.  A limitation of this decomposition, is that it may require premature definition of the outer boundary of the area.  For example, your decision to go from Oakland and Macomb into Lapeer may have been based on how you were arranging the Detroit area districts.  It might be produce a better Detroit-area plan by going into Monroe. Livingston, or Washtenaw to pick up the extra population.   But I am willing to risk that for the simplification that decomposition provides.

If there were a statewide redistricting commission, then there could be regional redistricting commissions as well.  There might also be local redistricting commissions.  For example, a Clinton County commission could refine the boundary within the county.  Even though they may think the division of the county between two districts sucks, they can still determine a least suck-iest division.

[muon2]

Microchops that are within the 0.5% deviation are penalty free. They just disappear, because the law allows for such population deviations. I would not attempt to distinguish between plans within the 0.5% deviation, as to which have the most perfect populations (way too complicated). The next 0.5% beyond that is what we are calling microchops. They should be penalized, and where they appear would be based on what creates the lowest erosity score (unless there is enough penalty vis a vis the quasi county erosity rule, with more highway cuts put in play - that is an empirical issue). You might balance more having more microchops to get rid of an intermediate or macro-chop elsewhere. I favor giving microchops a half point penalty.

I suspect when others weighed in, they might have been confused or mixed up, the microchops that just disappear (everyone I think agreed on that), from the next trench of microchops.

I fail to see the disadvantage of my approach here. We are having trouble communicating perhaps because of the conflation of the two kinds of microchops.

It is the sentence I have emphasized that I think brings us to disagreement. For the VA discussion I reposted my analysis of chop-free plans and found that a simple table captures the ranges of inequality with the same granularity as the chop score. If all chops count in the score then I see no argument for treating all plans equally with respect to population inequality. It is only when one could otherwise game the system with microchops, that it makes sense to me to relegate inequality to a tiebreaker role.

In this regard I strongly agree with IA where inequality is minimized while preserving counties and keeping compact districts. All share a role, and more compact shapes are routinely rejected in favor of greater equality, yet extremely erose shapes that are even closer to equality are also rejected. Without codifying a Pareto principle they have effectively sought a balance between compactness and equality. Without the potential of gaming scoreless microchops, then that balance is desirable in my view for any scoring system.

I don't understand what I put in red above. Would you elaborate please? Nobody is suggesting that all chops count equally. If I understand your system (yes, I have difficulty there I admit), macrochops activate chops of subunits that are more than microchops counting as a full chop, intemediate chops do not so activate but count as a county chop, and microchops are not chops but along with macrochops, activate more highways for the created quasi counties vis a vis the erosity score, for both subunits and counties. Is that right?

There are three non-political metrics by which I score a plan: inequality, chop, and erosity. I've tried to construct whole number scoring systems where individual points matter for each of those metrics so that the trade offs are meaningful. If the simplest scoring models work and are balanced in impact then those three metrics are the variables to optimize, and they ideally should have equal weight.

In the initial metric for the chop score, there was only the count of county chops. Microchops were introduced I believe when we were looking at CA plans a couple of years ago, and if they were used to make plans that go from within 0.5% to become more equal they didn't count in the score. I discovered that there were plans where for instance there were two microchops where either one might be eliminated and have a plan that was within 0.5%, but if both were eliminated it would be outside 0.5%. If both microchops are in the plan there is no direct way to score it unless either both get some value or both get no score, since the alternative is to guess the mapper's intent as to which should get a point and which shouldn't. That's not a good system, IMO. So I've been using a system where all microchops count zero, though one could restore them to some chop value.

As I looked at plans with microchops at zero value, I saw that one could drive the inequality score arbitrarily low without increasing the chop count, and thereby create a bunch of potential plans under the three variable measure. If one variable (inequality) was decreasing another could be increasing (erosity) but the plan was still Pareto optimal. However, it was far more likely that if two variables (chop and erosity) were both increasing, then another plan of equal inequality could beat it on at least one measure.

The plans that added microchops to an existing plan were all simple variations of the original plan. It didn't seem desirable to have them added to the mix as if they were new ideas. An easy fix was to relegate inequality to a tiebreaker role, so that only chop and erosity were the primary variables. Now the potential increase in erosity due to the microchops would be fatal to the plan despite the lower inequality.

There is a downside to inequality as a mere tiebreaker. In states where zero chop plans are easy to come by (such as IA), erosity becomes the sole metric and the Pareto frontier is reduced to a single point, eliminating any flexibility. If inequality maintains equal footing with erosity, then a set of possible plans balancing inequality with erosity will exist for those states preserving flexibility.

I want the basic system to apply to both MI and IA. I can argue for inequality as a tiebreaker instead of coequal measure when I get a bunch of trivially modified plans due to zero point microchops, putting the problem in MI ahead of the lack of flexibility in IA. I can't make that case once microchops have a non-zero value in the chop score.

And yes, this elaboration does not touch on macrochops. Macrochops themselves act as a regular chop in the chop score. In the simple definition of a chop the pieces of the chopped county are treated as quasicounties, replacing the original. Macrochops serve as a definition of when to replace the chopped county by the subunits of the county instead of the chopped pieces. Then subunits formed by macrochops may in turn be chopped. None of this affects the issues surrounding microchops that I have seen.

[Torie]

Thank you for all of that, but I repeat, is the statement below in any way inaccurate as to how your system works practically speaking?

"If I understand your system (yes, I have difficulty there I admit), macrochops activate chops of subunits that are more than microchops counting as a full chop, intemediate chops do not so activate but count as a county chop, and microchops are not chops but along with macrochops [intermediate chops as well], activate more highways for the created quasi counties vis a vis the erosity score, for both subunits and counties. "