Chops and Erosity - Great Lakes Style

[muon2]

I wonder how the map below would score under the new Muon2 scoring system. No township or city chops other than Detroit (although the avoidance of a chop between MI-10 and MI-11 was at the cost I suspect of more state or US highway chops, so not sure if the avoidance of a chop helps or hurts the score; ditto for the line between MI-10 and MI-09), no chops of major metro areas other than Grand Rapids and Detroit which are unavoidable, all CD's within a 1% deviation, and if competitiveness is a less than 5 PVI, everything is  competitive other than the two black CD's and MI-05 and MI-07 (well MI-02 has a Pub PVI of 5.02% per the 2008 numbers). I don't remember what the cut off was for an uber competitive CD. Was that 1.5 PVI or less? If so, MI-04 and MI-06 are uber competitive.


 

I'm not set up for the MI calculations at the moment, but the main thing to watch out for is the size of the chop. 0.5% or less of the quota is a microchop - no chop penalty, can add minimally to erosity. Over 0.5% up to 5.0% is a regular chop - chop penalty, but minimal erosity addition. Over 5.0% is a macrochop - chop penalty and it causes the map to look at townships as if they were counties so that the Detroit districts get treated on the same basis as the large rural ones. That increases erosity scores. You want to avoid macrochops outside the Detroit MCC.

edit: nb. I took a quick look and I think you can rid yourself of one macrochop by moving Missaukee and Osceola to CD 2 (reducing the Kent chop), then pushing CD 4 southeast. The Jackson macrochop is harder to deal with given the Lansing UCC.

[muon2]

Don't see how you can do that, without creating another macro chop or an extra county chop. But that doesn't mean it can't be done. I just don't see it.

If you reduce the chop in Kent, there will be an offsetting macro-chop into the Grand Rapids metro area elsewhere, plus maybe another county chop.

I found a number of choices that kept the raw chop count the same. By raw chop count, I meant not counting UCC chops. That count is somewhat relevant, since there is still the outstanding question as to how to count chops in single-county UCCs. Is a chop into the rural part of Clinton county (Lansing UCC) worse than a chop into Jackson (Jackson UCC)?

[muon2]

Your suggestion about UCC chops seems reasonable. When I looked it up, Lapeer was in the Detroit UCC by the way. That is why the chop was there - to make a good faith attempt to have but one chop into the Detroit UCC. Which suggests that while having one chop in a multi county UCC area should count as but one chop, having two chops into it perhaps should be penalized, and count for three chops. 

Lapeer is in the Detroit metro MSA, but not the UCC. To be in the UCC it must have either 25K or 40% population in an urbanized area. Lapeer has only 71 people, 0.08% in the urbanized area so it isn't included. I stickied jimrtex's master list of UCCs after he created it.

[muon2]

My metric was to keep the BVAP about the same for both majority minority CD's.  That might not fit the scoring system, but comports with the intent of the VRA. Something to think about with your scoring system. This is about the the internal lines within the city of Detroit right?

My understanding and observation is that to satisfy the VRA the districts must provide the opportunity for the minority to elect the representative of choice. In practice, without a lot of precinct data to analyze, a BVAP at 50%+1 is sufficient. In IL the VRA districts were not designed to equalize BVAP, just to provide sufficient opportunity in each district.

So since Detroit must be chopped, there needs to be some antigerrymandering protection within the city. Thus there are city subdivisions, which I took from the city's planning department. They look suitably neutral. Treat them like you would separate suburbs for scoring purposes.

[muon2]

A fundamental problem for measuring compactness (or erosity) is scaling. Districts should have a similar measure regardless of the scale of the district. If only the boundary is considered, small, but irregular districts are favored over large but rather compact ones. One solution is to compare the boundary to the area, and many compactness measures do that, but lack the ability to handle districts that have a dense area at one end and rural at the other. Erosity at the urban end doesn't get weighted appropriately.

The solution to this is to shift the scale of granularity based on the population density. Ideally as the density increases the measure of the boundary uses smaller units. Practically if one is using the granularity of counties in less dense areas one wants to shift to smaller units - townships and cities when the population gets too dense.

My specific solution is to recognize that in dense areas, the large population of a county will tend to support a large chop - a macrochop. The macrochop becomes the threshold to determine when the scale factor changes.

[muon2]

Thought Holly was a city. Didn't realize I chopped Lennox. Why is Lennox a microchop, and Holly isn't? Your rules as written are hard to understand. I wish they could be more clearly stated.

Definition: Microchop. A microchop is a chop where a district has less than 0.5% of the quota in a geographic unit.

Example: In MI for CDs 0.5% of the quota for one CD is 0.005*705,974 = 3,529.87. So if a district has less than 3,530 in the county, township or city, or neighborhood cluster under consideration it's a microchop.

I'm open to suggestions for how to write the definition more clearly, but the definition should apply to states that don't have the same geographic units as MI (eg. states without townships) and don't have the same quota.

In Lenox twp, I count 8,817 people in CD 9 and 1,653 in CD 10. So CD 10 is a microchop of Lenox twp.

In Holly twp, I count 5,238 people in CD 10 and 6,079 in CD 11. Both are over the microchop limit so it's just a regular chop.

[muon2]

Well, based on all of the above chit chat, and assuming no double chop penalty for a macro-chop into a multi-county UCC (and thus being able to chop Clinton County), I came up with the below, with the only intra-county macro-chop being one in Detroit that I don't think can be avoided.

It isn't easy, but you managed a double penalty in Clinton as you drew it. The threshold for a macrochop is 5% of the quota which is 35,298.7 so any chop larger than that into a geographic unit triggers the macrochop provision. The chop of the UCC by CD 3 is 35,478, so it is just barely a macrochop.

The chop into the UCC is a macrochop so it adds a point and causes the UCC to be treated as individual counties. If the chop was 200 people less then it would be a simple chop and the counting would end with one point for the UCC chop. I know it seems arbitrary, but the threshold has to be somewhere and 5% has some rationale, so it's slightly less arbitrary than say 6%.

Since the macrochop causes the UCC to be considered as counties, the chop in Clinton counts again in the score. Had the initial chop been 200 people less, then we wouldn't get to this stage. The two pieces of Clinton are 35,478 and 39,904. You managed to get both over 5% so it's a macrochop of Clinton, too, which is tough for a county with less than 11% of the quota. That causes the county to be treated as a collection of townships and cities, but there are no chops there so the chop count stops at two.

Double chops into a UCC are more likely to trigger a macrochop provision, so they are more likely to get a double penalty. You just happened to pick a chop right over the threshold so it impacts your map as well. If you shift one township in Jackson from CD 8 to CD 7, and one township in Clinton from CD 3 to CD 8, your chop score goes back to the single point chop of the Lansing UCC.

[muon2]

Does the below make the macro-chop for Clinton magically disappear by exploiting the 0.5% leeway in CD populations? If so, we have a problem, and such a result would never survive public square derision at this “anomaly.” Maybe having a 5% rule plus whatever play there is in adjacent CD’s given the 0.5% rule would resolve that. And if Jackson is a macro-shop, maybe a double one if a single county UCC generates a double penalty, does the second map “solve” that problem?  Don’t like that either, but I suppose that if single county UCC’s don’t matter, that is OK, and this version is more likely to have highway cuts, and a higher erosity score, which is good, so the prior map would have a higher score.

Can you avoid a macro-chop into a UCC by having two smaller chops into a multi county UCC, so instead of avoiding a second chop into a UCC, and searching for a chop elsewhere, you actually arrange the second chop to be into the UCC to get down below 25K plus per chop? If so, that rewards two smaller chops into a UCC. Don't like that one either. It should either be neutral, or penalized. I prefer just one chop into a multi county UCC myself.

This uber complicated system needs to be entirely scrubbed to avoid all of this game playing potential that I am exploring in this series of posts I am making, and needs to be taken seriously in my opinion.

I posed some other questions above which were not responded to btw. Cheers.

 

Your map here for Jackson was one of things I was looking at originally when I suggested you could reduce macrochops. I think this reduced chop of Jackson is preferable to your original version, and the swap of Branch and Hillsdale is a reasonable price to pay since CD 7 and 8 aren't significantly more erose with your new version.

The nub in Clinton is an artifact of the fact that you just happen to be right at the edge of the threshold. Any scoring system will have thresholds and strange effects right near their limit, due to the fact that we want to keep whole counties and whole townships when counties are split. It a problem common to any discrete system. Compared to the type of gerrymandering that otherwise occurs I'm not really put off by the nub.

I'll try to put together a flowchart of the scoring system. It's actually much simpler than one that has a long list of exceptions to try to accommodate every possible geography. The complexity is that it is recursive which helps me solve the erosity issue between rural and urban areas.

[muon2]

Here a word version of the process flow to get the CHOP score.

1. Begin with a map consisting of UCCs and counties not in UCCs as geographic units.
2. Set the chop count to zero.
3. Select a geographic unit that has not been tested.
4. Is there more than one district in the unit?
    Yes, continue to 5
    No, skip to 14
5. Count the number of districts in the unit, subtract one, and add that to the chop count.
6. Do any districts in the unit have a population less than 0.5% of the quota?
    Yes, continue to 7
    No, skip to 9
7. The districts in the unit with less than 0.5% of the quota are microchops.
8. Subtract the number of microchop districts in the unit from the chop count.
9. Subtract the district with the largest population in the district from the total population of the district.
10. Is that difference in step 9 greater than 5% of the quota?
    Yes, continue to 11
    No, skip to 14
11. The unit has a macrochop. Replace the unit by its subunits.
12. Treat each subunit as a new, untested unit.
13. Skip to step 15.
14. Mark the unit as tested.
15. Are there any untested units left?
    Yes, return to 3
    No, continue to 16
16. Report chop count as the CHOP score

And here's a diagram of the core process.

[muon2]

Let me address the microchop concerns you raise. One thing you may be missing is that CHOP is not the only measure used to judge a plan. Microchops don't increase the chop count, but they do impact erosity, and generally microchops will increase erosity.

If a plan added a lot of gratuitous microchops, it is highly likely that the erosity increases. Suppose there are two plans, but one adds a bunch of microchops. If they keep the same chop count, then by the Pareto rule the morearose plan is excluded. Thus the gratuitous microchop plan is eliminated.

If one plan adds a bunch of microchops, increasing erosity but reducing chops, then both can go forward as they are pareto equivalent. Rarely a microchop reduces erosity, but it does in your Lenox township example, but if you look at the shape that makes sense. The indent for CD 9 was reduced by the microchop.

That said, a modification that adds one chop when the sum of all microchops in a unit exceeds 0.5% of the quota may make sense.

On your concern about the 1% effect, that is valid on its face. However, there is also a measure of inequality used as a ties breaker if the chops and erosity are the same. A plan that stretches to 1% with microchops would be competingagainst plans that use microchops to decrease inequality, and could be eliminated by them.

[muon2]

Let me address the microchop concerns you raise. One thing you may be missing is that CHOP is not the only measure used to judge a plan. Microchops don't increase the chop count, but they do impact erosity, and generally microchops will increase erosity.

If a plan added a lot of gratuitous microchops, it is highly likely that the erosity increases. Suppose there are two plans, but one adds a bunch of microchops. If they keep the same chop count, then by the Pareto rule the morearose plan is excluded. Thus the gratuitous microchop plan is eliminated.

If one plan adds a bunch of microchops, increasing erosity but reducing chops, then both can go forward as they are pareto equivalent. Rarely a microchop reduces erosity, but it does in your Lenox township example, but if you look at the shape that makes sense. The indent for CD 9 was reduced by the microchop. I tend to doubt there are enough state highways around (that is your proxy still for erosity right?), to be a panacea for microchop city. You might ponder this one some more. There are two types of connectivity used for erosity - local which utilizes public roads, and regional which utilizes state highways. During our work over a year ago it became clear that in urban areas there aren't enough state highways to establish connections where there is high population density. Therefore subunit connections need only be local. This was the mechanism used in the New England town-based thread last year. Top level connections between counties and UCCs are regional.

That said, a modification that adds one chop when the sum of all microchops in a unit exceeds 0.5% of the quota may make sense.  That would help to fix the issue.

On your concern about the 1% effect, that is valid on its face. However, there is also a measure of inequality used as a ties breaker if the chops and erosity are the same. A plan that stretches to 1% with microchops would be competing against plans that use microchops to decrease inequality, and could be eliminated by them.  Is that in the text somewhere?  I am not sure just being a tie breaker is good enough, if a bunch of microchops are used to avoid a non microchop somewhere. I think this comment of mine should be taken seriously. It just won't be accepted in the public square. We've been using microchops in this fashion for almost two years now on multiple threads, so I'm not sure I see the problem. The use of the INEQUALITY score as a tie breaker for plans with matching CHOP and EROSITY scores is in the approved rules for the VA Forum Commission.

Still waiting for responses to my other questions which you keep ignoring. If I can get past one set of questions I can move on to the others. That includes an example of erosity in one of your mixed districts I've been trying to complete for over a day now.

[muon2]

Speaking of my dislike for two chops into a UCC, I also dislike any county that is not a UCC taking a two chop hit, and try to avoid that myself. It seems unfair to the county being cut up. Thus, it might be considered to levy an extra chop penalty when one does double chop a county or UCC. Find some other county to chop. And within a UCC, can one chop counties galore without penalty, as if one county, absent there being a macrochop into the UCC?  That really gives an incentive to do multiple microchops into a UCC, or at least intermediate chops. And the Detroit UCC has no chops into it at all, just a chop out into Lapeer County. So does that mean anything goes chop wise with the UCC, other than worrying about erosity?  If so, that is not good. Why was I spanked for not respecting Detroit hoods, if there is no penalty attached.

I can try to find the thread from a couple of years ago, but we looked at a number of models for chops back then. On one extreme one can count fragments which favors plans that put all the chops in a few counties. On the other extreme one can count the number of counties that are chopped. We reached a consensus that a chop into a new county has the same value as a chop into an already chopped county. The only change I've made since then is to say if the sum of the chops gets too large, it's a macrochop and the plan will take a hit in erosity and perhaps additional chops. That tends to spread the chops out but doesn't force the issue.

On your Detroit question, you have the minimum number of chops of the Detroit UCC so you get the lowest possible chop score at the UCC level (5 points in the flow chart above). Suppose instead a plan added Monroe to the Metro districts to get a better shaped CD. That would force a chop into the Detroit UCC and there would be at least one extra chop assessed compared to your plan. So it does matter.

Unlike VA where I was busy putting out neighborhood maps to guide any plans, you jumped to MI before I put out a corresponding Detroit map. So you are forgiven. Those neighborhoods can be chopped and points assessed just like anywhere else, so if time permits, you could tidy up those boundaries in the city. No harm, no foul.

[muon2]

I thought I did clean up the Detroit lines, but my issue here, is that I don't see in the text of your rules anything about intra county chops where there is no macrochop into the county or UCC, so internal units don't matter. And I interpret your text as treating a UCC as one county, so not only intra county subdivisions, but counties themselves within the UCC, don't matter. As to chop count you can draw the lines anyway you want, and only have to worry about erosity. Perhaps I am misinterpreting your text. Remember that I am a lawyer, so I am a text driven kind of guy!

I am still concerned about multiple chops into one county not being penalized. It seems unfair to the county or UCC taking the hit.

My focus here on this exercise of course, is not playing the game here on Atlas, but fashioning something with more universal application that will withstand all the nit picking. I have no doubt that is why you are spending so much time on this as well, no?

I suspect that part of the lack of clarity is due to my transfer of test from the VA effort here. In VA there are no state-defined subunits, so there was little point in putting in a general statement. Obviously MI not only has well-defined subdivisions, but it has explicit statutory use of those in redistricting. I would expect a general statement for all such states with statutory divisions, but I wasn't working on it when we switched to MI. My intent would be something along the line of requiring that in a chop of a unit, no more than one subunit may be chopped by a district, much like current MI law.

Multiple chops in a county are penalized, just no more than multiple counties chopped.

[muon2]

Here's a rather crude example of erosity measure taken from Torie's initial CD-10. The brown lines indicate the townships invoked by the macrochops of Oakland and Macomb. The blue line segments indicate connections between geographic units that span the district border. There are 37 such segments. If some parts are hard to read I can only suggest viewing it with a tool that has a zoom.



Since St Clair is not chopped there is only one regional connection to each of Sanilac and Lapeer counties. Lapeer is chopped but not macrochopped so it stays with regional connections to St Clair and Genesee, bu the chop creates two units with a local connection between them. The regional connection between St Clair and Lapeer is internal to CD 10 and the regional connection between Lapeer and Genesee is in CD 5 so neither create erosity. Without the Lapeer chop there would be connections to both Oakland and Lapeer, so this is an example of a well-placed chop that helps reduce the erosity.

In a macrochopped county, the townships and cities become the units of interest. In Oakland, Holly and Groveland each establish separate connections to Genesee, as well as to the townships to their south. The chop in Holly creates an internal connection between the two parts, but the connection to the south is to the Holly part in CD 11, so it doesn't effect erosity.

It's interesting to note that half the connections contributing to erosity here are in the Troy peninsula, but to my eye that is a fairly erose feature, so I would expect it to generate a lot of points. I found that all the contiguous units were also locally connected. That's probably not surprising in a flat area that is highly developed.

I mentioned before that the Lenox microchop is another example of erosity reduction. Without it the erosity would count connections to both Richmond township and Richmond city. With it there is just the link between the two pieces in Lenox.

[muon2]

Thanks for the comment and review. I suspect however that failure to penalize county chops within a UCC, equally applies to VA. That is a policy decision, but to ignore intra UCC county lines will be problematic, and controversial, in the public square. If ignored in MI, the array in the Detroit UCC would be different in all probability, particularly the lines of MI-09 and CD-10 and CD-11 where they intersect.

How can Macomb and Oakland be deemed "macro-chopped" if the Detroit UCC is treated as but one county for chop purposes?

The Detroit UCC was macrochopped, so it was replaced by its constituent counties (this must happen in any UCC larger than 105% of a CD). Then some of the constituent counties were macrochopped, specifically the three that must be macrochopped. Those then are replaced by townships and cities. Note that any large UCC will have to consider the internal county lines and the public square worry is abated. (nb. all the VA UCCs were large in this sense, or are one county plus an IC.)

The only time county lines might not come into play is for a small UCC like Lansing. There one could have a situation where a chop into the rural part of a county counts the same as a chop of a whole county. It was exactly this scenario in the Lansing area in our 2013 discussions that led to the idea that a splitting of a whole county off of a small UCC is just as undesirable as a chop into a county. This system preserves that sensibility.

[muon2]

The Detroit UCC was macrochopped by going out into Lapeer?  Is it clear in the text that chops out, are the same as chops in, for penalty purposes, when it comes to UCC's? If below the 105% threshold, than is it open season to chop within the UCC without penalty per the text?

No, the chop out kept you at the minimum score for the Det UCC. A chop in would add. The text say count the number of districts in the unit and subtract one, that's the chop count.

If a unit is within 105% of the quota then it is possible to chop (with penalty) but not macrochop. If a unit is under 10% of the quota it is not possible to macrochop, but chops are still possible. The math in the text implies that under 10% you may only have chops (and microchops) within a unit, from 10% to 105% you may have chops and possibly a macrochop depending on the division within the unit, and above 105% you must have chops that form a macrochop. Chops that are not micro always count towards the penalty. Macrochops simply flag those units that must be inspected at a higher granularity for chops within chops.

States like MI can have a rule that a district in unit not macrochopped shall not have move than one subunit chopped between the same two districts. The rule could also be that if two districts chop two or more subunits in a unit then the chop becomes a macrochop. The first form of the rule is a bar and second form is not a bar, but creates a penalty.

[muon2]

The above text is almost impenetrable. What exactly happens that causes a penalty with lines within a UCC when a macrochop is in play, that would not happen absent a macrochop?  Exactly how is the penalty free leash shortened?  Where in the text are you penalized for chopping counties within a UCC?

In the flow chart, a macrochop does not add a point to the score, only chops add to the score. There must be one or more chops to trigger a macrochop, so one cannot escape the penalty for chops. Note that a macrochop is a statement about the whole unit (UCC, county, etc) not about individual districts in the unit. What a macrochop does is determine that the plan must be assessed at the next finer level of geography as well as at the coarser level that triggered the macrochop.

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The second paragraph is a restatement of the numbers in the flow chart. The rule is the flow chart, and the second paragraph is an example derived from the rule.

As I said earlier, the rule is written for all states and not all have statutory subdivisions like MI, VA being one of them. Certain groups of states may have additional rules to recognize the more important status those subdivisions have. That would include a special rule to recognize towns instead of counties as the primary unit in New England states. It would include special rules for the statutory subdivisions in many of the Midwest and Northeastern states. I mention two such versions of a rule in my previous post.

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State highways are not particularly relevant at the township level, since there aren't enough to establish connectivity. If there is one it might be the preferred connection, but it isn't a guarantee. A state highway on the border of a unit counts in both units for establishing a path for a connection. In your example, I see no reward for the microchop, it looks to me like it increases erosity by one. On your other point I'm still assessing my suggestion that the sum of microchops in a unit be considered as a potential chop, which would address the case you raise when turning Troy into a microchop.

[muon2]

Let me roll back to the basic description of erosity as a cut set from a graph. One starts with the identified units in the state and determines which are connected. For example here's one I did for RI a year ago.



The units here are towns so the links in blue are based on local connections.

Here's the AL one from 2013. Connections are based on state highways since it's a county level map.



The map defines the connections at the top level (the UCC concept was not fleshed out), but some measure is needed to determine when should connections at a lower level be used. That is the job of the macrochop. So some plans will need different areas expanded in more detail based on the chops they create.

The erosity comes by overlaying the district boundaries on the connection map, like this one I put together in 2013 for AL. The erosity in the early simple version in just a matter of counting all the lines intersected by the boundaries - that's called the cut set.



The problem is that in a dense county like Jefferson AL there is nothing in this original method to prevent some serious gerrymandering nor to measure erosity effectively in the urban area. However, AL doesn't have statutory divisions between the county and VTD available like MI does. If I were repeating my work for AL today, Jefferson would be replaced with some reasonable graph of munis and planning zones that would have its own internal and external connections and the erosity would be recomputed based on that. AL doesn't have townships, but there are sources from Jefferson that would be a reasonable substitute. This is what I was posting for the larger VA counties and ICs on that thread, so when we got to connections and erosity there would be detail in place.

I suspect part of the miscommunication is that I've been working on this continuously for two years and I don't tend to roll back to the beginning every time I start up on a thread. In fact, when I did roll back to the start on the VA thread, I couldn't even get to this point before the participants lost interest. On this thread I've been stuck starting in the middle of the content and trying both to push backwards to go over material from one more years ago and to push forward to the detail able to actually produce a score.

[muon2]

I had hoped in the VA exercise to take everything from the start, go step by step, get a group consensus on what to test, then let mappers with different visions produce plans governed by the rules. Each mapper has different designs that they favor, and having a bunch of mappers would expose weaknesses in a set of rules that I alone might not find.

But some rules can support variations and in VA the independent city is the unique feature that needs special rules. I just couldn't keep the interest up long enough to come to consensus on what rule to test. That meant I never got as far on chops and erosity as I wanted to. Nor of course did I address the township/city structure in MI which is shared by a few states, but needs special handling when it occurs.

[muon2]

WI was one of the original states for which I posted a regional county connection map back in 2012. All of the links are based on state highways between county seats.



The first Torie map in the previous post cuts 11 links on the boundary between CD 3 and CD 7.

The second map cuts 9 links not counting the chop in Dunn. The chop in Dunn has a link between the two parts which is also cut, bringing the total to 10 that is how it would be judged on the one interpretation (used in the AL discussion). If viewed this way it has one more chop and one less erosity and is Pareto equivalent to the first map.

However, depending on how one counts the links, a secondary path between Dunn and Chippewa follows WI-64. If that link becomes active after the chop then the number of cut links rises to 11, the same as in the first. IIRC this interpretation was used in a discussion of MD plans way back when, in part because there was no concept of macrochops. If this view is used, then the second map loses to the first one. It's worth exploring the relative merits of the two interpretations.

I've been putting together a detailed example for MI which I hope will illustrate the counting in a chopped county as well as provide some justification for the crossover to macrochops for large chops.

[muon2]

As an example of the relationship between chops and erosity, consider CDs 2 and 3 in following plans. In all of the plans the two CDs together are made up of 10 whole counties and exceed the quota for 2 CDs by 1161. All three plans place a chop in Kent county and have 2 CDs for the Grand Rapids UCC, which is the minimum possible.

Suppose in a crowdsourcing exercise to produce a map Plan A is submitted. The deviation in CD 2 is +1854 and CD 3 is -693, an average deviation of 1274, a range of 2547 or 0.36% of the quota.



The county connections for the plan are in the next image. There are blue dots for the 10 count seats, orange lines to show the links internal to CDs 2 and 3, and yellow lines to show the links to counties in other CDs. Those yellow lines contribute to the erosity. A red line shows the link between the two parts of Kent, and since it is split between CD 2 and 3 it is cut as well and adds to the erosity. Using the red and yellow cut links, CD 2 has an erosity of 7 and CD 3 has an erosity of 13.

The path used to form the link between Ottawa (Grand Haven) and Kent (Grand Rapids) follows I-96 roughly where the yellow line is between them. Holland is the largest city in Ottawa, and if it is considered as the node for Ottawa the connection would be along I-196 at Grandville. A longer path than either of those exists that follows MI-6 near the southwestern corner. If the two parts of the Kent chop were treated as if they were individual counties for connection purposes then there would be a link that connects the CD 3 part of Kent to Ottawa along MI-6, and that is shown in pink. If it is counted then the erosity of both CDs increases by 1.



End part 1 of 3

[muon2]

Chops and Erosity, part 2 of 3

Since this exercise assumes a public process to produce a map. Someone observes that the population inequality can be reduced without changing anything other than the choice of cities and townships to include in the chop. This results in plan B where CD 2 is +172 and CD 3 is +989, an average deviation of 581, a range of 817 or 0.12% of the quota. It has a significantly lower inequality, but the CDs have the same set of cut regional county links, so at the county scale the erosity is the same. Using inequality as a tie breaker plan B would knock plan A out of consideration.



Another mapper also makes some changes to the Kent chop and comes up with plan C. Here CD 2 is -588 and CD 3 is +1749, an average deviation of 1169, a range of 2337 or 0.33% of the quota. The CD boundaries involve the same cut links as in plan A and B so it would be tied in chops and erosity. The deviations are slightly better than plan A but probably not enough to have a different inequality score overall. The internal boundary of the chop is clearly less erose than either plan A or B. However, since inequality would be the only tie breaker plan C would fall to plan B.



It seems that there should be some consideration for the lower erosity visible in plan C. Even conventional compactness measures might not see much difference between B and C since they would be dominated by the large perimeter of the whole counties and would lack sensitivity to the smaller changes in Kent. The best way to give consideration to the shape within Kent is to zoom in on that county and apply a similar technique to the on used between counties.

An important question is whether to zoom in on all chops to consider erosity. There is already an increase in counted links created by the chop, and that counts towards erosity. Population deviations within 5% of the quota are considered substantially equal by SCOTUS, so it is reasonable to consider chops that partition off less than 5% of the quota small enough to incur only such erosity changes that are due to the link between pieces. Adding the effect of secondary highway links, like the pink line in part 1 would work to penalize chops that were added solely to cover only primary links to reduce erosity.

Removing larger chops would result in districts that were substantially unequal. Such chops will most frequently occur in high-population counties that have areas of high density. Zooming in on only those counties where the chop is substantial provides a mechanism to consider the erosity in those high-density population areas where the boundary will be small compared to county-based boundaries.

This becomes the definition of a macrochop. This is not the opposite of a microchop which functions in all ways like a regular chop in regards to erosity. A macrochop sets a point at which it is worthwhile to zoom into a map to see detail that is relevant to scoring.

End part 2 of 3

[muon2]

Chops and Erosity part 3 of 3.

A zoom into Kent replaces counties with cities and townships, and seats of county government with the town hall or city hall. In the map below the halls (nodes) are marked roughly with blue dots and the lines indicate the connections (links) between them. Villages are shown on the map, but they are not considered subunits of the county for redistricting purposes.

Yellow lines indicate connections where the path between units uses a state highway at the boundary, and blue lines are connections made solely with local roads. Yellow lines generally include local roads to get from the hall to the nearest state highway, so what matters is the status of the road as it crosses the town or city line.



One thing to note is that there are a number of nodes that are only connected locally. This includes the cities of Cedar Springs, Rockford, and East Grand Rapids, and the townships of Spencer, Vergennes, and Bowne. These units are considered to be connected by local links, so cutting those links would also count towards erosity. For other local links, scoring can either count them or not, so the calculation will be made both ways.

Here's a zoom into plan A. The chop cuts 8 yellow lines (including the Byron Twp to Ottawa link), and 3 blue lines to locally connected units for a total of 11 cut links. If all local cuts were counted the total would rise to 19.



Here's a zoom into plan B. The chop cuts 9 yellow links (including two to Ottawa), and 5 required blue links for a total of 14 cut links. If all local cuts were counted the total would rise to 20.



Here's a zoom into plan C. The chop cuts 8 yellow links (including one to Ottawa), and 2 required blue links for a total of 10 cut links. If all local cuts were counted the total would rise to only 12.



Counting the chops in Kent would keep plan C in the mix with a lower erosity than plan B or A. Since all three plans have the same chop count, but differ in erosity, the inequality doesn't come into play. Plan C is the lone surviving plan of the three.

[muon2]

I see what you are doing now, and how a chop (other than a micro one?), creates a quasi county that then has independent highways connecting it to adjacent county seats. And apparently state highways to don't go to the county seat of an adjacent county, or quasi county, don't count. Not sure that is good policy, to exclude some state highways that way, and it adds complexity, but at least now I understand the highway cut issue.

Microchops also create quasi-counties. They are chops in all respects, except that they don't add to the chop count (unless the total of all microchops in a county exceeds the 0.5% threshold).

The regional connection rule is that one must be able to trace a path from one node to another on numbered state or federal highways without crossing into a third county. Highways on a border count in both counties. For counties, I used the address of the seat of county government and the point on a state highway nearest it as the node. A case can be made to use the seat of government of the largest city in a county, if that is not the county seat.

[muon2]

I see what you are doing now, and how a chop (other than a micro one?), creates a quasi county that then has independent highways connecting it to adjacent county seats. And apparently state highways to don't go to the county seat of an adjacent county, or quasi county, don't count. Not sure that is good policy, to exclude some state highways that way, and it adds complexity, but at least now I understand the highway cut issue.

Microchops also create quasi-counties. They are chops in all respects, except that they don't add to the chop count (unless the total of all microchops in a county exceeds the 0.5% threshold).

The regional connection rule is that one must be able to trace a path from one node to another on numbered state or federal highways without crossing into a third county. Highways on a border count in both counties. For counties, I used the address of the seat of county government and the point on a state highway nearest it as the node. A case can be made to use the seat of government of the largest city in a county, if that is not the county seat.

So Hwy 29 counts as two cuts, because it goes from one node to another, just not to nodes in adjacent counties? I still think I prefer counting all highway cuts, but in the end it comes down to what is the best proxy for erosity.

We will just have to disagree on whether microchops are penalized. I won't support that. It creates an incentive to have one microchop per county (or more up to 0.5% as you say), so that there are no bigger chops, making a mess of a map. The public square won't stand for it.

I still contend your microchop fear is a non issue. Those microchops are quasi-counties, too, so they tend to increase erosity. If they didn't one would place enough of them to drive the inequality to near zero and drive out competing maps with the inequality tie breaker. When I've tried that I usually find that my erosity rises and I'm left with a Pareto equivalent, but not superior plan.

The original driver for microchops was your concern about sufficient flexibility, since we saw areas of states where the constraints otherwise left few choices. Flexibility cuts both ways, and a set of rules that are too inflexible may not sit well with other parts of the public than the ones who would disfavor microchops.