Talk Elections

Compactness is a requirement of many redistricting plans, and generally either bases a measure on squareness or on boundary length. IA uses one of each type of measure, one to minimize the difference between the north-south and east-west dimensions of the districts, and one to minimize the total perimeter of the districts. These measure compactness, but they can fail to address erosity, that is how ragged the edges are.

In describing various standards for assembling counties into a district I suggested that counties should be contiguous only if one could drive between county seats on numbered state or federal roads (or regular ferry service) without going into any other county. Highways along a border count in both counties. This would eliminate counties connected over mountain ranges, largely unpopulated forests, or unbridged rivers, and also eliminate many occurrences where the border of contact was small.

If one counts the number of contiguous county borders that make up a district border then one has a simple measure of erosity of a district. This measure only counts the borders with the state. For a state the sum of those borders divided by two become the measure of the erosity of a plan. The plan measure divides by two since every border appears on two districts.

As an example I'm going to use WI. I took a county map and drew lines to show all the connections between counties using my rule above. Edit: There should also be a link from Racine to Waukesha.

()

Now to apply it, here was a map version that minimized all the district deviations by using a single three-way split of Milwaukee. It's not very pretty and the county erosity is 67. In addition Florence county in the northeast is not connected to the rest of CD 8.

()

Now by comparison here is the map put together by traininthedistance. It has an erosity of only 51 compared to 67 for the map above.

()

I previously posted a modified version of his map to reduce the population deviation and improve the shape of CD 5 and 6. The changes reduced the erosity to 49. So by both population deviation and erosity it is a better plan.

()

Now to compare to the map in the previous post. That map in the previous post can be characterized by a population range of 673, an average deviation of 195.25, 74 county pieces (72 counties plus 2 extra in Milwaukee), and an erosity of 67. The map immediately above can be characterized by a range of 4623, average deviation of 1035.25, 73 county pieces, and an erosity of 49. It has one fewer fragment but much less erosity in exchange for a wider population range and deviation, though still within the 1% range limitation and 0.5% deviation for microchops.

That seems to work well with reasonably shaped counties, but maybe not so well with erose counties. An example would be Inyo County, and then appending counties to the north and south, which creates a CD which looks erose, and is erose, but might not be erose by your standard. It also does not measure erosity within counties, which obtains for LA County, Cook County, and NYC in particular (do the boroughs in NYC count as counties?).

If the county is erose, it will still be fine. I'll see if I can get some examples from southern states to show, since they tend to have quite a few strange shaped counties. CA shouldn't be worse than any southern state in that regard. What this can't measure is when a state highway connection shouldn't be used such as in a mountain pass. I haven't worked out a way to discriminate those.

For in-county splits, I am leaning toward the MI rule for township splits. It would provide that the split should not increase the bounding circle size around the district, and if it must it should do so minimally. For microchops into a county I'm content that the microchop be connected by any local road to the other county and not unduly create more municipal splits.

Well one can create an erose CD within a County, or part of a county, without city or township splits, so some rule may be needed for that, if there is a viable one. Do you remember my action in Oakland County, MI, where CD's wrapped around three sides of Pontiac to pick up the Pubs and keep out the Dems except via a chop of one of the Dem townships?  You also still need a rule for splits within cities, in particular Chicago, LA, and NYC. We are osculating here between erosity and splits, which are not the same thing. I still wonder how this rule would work with the Owens Valley and Inyo county, with an erose county and topographic issues.

In-county erosity will need its own set of rules, but they are complicated by the different types of county subdivisions that exist. Half the states have well defined subdivision and half don't. Of those that do some have overlapping layers, eg municipalities vs townships in IL.

Since the model I'm working towards is a multiphase process, and the first phase concentrates on whole counties, I need an erosity measure for that step. IA counties look very different than ID counties. Even within a state like OH there's quite a difference in the shape of the northern and southern counties. I observed that standard compactness measures were not working well in the OH competitions since some types favored the straightline counties of the north and others were forgiving of the erose river boundaries of the south.

My model here frees the measure from the type of shapes that are prevalent in the counties. It also allows double duty as a stricter requirement for contiguity than the usual legal definition.

Anyway, for the boring county states, I think your rule is most excellent Muon2. Well done. I of course want it all, and want it now, so thus my comments. 

My inclination would be to take the smaller county's area divide by pi, take the square root, and multiply by two times pi (circumference of a circle with equivalent area, and then multiply by some factor, say 5%.   If the distance between the two end points of the boundary is greater than that, they are connected.   And possibly consider local opinion.  I suspect Florence County may consider Marinette County more of a neighbor, even though you have to go through Michigan and Norway to get there.

WV is an interesting state to test this erosity measurement. There is a wealth of whole county plans submitted to the court during the challenge to the WV maps, and the acknowledgement in SCOTUS's decision that larger population variances are permissible. The state also has plenty of interesting geography to please Torie.

This is the map showing connected counties using the rule that connections exist when one can travel between two county seats only on numbered state or federal highways without entering any other county.

()

As before, the erosity of a district is measured by the number of connections that are severed by its border to other counties in the state. The total plan erosity adds all the districts and divides by two to avoid double counting.

This is the approved plan:
()
The plan has a population range of 0.79%. The district erosities are 14, 27, and 13, or a statewide total of 27. However, note that the plan is not contiguous by this measure since there is no state highway connection between Hardy and Pendleton in CD 2.

Lewis had a WV plan with a population range of 0.09%.
()
The erosity is lower than the approved plan with districts at 17, 16 and 13, for a statewide total of 23. However, it also used the Hardy-Pendleton connection which violates contiguity by this measure.

The best plan submitted for population deviation was Cooper 3 with a 0.04% range.
()
The district erosity is 17, 16 and 23, for a statewide total of 28. This is worse than the Lewis plan or the approved plan, but it doesn't violate contiguity.

Next, I'll post some of the other public submissions with their erosity analysis.

()

The above diagram includes 4 plans referenced in the federal redistricting suit. Cooper was the plaintiff in that suit and filed three plans, initially before the redistricting committee then to the court. The federal case also referenced the Facemire-Snyder plan which is a whole county version of an exact plan filed by Sen. Snyder early in the redistricting process.

Cooper 3: Range 0.04%, Erosity 28 (17, 16, 23).
Cooper 2: Range 0.06%, Erosity 32 (17, 32, 15). CD 2 is discontiguous by this method.
Cooper 1: Range 0.09%, Erosity 28 (22, 19, 15).
Facemire: Range 0.42%, Erosity 21 (17, 12, 13).

Using the Facemire plan as a starting point I reduced the erosity to 20 (19, 9, 12) while the range rose to 0.93%. That plan is the large one in the image above.

One way to use this form of erosity with the range is to look at a Pareto optimal plan. To be Pareto optimal the plan should not be able to be improved in one measure without getting worse in another. By that criteria, both Cooper 1 and 2 fall to Cooper 3 since it has a lower range without making the erosity worse. However, both the Facemire and my reduced erosity plan would be equally valid as Pareto optimal choices within a 1% maximum range limitation.

Muon2, how do you get to a count of 9 for your WV-2 CD in your "perfect" map?  I count 6 internal border counties myself. I read this entire thread and your explanation of your method, and my mind came up short. I just could not parse how you got your numbers. I feel like an idiot! :(

It does seem like the 1% variance rule is alive and well assuming that it is justified to avoid splitting stuff. You must be very happy.  :)

It's not the number of border counties but the number of connecting segments that must be broken to partition the district from the rest of the state. That's why I posted the connectivity map first. CD 2 in the low erosity plan cuts nine of those links. One feature I like in this method of counting is that there is no penalty for separating contiguous counties over a mountain that don't have a highway link, for example between Webster and Pocahontas.

The other interesting feature is that it naturally sets up a Pareto choice between erosity and population range. This can be extended to add a choice related to the number of county splits in state with more larger counties.

Oh, it is driven by where the roads are eh?  How creative. Is that your personal little invention? And if you don't like the result, why just build or remove an inter-county road!  Or upgrade/degrade it from a state highway/county road or something. :P

I might add that this is not so much an erosity test, as a communities of interest test, no?

The concept arises from network theory. The counties can be viewed as a network with connections between any two that share borders. One network technique is to prune connections before working with the network as a whole. In graph theory the size of a cut set (eg. the number of broken links) is a relevant parameter that points to the internal compactness of partition compared to the graph as a whole.

In our CA maps, many frequently noted weak connections between counties that really weren't as valid as others. These typically fell into two types: counties with small lengths of near point-contact and others that had a significant natural barrier that should discourage linking. I could try to prune these on a case-by-case basis, but I noticed that in most cases there was no more than a local road providing a link. By restricting links to state and federal highways, the map was generally pruned of those two types of weak connections using a neutral standard that could be uniformly applied.

Communities of interest are generally a squishy subject that are hard to establish in a uniform way. If I've accomplished that in part with my pruning algorithm, then I'll take that as a success. :)

I find nothing in the content or your post with which I disagree or question. Well done! Yes, all that "hard science/the maths thing" jargon about links theory that I had already penned in the prose poetry of a lawyer more suitable to the noisomeness of the public square, was noted but ignored with mens rea.  :)

Erosity is a measure of compactness.   Non-compact districts are disfavored because they tend to divide communities of interest, or link distant communities of interest, typically for reasons of political, racial, or incumbent-protection gerrymandering.

Traditional measurements of compactness based on perimeter length tend to disfavor use of boundaries that are coincident with rivers and mountain ranges, even though they often create transportation and communication barriers.  They also have scaling problems, and problems determining aggregate compactness (i.e. we can measure how compact a district is, but how compact is a map?)

1992-1996 congressional map (http://www.tlc.state.tx.us/redist/pdf/congress_historical/c_1992_1996P.pdf)

The districts in west Texas don't look too bad, until you start noticing all the chopped up cities (zoom in on Lubbock).   But that little excursion to pick out the minority parts of Lubbock adds very little to the perimeter or area of TX-13, and a mathematical formula based on the ratio of perimeter to area is fooled, just as much as the eye is.   By requiring whole counties, Muon avoids that problem.  When you consider the amount of population, put all of Lubbock County in one district or the other would require substantial additional changes to re-equalize population.

On the other hand, his method would not notice that Amarillo was sliced down the middle.  Amarillo has grown southward so about half is now in Randall County, with the downtown and minority in Potter County.  The map is attempting to make TX-19 about 95% Republican, and make TX-13 winnable by a Blue Dog Democrat (Bill Sarpalius).

In the first Ohio redistricting contest it made sense to draw a district along the Ohio River which had a low compactness score because of the meanders of the river, but it in effect plastered off that area of the state, permitting "smoother" boundaries.

What Muon in effect has done is defined a unit of area, "typical county area" and a unit of perimeter length "typical county-county boundary length" and uses the ratio of the the length of inter-district boundaries defined by the total area.

By using road-connectivity to determine whether a county-county boundary is counted, it eliminates a lot of corner or near corner boundaries.  A boundary that follows jogs in county boundaries is not erose.   It is just maintaining whole counties.   A district that wraps around a whole county on three sides is probably erose.

It also recognizes real transportation and communication barriers.

()

The ferry connections in WA probably need some special rules. For example, the Whatcom-SanJuan connection goes through the waters of Skagit. Should that disqualify it much like a road that briefly goes through a part of another county even if there is no population along that stretch? That's the case with the Snohomish-Chelan link which goes through King.

OTOH there is similar ferry service from Jefferson (Pt Townsend) to San Juan that appears to stay in the waters of those two counties, except when boats might choose to take a more easterly line.

I added the outline map which shows boundary relationships as a triangular mesh.  The links are county seat to county seat and don't necessarily cross the boundaries.  I also fused Benton and Franklin (Tri Cities), Chelan and Douglas (Wenatchee-East Wenatchee) and King and Pierce (Seattle-Tacoma).   I don't recall my reasoning on the last pair.  It could have been something like Everett and Bremerton being separated somewhat from Seattle, even though the suburbs run across the Snohomish-King line just as much as they do King-Pierce.

Stevens Pass links Everett and Wenatchee.  I don't think the NE corner of King is accessible from the rest of the county by any sort of ordinary road.  I'll bet if they need a sheriff, that Snohomish County will send someone up.

The Bellingham to San Juans ferry service by the Washington State Ferry is discontinued.  There is whale watching tour from Bellingham that has a two hour stop in Friday Harbor.  I would thus cut the Whatcom-San Juan link.  I think the only regular service to San Juan is now from Anacortes (which continues across to Victoria, BC.  I don't think there is service from Port Townsend.

The Stevens Pass issue is tricky. I recognize that the 627 people in King along US 2 can't get to any other part of King without leaving the county, but I'm hesitant to split the county unnecessarily if that is needed to link Snohomish and Chelan. Once King is to be split as part of a grouping that includes either of the other two counties then I would insist that it stay attached by road. The problem is that there are lots of instances like this in other states, including flat midwestern ones.

I looked at the WA ferry schedule and it appears that the Bellingham-Friday Harbor boat only does tours. However, the Port Townsend-Friday Harbor does offer one-way passenger fares as part of its regular service. It's not year-round, but many of the mountain passes aren't either. That seems like a connection that should exist unless ferries are restricted to those that can carry vehicles as well as people.

One other connection should be added between Franklin and Columbia. WA 261 crosses the river between the two counties. I doubt it would matter for any whole county map.

What is the definition of a "micro-chop" again?

I used a micro-chop as a portion of a county whose population was less than 0.5% of the ideal district size. If for example the court ruled against a whole or minimal county split plan based on a 1% maximum range, then use of microchops would bring the plan into exact equality with a minimum of population shifts.

The Port Townsend-Friday Harbor whale watching tour has ended for 2012.  I think that if you need anything on the mainland, you take the ferry to Anacortes.   You can drive from Bellingham to Anacortes, and you can take a ferry from Port Townsend to Coupeville and drive to Anacortes.

I would disqualify the connection between Franklin and Columbia as a near-corner touching.  I gave a definition upthread for that.   I am also looking for links between the major population centers.   Pasco to Dayton is on 123 up the Touchet river, and not a circuitous route.

My criteria would be:

(1) A minimally significant boundary;
(2) A direct non-seasonal transportation link between population centers.  The link need not cross the boundary between the two counties, and may also pass through other counties even if it does;
(3) Local sentiment.   This could be either to include a link or exclude a link.

I have updated the maps to remove the Whatcom-San Juan link, and to indicate the Columbia-Franklin link as a physical boundary, but not a significant linkage.

I'm quite willing to include/exclude the folks in Skykomish as part of a Snohomish-Chelan district.   The link is between Monroe and Wenatchee.  Skykomish is not a necessary part of the linkage.

By the way, the Skykomish school district is included in a Skagit-Snohomish-King legislative district.  

 LD 39 Washington. (http://www.redistricting.wa.gov/assets/maps/FINAL_010112_Plans/Legislative/MapBook/LD_39.pdf)

As Torie well knows my goal is to find proxies for communities of interest that avoid the problems that arise when one selects a panel to judge them on a qualitative basis. This prevents me from using (3) as a rule.

On (1) I would want a firm definition of what is significant. Columbia and Franklin are connected by a bridge over the Snake with a state highway. I understand that the boundary isn't very long, but with proxies there are going to be situations one would normally exclude that get in. In reverse Stevens Pass represents a case where one would normally include it but it gets excluded by the neutral proxy.  I'm willing to accept those few instances of each type because they work effectively in far more cases than not.

On (2) I presume that you are excluding all the mountain passes that have regular seasonal closures, but not those that only close for occasional storm events. Seasonally closed roads knock out the connections between Pierce and Yakima and between Skagit and Okanogan (by similarity with Stevens Pass), which you seem to have consistently done. This seems like a sensible rule, consistent with my sense of proxies.

What does "proxies" mean in this context?

 I might note i passing that Interstate 5 occasionally closes due to snow over the Gorman Pass between LA and the Central Valley. I once had to drive from Bakersfield back to LA via driving to Santa Maria, and then down 101 due to a closure - three times the distance and all in the pouring rain, but no snow that way.  So temporary closures to me indeed should not count.

By a proxy I mean a substitute metric that stands in for an intended goal that is difficult to measure. In this case I refer to local sense of community. County and municipal integrity are proxies for this. So too is the notion of connectivity by means of a highway.

It is not necessary to have an panel to determine local sentiment.  The governing board of the counties is sufficient, if done in advance of the actual redistricting process.  The problem is when a redistricting panel is determining community of interest on an ad hoc basis guided by the advice of self-interested individuals or groups.

My inclination would be to take the smaller county's area divide by pi, take the square root, and multiply by two times pi (circumference of a circle with equivalent area), and then multiply by some factor, say 5%.   If the distance between the two end points of the boundary is greater than that, the boundary is significant.  For 30x30 mile counties, this is about a 5-mile overlap.  The Columbia-Franklin border fails this test.

And the transportation link is not between any population centers.  If there is a community of interest between two counties, it is the people who form the community, not the areas that contain them.   I think you have a better argument for a Franklin-Whitman (Pasco-Pullman) link, though it is probably better considered a Franklin-via Adams-Whitman link.   OTOH, 78% of the Adams population is in the Othello panhandle.

For now I'll adopt the criteria that roads and ferries operate all year, but I'll stick to my definition that a connection exists when there is a continuous path between two county seats that stays within the two counties and uses only numbered state or federal highways or regular ferry service. I still like that better from the perspective of minimizing political shenanigans and from the perspective of a neutral observer judging maps.

That gives me this graph for WA.

()

I've included populations as a percentage of the population needed for a CD. One approach is to partition the graph such that each partition is within 0.5% of a whole number of CDs so that the range is less than 1%. WA has 3 counties larger than a CD, and King is larger than 2 CDs so the maximum whole county partition is into 6 regions. WA has 39 counties, but there are 4 counties that have only one connection so the graph has only 35 multiply connected nodes.

A partition into 6 regions would result in an average of fewer than 5 counties per region. Based n the analysis of best configurations, regardless of erosity, that would not be expected to produce regions within a 1% range. The observed threshold for a 1% range is about 9 counties, so 4 regions is a likely target for WA. A division into 4 regions requires two additional counties to be split, and a division into 3 regions would require three additional splits. A division into 3 regions instead of 4 could be done to decrease population range or decrease the erosity.

I'll try to post some examples.

Here's and example of how the erosity measure can inform a choice. I used my WA map with connections and divided it into four regions.

()

The Spokane region has 100.3% of a CD and an erosity of 6.
The Seattle region has 399.6% of a CD. The four CDs in it would have an average size of 99.85%. The region has an erosity of 7 and I'll come back to the internal erosity of its districts in a later post.

The remaining area can be divided into two regions two ways. Using the orange line creates a Bellingham region with 99.9% of a CD and the remaining Tacoma region has 400.4% of a CD. The boundary between these two regions has an erosity of 7.

An alternate division is shown by the purple line. This creates a Tacoma region with 300.6% of a CD and a Vancouver region with 199.8% of a CD. This is a greater population deviation but both regions can be split to keep the individual districts within 0.5%. However, this split creates a boundary with an erosity of only 4. This split would be equally acceptable from the view of Pareto efficiency.

I think major metropolitan areas should be treated as a single entity.

Iowa-style redistricting as you are defining it says to not split counties.  But it is based on a State where the most populous county is far from a whole CD, and are there few highly populated clusters of counties.

If a larger county is split, I don't see the necessary advantage of splitting it as multiple whole districts and a smaller fragment.   It is not derivative of the Iowa rules.  On the other hand one could restrict spanning.   A district spans two counties if it contains all or a portion of both counties.  The counties need not be adjacent.  Multi-spanning is when two or more districts span a pair of counties.   In terms of erosity, while they eliminate the inter-county link, they introduce two intra-county links between the districts in each county.   If only one district spans a border, it in effect shifts the link between the counties/districts.

By not splitting counties, Iowa avoids multi-spanning.  A ban on multi-spanning can be considered to be derivative of the Iowa rules, in better maintaining compactness and communities of interest in States which have a more concentrated population of than Iowa.

If one attempted to create 6 regions in Washington, the problem is not only the small number of counties, and also the limited transportation connections, but also that the three large counties, Snohomish, King, and Washington are contiguous.  To create 6 multi-county regions, the three must be in different regions.  This forces the region that includes King to go west to Kitsap or east to Kittitas.  In addition, because King is fairly close to the equivalent of 3 districts, it must have added a small amount, which rules out Kitsap as a partner, and Kittitas as well since there is nowhere else to go.

Metropolitan areas and urbanized areas represent communities of interest.  Metropolitan areas are defined on the basis of commuting patterns.  A significant share of the population crosses the county boundary on a daily basis.  Urbanized areas are defined based on dense population settlement.  When they cross county boundaries, they blur the county boundary as people go back and forth across the boundary, and there are likely multiple transportation routes between counties.

In Washington, there are three Urban Areas that cross county boundaries.  Seattle, which stretches from the Edmond peninsula, south through Seattle and Tacoma, and just barely enters into Kitsap County from the south.  Marysville is a separate urban area, as is Bremerton.  A definition based on population share would restrict the "county urban area" to Snohomish-King-Pierce.   Other cross-border urban areas are Kennewick-Pasco (Tri-Cities) linking Benton and Franklin counties, and Wenatchee linking Chelan and Douglas.  It is possible that the Wenatchee Urban Area does not include enough of Douglas to be considered significant.

Other urban areas cross state boundaries (Longview, Portland OR (Vancouver), Walla Walla, and Lewiston, ID (Clarkston)), but this is not significant for our purposes.

Proposed rule: County urban areas may not be split unless necessary to avoid splitting other counties.

Proposed rule: Large county urban areas (population greater than 1.0 districts) should be treated as a unit for defining apportionment regions.   County boundaries need not be respected, other than to avoid multi-spanning.

Proposed rule: Larger counties (greater than 1.0 districts) should not be divided among more than: round(quota) + 1.   In the Washington case, this permits King County to be divided among 4 districts, while Pierce and Snohomish would be limited to 2.  Limiting King to 3 districts overly constrains our solutions.

Proposed rule: If counties must be split, it is preferred to split larger counties.

Proposed rule: When dividing a state into multi-county apportionment regions, where some regions have more than one district, preference should be given to creating more single-district regions (i.e. Iowa-style).

Proposed rule: Districts may deviate from the ideal population by up to 1.0%.   Multi-county apportionment regions may not deviate more than 1.0% times sqrt(N).

Using the maximum range misses the objective of equal population, which is to hit the center of the bulls eye, though all shots within the inner circle count 10.  The maximum range is the equivalent to shooting the arrows, and then shifting the target.  It encourages systematic bias.

Minority Report: I would restore links between Snohomish and Chelan and Whitman and Asotin, and sever the Columbia-Franklin link.   There is a numbered state highway between Whitman and Asotin counties.  That you would later have to pass through Idaho if traveling between Clarkston and Pullman is irrelevant to the matter of whether there is a community of interest.

Retrospective: If the County Urban Area concept were applied to Iowa, it would require that Dallas and Polk counties be in the same district, unless this would necessarily require splitting other counites.   It is conceivable that Warren would also be included based on the precise rules.

The other multi-county urban areas are all across state borders (Davenport (Quad Cities); Dubuque; Omaha NE (Council Bluffs); and Sioux City.  Sioux City UA minimally crosses into Lincoln County.

muon2: your regions for Washington force a very ugly split of Yakima, ftr.

I contest the presumptions made by a few posters that splitting large counties is preferable to splitting small counties. In fact, I am generally of the view that splitting small counties is nearly always preferable to splitting large counties. If we take counties seriously at all (a debatable concept), we have to assume that all residents in the county have some sense of commonality not shared with residents of other counties. Taking that assumption, splitting a large county destroys the links between more people than splitting a small county does. I would much rather 10,000 people in a small county feel displaced than 200,000 people in a medium-to-large county feel the same way.

The residents of the small county are likely to be more homogenous with a single common interest.  If there is a single high school, the son of the doctor attended with the daughter of the clerk at the DQ.  The banker and the farmer would attend church together.  Everybody would know everybody.

The voters in the small county will be even more ignored than they are now.  The district offices could be 100s of miles away - in opposite directions.  The representatives are unlikely to have the editor of the Faraway Falls Farmer-Mimeograph on speed dial. 

In larger counties, the office may still be in the county.  The representative is going to pay attention to his constituents, as well as those in the adjacent districts.

Administration of elections will be harder for the small county, as they will need more ballot faces.  The larger county already is set up to handle this.

The very largest counties will be split anyhow. 

And the objective is to not split counties at all.  You are presuming that there is somehow goodness in splitting counties, when it is at best a necessary evil.

'

First off, this is an absurd romanticism of small towns, and it is therefore frankly shameful as a justification for policy. However, I will take your assumptions about small counties for granted.

You are ignoring that the ostensibly lesser harm to large counties is spread across a much larger number of people. I don't care about counties qua counties; they don't have feelings. I am concerned about the impact a county split has on individuals. Even assuming that the harm to individual persons from a county split is greater in a small county than in a large one, there are a lot more people in a large county by definition. Thus, that lesser harm is experienced by a much larger group of people. At best, it is debatable whether one harm is overall larger than the other. Even if the harm from splitting a county to individuals in a county of 10,000 is ten times the harm from splitting their county to individuals in a county of 100,000, the aggregate harm is equal. And that's still assuming that the difference is as comically large as you paint it.


True, but irrelevant.


Er... No, I'm not? I never said anything about splitting counties being inherently good. I don't even believe that.

I do, however, believe that splitting counties is sometimes superior to not splitting counties, though that is irrelevant to my previous point (and my previous point starts from the assumption that splitting counties is bad). Counties frequently do not represent communities of interest; they were, after all, drawn many decades or centuries ago in contours of terrain and population that are, in nearly all cases, irrelevant to modern society. In some places, counties have preserved their coherence, but many counties are incoherent blobs with no uniting force. And this is true of small counties as much as large; since we're discussing Washington State, take Adams County as a prime example of an incoherent county.

There are other viable reasons for splitting a county as well, for example if not splitting a county requires extremely oddly shaped districts. That is, it may be that each individual county is internally coherent, but by preventing county splitting, you are forced to draw districts that do not follow communities of interest. (A good example of this is the current West Virginia map, though there the ridiculous bacon strips are not actually necessary to avoid county splits.)

Thus, splitting counties is only a necessary evil in the sense that it is evil that counties are not instead rearranged to match communities of interest (or when population equality requirements mandate a split of a county that happens to represent a community of interest).

How is it a romanticism?  Harris County has 2 million voters, who during early voting may vote at any early voting location in the county, where they may vote using any of 100s of ballot faces.  Whether Harris County is part of 7 or 8 or 9 congressional districts does not matter with regard to administration of elections.  A smaller county that suddenly is divided between two congressional districts may have to produce twice as many ballot faces.

Rep. Michael McCaul has a full time office in Austin and a a full time office in NW Harris County.   He has a 2-day per week office in Brenham.   If you split Washington County, are the two representatives going to each maintain a one-day-a-week office in Brenham?   Washington County has about 5% of a congressional district.  Enough to at least acknowledge it is in the district.   Split in to two parts, 2% and 3% and there is less reason to pay attention.

This is harsh reality.  Why do you call it romanticism?

Let's keep our discussion to the substance of your argument, that the people in a larger county suffer a greater collective harm.  Your blustering about not caring about dirt qua dirt, or your assertion of comical assumptions on my part really are not helpful.

If King County is divided into more than one district, each of the parts will be well represented.  Voters will not have to travel far to an office.  The representative will understand their concerns.  The division into districts can better respect city boundaries, since there will likely be options.

If Wahkiakum is split in two, you will get the area totally ignored.  The division may be quite arbitrary because it must meet some numeric value.

Splitting a larger county will be more likely to leave areas where the residents still get recognition by their representative.  If 20% of your district is in a county, you don't ignore them, even if it is only 57% of the county is in your district.  If 3% of your district is in a county, it is easier to overlook, or give them less attention.

People thought that congressional districts had to be identical in population.  A federal district judge in West Virginia even quoted Bob Dylan to support his belief that this was now true.

The SCOTUS overturned him.  But a State that seeks to have variation in district size is going to have to justify it with a consistent policy.  Remember that some folks thought that Pennsylvania had committed unconstitutional political gerrymandering because they had a variation as much as 17 persons while indiscriminately cutting across neighborhoods.  Pennsylvania fixed that up, and everyone was happy as clams.

Imagine the deposition:

Q. When do you cut county lines?
A. Sometimes we do, sometimes we don't.
Q. Why do you do so?
A. Sometimes its politics, sometimes we are trying for population equality.  It differs.

That state loses its case in court.

Q. When do you cut county lines?
A. We try to avoid it whenever possible.   We believe that counties represent an important community of interest, and limits mischievous gerrymandering.
Q. But you do cut county lines?
A. Yes.  In the case of the largest counties, King, Pierce, and Snohomish, the population was greater than that needed for a single district.  We also recognized that there was greater variation within these counties, multiple communities of interest.   And that the flexible placement of the boundaries in these must-split counties permitted us to avoid splitting of other counties.
Q. But you did split other county lines.
A. One, we split Thurston.
Q. How did you choose Thurston?
A. We created three whole-county districts, one based in Spokane and generally in the northeast part of the state; one including Yakima, the Tri-Cities area, and Walla Walla generally in the southern part of the State east of the Cascades; and another linking the outer Puget Sound - Bellingham, Mount Vernon, the islands, Port Angeles, Port Townsend, Hood Canal, and then extending down the coast.  As it turns out the deviation of these districts is well within the 1.0% maximum deviation, in fact the largest is 0.2%.
Q. And what about the split counties.
A. We eventually realized that in addition to the cuts in Snohomish, King, and Pierce, one additional split was needed.   As it turns out, the closest whole-county district that contains Clark County would have 2.0% deviation of the ideal, outside the limits we expected that the courts would accept for a congressional district.  It would also require another county to be split.
Q. So how did you choose Thurston?
A. We considered two choices, Clark and Thurston.   Clark was considered the preferred choice because of its size, almost 2/3 of a district.  But the split necessary to prevent splits elsewhere would have required a King-based district to stretch to Spokane, and would have reduced the number of whole county districts to 2, since the Yakima-Tri-Cities district would include part of Vancouver.  Clark is essentially in the corner of the State which reduces our flexibility.  This essentially left us choosing Thurston.
Q. Any other advantages?
A. The 4 county metropolitan area, including Tacoma, Everett, and Bremerton is entitled to 5.489 representatives, essentially, 5-1/2.  We could either trim off 1/2 a district, or add areas with 1/2 a district population.  Since we had added the 3 Trans-Cascade counties of Chelan, Douglas, and Kittitas, the addition of a part of Thurston gets us to 6 districts.  We provided maximum possible separation between the Seattle metro districts and the rest of the state.
Q. What about smaller counties.
A. Really no viable options, unless you want to split several.  And if it really doesn't hurt to split a small county, to get within 1% deviation, it begs the question why not just go ahead and divide several small counties to get within 0.0001% (plus or minus 1 person).


You are going down a path of ad hoc justification of your redistricting decisions, which is simply begging a court to intervene.

60% of Adams County is in Othello.   So what you are doing when you include Adams County, is adding the people in Othello along those that live in more rural settings.  That they don't live in a circle  around Othello matters not.  If you were to split Adams County, there is a good chance that you would also have to split Othello.

I think there is a world of difference between "small counties" in Texas and New Jersey.  It's entirely reasonable to expect that in rural TX, county lines matter because everything (school districts, police forces, etc.) are based on them, and there's a lot of unincorporated area that's administered by no subdivision smaller than counties.  This isn't the case in an urban state like NJ, where everything is part of an incorporated municipality.

In general, county lines are probably going to be less important in urbanized areas where municipal lines, urbanization patterns, VRA concerns, etc. should take precedence.  But in rural areas, counties matter.  It's really an urban versus rural difference rather than large versus small- I mostly agree with the gist of jimrtex's argument (even if he does romanticize small towns somewhat), but I'd much rather see small Bristol County, RI split than large Sedgwick, KS, just for instance.

And of course, in some states there's a sufficiently low county-to-CD ratio that every district is going to have to split enough counties that it starts to just not matter anymore.  Arizona and New Jersey are probably the best examples.  (CT and MA would be too, except that counties are already meaningless up there.)

Here is how urban areas can be used in the redistricting process.   The census bureau defines urban areas based on dense population settlement.  From 1910 to 1940, defined as urban, incorporated towns and cities with a population of 2500 or more.    Beginning in 1950, the census bureau began defining urbanized areas based on cities of over 50,000 and adjacent densely populated areas whether in smaller cities or in unincorporated territory, recognizing that suburban growth often disregarded formal political boundaries.  Towns with a population over 2,500 outside urbanized areas continued to be classified as urban.

Beginning with the 2000 Census, the census bureau began defining urban areas based on dense residential settlement, without regard to political boundaries.   Essentially what we would recognize as a city or town as we drove in, without regard to where or if there were a city limit sign.  To maintain consistency with past definitions, urban areas with greater than 50,000 population are classified as urbanized areas.   Urban areas with less than 50,000 population are classified as urban clusters.   Urban areas must have a population greater than 2500.

Urban areas form a local community of interest.  The residents live near each other and interact with one another on a regular basis.  They may also be used to help identify transportation links.  People near a county boundary may cross it on a regular basis using country roads.  Some may do this for employment, a farmer may have fields in both counties, a school bus may collect students on both sides of the line to attend a rural school.   But for large scale interaction, it has to be between the population centers of the counties.  When using transportation links to define regional communities of interest, what matters is whether there is a direct transportation link between the population centers.  It really doesn't matter that the link clips the corner of another county.  It is interchange that forms the regional community of interest,

Core-based statistical areas (Metropolitan Statistical Areas and Micropolitan Statistical Areas) are groups of counties that include n urbanized area or urban cluster as their core.  These statistical areas are defined on significant commuting between the counties, typically residents of an outlying county commuting into a central county, that contains the core urban area, but sometimes when the outlying county contains employment centers that attract workers from the central counties.  Thus the urban areas help identify economic community of interests.  

When these urban areas cross county boundaries they tend to blur that boundary.  You can't readily see the boundary, and it is not located in a rural area away from the county seat.  The counties are not as significantly separate local communities of interest.  Preference should be given to not separating these counties, if population equality and not splitting counties can be achieved in other ways.

The first use of an urban area is to determine the most significant population concentrations in a county.   Definition: Any urban area that represents 10% of the population of a county is considered a significant population concentration for determining transportation linkages between counties.   If no urban areas are above 10%, then the urban area with the largest share of the county population.  If no urban areas, then the county seat.

For Washington:

Adams - Othello (60% of county population)
Asotin - Clarkston (part of Lewiston, ID UA) (93%)
Benton - Kennwick (part of Kennewick-Pasco UA) 83%
Chelan - Wenatchee (54%)
Clallam - Port Angeles (32%) and Sequim (27%)
Clark - Vancouver (part of Portland, OR UA) (85%)
Columbia - Dayton (66%)
Cowlitz - Longview (60%)
Douglas - East Wenatchee (part of Wenatchee UA) (73%)
Ferry - Republic (county seat)
Franklin - Pasco (part of Kennewick-Pasco UA) (81%)
Garfield - Pomeroy (county seat)
Grant - Moses Lake (37%)
Grays Harbor - Aberdeen (41%) and Montesano-Elma (12%)
Island - Oak Harbor (42%)
Jefferson - Port Townsend (28%) and Port Hadlock-Irondale (14%)
King - Seattle (96%)
Kitsap - Bremerton (79%)
Kittitas - Ellensburg (51%)
Klickitat - White Salmon (part of Hood River, OR UC) (20%) and Goldendale (16%)
Lewis - Centralia (39%)
Lincoln - Davenport (county seat)
Mason - Shelton (31%)
Okanogan - Omak (20%)
Pacific - Raymond (20%) and Ocean Park (16%)
Pend Oreille - Newport (17%)
Pierce - Tacoma (part of Seattle UA) (93%)
San Juan - Friday Harbor (county seat)
Skagit - Mount Vernon (54%) and Anacortes (14%)
Skamania - Stevenson (county seat)
Snohomish - Everett (part of Seattle UA) 66% and Marysville 20%
Spokane - Spokane (83%)
Stevens - Colville (11%)
Thurston - Olympia-Lacey (70%)
Wahkiakum - Cathlamet (county seat)
Walla Walla - Walla Walla (80%)
Whatcom - Bellingham (57%)
Whitman - Pullman (62%)
Yakima - Yakima (58%)

The 10% threshold appears to discriminate fairly effectively against urban areas that barely lap over into a county (eg Kennewick-Pasco into Walla Walla, Seattle into Kitsap, Centralia into Thurston).  It also cuts out a lot of smaller centers in counties such as Whatcom (Birch Bay, Everson, Lynden, and Peaceful Valley) where Bellingham is the focus of the county economy.

Rule - Adjacent counties are transportation linked if there is a direct non-seasonal highway, ferry, or passenger rail link between significant population concentrations.  The link need not be entirely in the two counties.  It is better to cut fewer transportation links with district boundaries, and districts should not be formed on the basis of contiguity where no transportation link exists - adjacent non-linked counties could be part of the same district if they are linked via other counties.  

The second use of urban areas is to define counties where the urban area forms a strong link between the counties based on shared continuous residential settlement.  Rule: Counties that share their largest urban area are considered to be strongly linked.  Drawing district boundaries that split the counties should be avoided if possible.

In Washington, the following such groupings exist:

King-Pierce-Snohomish (Seattle UA)
Chelan-Douglas (Wenatchee UA)
Benton-Franklin (Kennewick-Pasco UA)

Rule: Counties containing large urbanized areas should be treated as a unit in apportioning congressional districts.   This will tend to reduce districts where the metropolitan areas dominate less populated areas outside the metropolitan area.

King-Pierce-Snohomish have a population equivalent to 5.116 districts, so should be contained in an area with 6 districts, or have areas trimmed off get them to 5 districts.  If Kitsap is included it is at 5.489.

I could see using UAs to replace county seats for determining connections. I would like something less arbitrary than 10% such as the actual Census definition, but that may open too many doors. I'd also like to identify a single point for computing connectivity, so perhaps using the largest UA would be an appropriate substitute for county seats. Multiple points within a county make the application of graph theory more difficult. It effectively converts the problem to one where some counties act as multiple counties, but provides little or no guidance as to where the boundary within the county might be. For counties that are split I like the idea of using the largest UA in each county fragment to determine centers for connectivity.

I would not insist on county size as a criteria for when to split. I would prefer using erosity to decide which county to determine if a split is more or less optimal that a different configuration. I don't require that whole districts be nested in large counties, and used WI as an example at the beginning of the thread to show how a split of Milwaukee county with no district wholly within could be part of a plan to reduce erosity.

I still strongly think that one has to keep the connection within the two counties in question. I understand that you are allowing one to ignore a de minimis connection through a third county (or other state), but I think that leads to qualitative judgements as to connectivity. If a state were individually to adopt a connectivity standard they might well want to specifically define their sense of connecting links that could include exceptions both way such as in your post. As a general rule I would not want to second-guess what a state would select.

I'm not sure what you mean by census bureau definition with regard to the 10% threshold.  The 10% threshold is only used to determine whether there are urban areas that have enough of the county's population to be considered significant population centers for their county.   If you don't have 10% of the population, you are not significant, unless you are the only opulation center.

I am proposing defining "transportation link between counties" as "transportation link between significant population centers of the counties."

We are in essence saying that Whatcom is not a neighbor with Okanogan despite their common border, because people don't travel directly between Bellingham and Omak on a regular basis throughout the year.

But if someone is traveling to Clallam County, it could be to either Port Angeles or Sequim.  We aren't creating a transport network.   We are pruning a county adjacency network, based on whether the people of the counties are really neighbors.  Someone who lives next door who you never see is not really your neighbor.   Someone who lives on an adjacent property can be your neighbor even if you cut across the corner of another property when you are visiting his house.   There may be instances when two households live on an adjacent property.  You can be their neighbor, even if you only interact with one household.

Clarkston is on the Asotin-Whitman border.  93% of Asotin County is in the Lewiston Urban Area.   There is a bridge across the Snake river from Clarkston into Whitman County that is a state numbered highway (Washington 128).   You can turn left and eventually get to Pullman.  You can continue on Washington 128 until it becomes Idaho 128 and then turn north to Pullman.  It's a 46 minute drive.

If you Google Vancouver WA to Kennewick, it want you to take the I-84 on the Oregon side of the river.   You have to drag the route 4 separate times to keep it in Washington, and increase the time by 16%.

If you Google White Salmon to Kennewick you cross into Hood River, OR.  If you Google Goldendale to Kennewick you go via Toppenish.  Does this mean that you can't get from Klickitat to Benton?

This map shows the most significant UA in each county, highlighted in solid red.  Other UA are outlined in red.  When there are multiple significant UA in a county (eg Port Angeles and Sequim in Clallam) each contains more than 10% of the county population. 

In cases where the largest place in the county that is in a UA is not included in the name of a UA, it is indicated in parentheses: P(V) for Portland (Vancouver), HR(WS) Hood River (White Salmon), Lewiston (CLarkston), Wenatchee (East Wenatchee), Seattle (Tacoma), Seattle (Everett)

In the 6 counties without a UA, the county seat is indicated in blue.  In all 6 counties, the county seat is also the most populous place (Carson River Valley CDP is more populous than Stevenson, Skamania)

The portions of the Seattle UA that are in Kitsap; Centralia UC in Thurston; and Kennewick-Pasco UA in Walla Walla represent less than 10% of the population of those counties, and are not considered to be a significant UA for those counties.

Seattle UA, Wenatchee UA, and Kennewick-Pasco UA cross county boundaries a significant portion of the population of each of the counties (a majority in all).  These UA are highlighted in green.

()

This is definitely an exercise in pruning the edges from the initial graph based on connections alone. But there is a difference between how an outside neutral observer would prune, and how a local entity would prune. My goal, in the spirit of the IA mapping process, is to apply criteria that are above the influence of local considerations. If the local jurisdiction wishes to weight other neutral factors differently, I provide for that by looking at the Pareto efficiency of the plan, so that more than one can be valid. My sense is that the changes you have suggested would fall within the scope of my smaller rule set and the Pareto choice. We can measure erosity both ways for a plan and see if there is an outcome-affecting difference.

In the case of links that you would leave but I would cut this only becomes relevant if the overall contiguity of a district relies on your link (eg Chelan-Snohomish). If such a plan looks otherwise Pareto efficient, it might be worth using a variable to measure unlinked pieces in my sense of unlinked. This is not desirable because the graph now must track two different types of edges. If there's a specific plan that looks like it really should be considered but I would exclude we can revisit this question.

For this graph-oriented algorithm, whether it's the county seat or the most populous UA, it only makes sense to have one node identified for each county. When a county must be divided it makes sense to associate each piece then with the largest UA within it for the purposes of calculating connecting edges. It doesn't make sense to start with two or more UAs at the first stage of pruning.

Forcing some counties to precombine when their largest UA is share between them is an interesting notion, and makes a judgement to sacrifice whole counties to maintain the UA in compact districts. I think I cover this with the Pareto choice. Generally counties that share a UA will be better connected and splitting the UA tends to create erosity so the trade of lower erosity for whole counties should work. This is worth testing and if it fails in more than rare instances a specific variable to count split UAs should be introduced separately from split counties and split municipalities.

Every state should do redistricting like Iowa or Washington state.

Fixed.

IA is hard to replicate without some changes to the criteria. They have a detailed list of criteria that works well given the homogenous demographics, grid-like geography and relatively dispersed population. Given their rules they have their state legal bureau draw a map and then ask the legislature to approve or reject it. If rejected they can try two more times based on specific comments from the legislature as it relates to the statutory criteria.

There's been a robust debate about erosity ad connectivity on another thread (https://uselectionatlas.org/FORUM/index.php?topic=166217.0), but it brings up an issue to consider here. If we assume some sort of connection map, whether by the definitions above or on the referenced thread, there is the question about counting the severed connection (erosity) or the maintained connection (connectivity). I think there are two strong reasons to support erosity as the better measure.

First, if there are no split counties or a plan makes a split into a number of apportionment regions of a whole number of districts then there is no relevant difference. For example if there is no weighting then a plan that has L total links and the regions cut x links, then the erosity is X and the connectivity is L-x. Comparing two different plans with the same number of regions is just a matter of preferring a lower number for erosity or a higher number for connectivity, but the ranking is the same. Adding weights to the links doesn't change this math. One suggestion was to modify weights based on the need to split certain counties, which could change the math by changing L for the plans. I think this creates an unneeded distortion and the ranking on that thread did not change based on the weighting method.


However there are reasons why measuring the cut links could have an advantage over the measuring the remaining enclosed links. Part of this starts with some geometrical theory. The links in the map are associated with boundary pieces between counties. As such they correspond to line segments on a map rather than areas. Suppose one has a nearly circular area to divide into a number of nearly circular pieces (I'm using nearly circular as approximate language for compact and when the number of divisions is large it's a very good approximation.) The total area enclosed does not change with the number of pieces but the total length of the boundaries do. In fact in the limit of a large number of circular pieces N the area is fixed but the boundary increases by a factor equal to sqrt(N) - 1. Since the erosity is tracking the boundary one would expect it to follow that relation based on the number of regions. Connectivity has no analogous formula related to the number of regions since it requires knowledge of the total link count L as well as the number of regions.

Of course in the case of equal region counts I've already claimed it doesn't matter, but if one wants to compare plans with different region counts it does. Here the recent discussion about TN (https://uselectionatlas.org/FORUM/index.php?topic=127884.msg3568212#msg3568212) is a good illustration. And for simplifying the discussion I'll use an unweighted set of connections as I did in that thread.

()

There were two plans of seven regions with erosities of 63 and 60 respectively.
()
()

However, Torie made a case for an eastern region that was based on the natural geography. My result was a plan that only has five regions but an erosity of 42.
()

Forcing a choice based on a higher number of regions excludes a potentially reasonable 5-region plan. Connectivity offers no direct way to compare a five-region plan to a plan with seven regions, but erosity does. Since the erosity should scale by sqrt(N) - 1, all the plans can be projected to the equivalent 9-district erosity by using that scale factor. The 7-region plan with erosity of 60 scales to an equivalent 9-district erosity of 72.9 (60 * 2 / 1.646). The 5-region plan with an erosity of 42 scales to an equivalent 9-district erosity of 68.0 (42 * 2 / 1.236). The scaling shows that though two additional county splits would be needed in a 5-region plan, the natural division of the east does provide a reduced erosity when scaled appropriately.

What if you include 1/2 the state perimeter?  Then the proportionality is sqrt N?  This would also permit you test the proportionality relationship going from a 1-region plan to a 95-region plan, where each county is a region (Tennessee has 95 counties).  Tennessee itself is not particularly compact, long and narrow, with an eastern border that is more east-west than north-south.  So I may be able to get low erosity estimates by north-south splits among fewer regions.

I think there is a conflict between adding unnecessary county splits, and reducing erosity.

In Iowa, if I merge two of the four districts, I am entitled to a scale factor of

(sqrt 4 - 1) / sqrt 3 - 1) = 1 / 0.732.

That is, if I reduce the erosity by more than 27%, I have improved the erosity estimate.  If there are 4 inter-district boundaries (it is unclear whether IA-4 (Story) and IA-2 (Jasper) connect), I can simply choose the most erose and am guaranteed a 25% improvement even if all 4 inter-district boundaries are identical.

So let's take the current Iowa plan, which appears to have an erosity measurement of about 35, and combine IA-2 and IA-4, reducing the erosity to 22.  My adjust erosity is 30 (22 * (1.00 / 0.73).

If I actually adjusted the boundary, say swapping Marshall and Worth, for Grady, Chickasaw, Floyd, and Butler, I could get an erosity of 28, with not horrible inequality (around 1%).  If I split a county, I could get the inequality down to a minimal amount.

For that matter, I could straighten out the boundaries of IA-3, and compensate IA-2 and IA-4 with a nibble of Polk and further reduce erosity.

The adjustment may favor combining larger districts, without any regard for how convoluted the boundary is.  Imagine a state with 12 square counties, 3 east-west and 4 north-south.  The state is divided into 6 districts, all square.  In the more arid west, the districts each contain 4 counties, while in the east, each county is a district.

The erosity is 9.  I combine the two western districts, reducing the erosity to 7.  Multiply this by 1.449 / 1.236 and my estimated erosity improves to 8.2.

It may be that it is better to estimate the erosity of each region, including its perimeter, and estimate the erosity of a N-district plan, rather than doing a global estimate.

In the Tennessee plan, I can reduce the estimated erosity of the 3rd plan by combining district four with the 3-district eastern region.  In the second plan, I could combine districts 5, 6, and 7; and 3 and 4 into two regions and improve estimated erosity.

Perhaps we are better off insisting on non-splitting of counties; rather setting overly tight population equality standards.  IIUC, you claim to be able to find the best trade-off between erosity and equality.  So why set an independent limit on equality.

Why not:

()

Or (avoiding the direct joining of Grays Harbor and Jefferson, at the expense of slightly more inequality).

()

On this last point, I wouldn't trade off that much. I think any neutral criteria is still constrained by federal law for population equality. Beyond a 1% range I don't think the court would go for CDs. For legislatures the range can be wider, 10% or more, but could be restricted by the state. For instance the IL court has ruled a 1% range to be the maximum for the legislature, but local jurisdictions can go to 10%.

Your earlier point about combining districts into a few large regions to lower erosity could be problematic. However, as I noted in the TN thread, one must look at the actual district erosity as well as the region erosity. Here each county split creates a new node in the county and it then has attendant links to other nodes. The link between nodes in one county adds to the erosity, as do any links that now would separate districts.

For instance the first 7-region plans has a district erosity of 68. It has cut links that appear between Williamson and Rutherford and between Davidson and Rutherford and there are now two links from Davidson to Cheatham instead of one. That plus the splits of Davidson and Shelby get it from 63 to 68. The second 7-region plan adds a second cut from Davidson to Williamson as well as the cuts for the split counties to go from 60 to 63. The 5-region plan add a number of additional cuts to go from 42 to 54, but it shows that it is not just better at the scaled region erosity, but is really better at the district level, too.

I haven't tried it yet, but I suspect that your IA proposal would show a better scaled region erosity, but would not improve the district erosity. Since every split county creates at least one new cut link, the TN combinations you suggest would result in higher district erosity. Only if those combinations could result in lower ranges would they make sense for consideration, since they would not improve either splits or erosity.

Because 18.3% deviation pretty blatantly spits in the face of "one man, one vote".  I could see going up to, say, 3 or so percent deviation in the name of keeping political and natural boundaries together (and no, counties are not the be-all and end-all of boundaries), but when you get to double-digit variation I would hope that gets recognized as obviously unfair in all quarters.

I don't believe your erosity estimate is robust enough or accurate enough to be used as a metric.

First, Tennessee is not shaped like a circle, and 10 is not a large number.

And while the total perimeter of a large number of circles does approach a limit of sqrt(N) * 2*pi*R, where R is the radius of the containing circle, it might not do so monotonically.   7 small circles can be placed in a hexagon shape, with rounded vertices that would have a relatively small outer bounding circle.  Add in an 8th and the bounding circle will expand quite a bit, remove a 7th, and the shrinkage will be small.  Going from 6 to 7 had little cost, going from 7 to 8 a large amount.

This can be illustrated with an idealized-Iowa-like state which has infinite granularity but where boundaries are restricted to North-South and East-West lines.   We draw equal sized districts (the population density is non-varying).

()

P is the total perimeter of the districts, so the shared (green) perimeters are counted twice, and the outer black perimeter is counted one.  This allows us to use sqrt(N) for our estimate, rather than sqrt(N)-1.   This has the advantage that sqrt(1) is non-zero, while sqrt(1)-1 is zero.

So the perimeter of our single district state is 4.00, while for a 2-district state it is 6.00.  But E, the estimate of the 2-district perimeter, is sqrt(2/1)*4.00 or 5.66.  We grossly underestimated our total perimeter.   This is not that surprising.   Our districts are non-compact by Iowa standards, with a length twice their width.

We continue to 3 district, where P is 7.33.  Our estimate is based on the 2-district case.  That is E(N+1) = P(N)*sqrt((N+1)/N).   E(3) = 6.00 * sqrt(3/2) = 7.35.  In this case our estimate is good.   We have the very-elongated district L:W = 3:1, and two somewhat compact districts L:W = 4:3.  This really says that 3 districts are about as good or about as bad as 2 districts.  If we based our estimate for 3 districts on the 1 district case, our estimate would be 4.00*sqrt(3/1) or 6.93, which would have been too optimistic.

But for a 4-district case P is 8.00, but our estimate of P(3)*sqrt(4/3) = 8.47 is much higher.  Our districts are all maximally compact.

As we continue on, the estimate for 5 districts is quite low, followed by slightly high, slightly low, slightly high, and then quite high when we once again reach a compact symmetric case for 9 districts.

There's no question that for any particular shape of a state and the type of cuts that are allowed then there will be exactly that type of oscillation with respect to the large N ideal. However, when I started with some other shapes, such as hexagons or rectangles, I found that the oscillations peaked for different values of N.

If I consider the wide range of state shapes and types of county shapes as cuts, the right value to use is the weighted average of all possible choices. A more detailed study would involve modeling all 50 state shapes and their county shapes to get the best statistical average. Since I observed a smoothing with just three choices, I'm willing to speculate that the smoothing will persist when applied to a larger set.

The first example really wasn't how we were subdividing regions.  In this 2nd example, we nibble off smaller regions one at a time.

()

Somewhat surprising is the two-district case, where we do better than estimated, and much better than where we split into two equal areas.  But it less surprising when we recognize that both regions are fairly compact, with only a corner missing from an almost square larger district.  And of course smaller areas have smaller perimeters (P1/P2 = sqrt(A1/A2) for two similar polygons.  The perimeter of the small region with 1/9 the of the total area, has a perimeter of 1/sqrt(9) or 1/3 of that of the total area.

Adding a 3rd district is not quite as compact as expected base on the two-district case.  But our larger regions is beginning to become less compact.  Our estimate based on two-districts was 6.53.  We only did 6.67.   But an estimate based on 1-district is 6.93, and when we create 3 equal area districts the perimeter is 7.33.

Adding a 4th district is quite efficient as we only have to chop off the panhandle of our large region.   Our 4 districts are are as compact as 3 equal area districts can be, and better than the 4-district symmetric equal-area case of our first set of examples.  That is, we can get a smaller perimeter by having greater variation in region size.

As we continue on, we are worse or better than the estimate based on the previous case E(N+1) = P(N)*sqrt((N+1)/N), depending on if we are able to cut off a panhandle of larger district, or the new district is cut off from the body. In all cases, other than for N=8, we are better than the equivalent equal-area version for the same number of districts.

It is instructive to look at the 4-district case in more detail.  An estimate based on 1 district would be that we would have a perimeter of 8.00.  We managed to do better, with a 7.33.  And based on the 4-district case, we should be able to do 9-districts in 7.33*sqrt(9/4) with a perimeter of 11.00.   We managed to do only 12.00 even with maximally compact districts.

What went wrong?  We essentially gamed the system by creating districts for grossly different size, and then were unable to maintain that when subdividing the larger region.  

If we use your proposed estimate method, which only measures interior perimeters, our estimate is even worse.  The interior perimeter of the 4-district case is 1.67.  The estimate for the 9 district case would be 1.67 * (sqrt(9) - 1)/(sqrt(4) - 1) = 3.33.  But in reality it is 4.00.

If we adjust these numbers so they are equivalent to my values for total perimeter, the estimate for the  9-district case would be 10.67, which is even further from my estimate of 11.00, and the actual value of 12.00.  Your estimate is based on how well you did in creating the first three small districts, with no knowledge of the area in the larger district beyond other than one side and an idea of its area (because we knew that it could be divided into 6 more areas).

We could do better, by making an estimate for dividing the larger area into 6 districts.  The total perimeter for the 4-district case is 7.33, 4.00 of which is the total perimeter of the 3 small districts, and 3.33 for the 4th district.  The estimate for the total perimeter after dividing the larger area into 6 districts is 3.33 * sqrt(6/1) = 8.16.  Add in the 4.00 for the original 3 districts, and the estimate is 12.16, which is just above the actual 12.00.   We went from a slightly non-compact area (length:width of 3:2) to 6 maximally compact districts.

In your Tennessee example, a better estimate of dividing the eastern edge of the state into 3 districts could be obtained from using that area alone, and not basing it in part on the rest of the state which had a mix of district sizes between smaller ones in the Memphis and Nashville areas, and the more rural areas between the mountains and the Mississippi.

If we use county-county links to measure erosity, this is problematic, since it would require determining links between counties in Virginia, North Carolina, and Georgia, and those in Eastern Tennessee.  There may be scaling problems because of county size styles.   And how well we divide eastern Tennessee is probably only marginally related to how easy it is to travel across the mountains between Tennessee and North Carolina.   A better approach would be to calculate the internal erosity among all counties in the area being divided, and estimate downward.

There are 25 counties in your 3-district Eastern region, 6 counties in your greater Nashville region 2-district region, and 14 counties in your Western-Memphis region.   The inter-county link counts for these 3 regions is 50, 9, and 26 respectively.

We can estimate the erosity if each of these county areas were consolidated into N regions:

Erosity(N)/County_Links =  (sqrt(N) - 1) / (sqrt(ncounties) - 1)

For the eastern region:

Erosity(3) = 50 * (sqrt(3) - 1) / (sqrt(25) - 1) = 9.15

For the greater Nashville region:

Erosity(2) = 9 * (sqrt(2) - 1) / (sqrt(6) - 1) = 2.57

For the western-Memphis region:

Erosity(2) = 26 * (sqrt(2)-1) / (sqrt(14) - 1) = 3.93

These are reasonably close for the first two, if we ignore your proposed county splits (10 and 3).  As to be expected, we missed badly in the 3rd, because there we simply chopped part of Shelby County from the region.

Add these to the 5-region erosity for a final estimate:

42 + 9.15 + 2.57 +3.93 = 57.65

BTW, why are links preferred over perimeter?

There are three reasons to consider links.

1) Perimeter-based compactness measures have a well-known bias against subdivisions based on irregular natural features such as rivers and mountains. County-based division runs into this problem when counties use those same natural divisions. The other large class of compactness measures are based on bounding shapes like circles or polygons, which are weak at penalizing peninsulas jutting into a larger district. Links don't penalize natural irregular boundaries, but do weigh against jutting peninsulas.

2) County or municipal integrity are proxies for communities of interest. Links also are proxies for communities of interest with respect to associations between counties. Perimeter and area measures don't add anything to identify potential communities of interest. Links can add more than just a compactness measure.
 
3) Perimeter calculations or bounding methods require GIS software with enough sophistication to calculate lengths and areas. That's fine for those who are experts in the field, but I'm looking to make redistricting more accessible to the public. I'd like users to be able to proceed with a spreadsheet and standard mapping software like Mapquest or Google Maps. Links can be simply counted by the mapper using nothing more sophisticated than a highway map.

Any redistricting process that allows meaningful citizen input will have to provide the GIS and demographic data, and likely at least simple software.  The Census Bureau has road and street layers, and it image data can be meshed in as well.  The software used in the Ohio redistricting commission supported image data, but the Ohio sponsors did not wish to pay the licensing fees to Google/Bing.

There are ways to simplify borders for calculating perimeter.  This has the additional advantage of resolving whether one can directly link Lincoln and Okanagan counties, since there is an extreme penalty for doing so.

()

It's not only the cost and availability, but also the level of sophistication needed by the user. Most perimeter calculations are not simple to understand without a reasonable background in geometry. Counting links is not hard, just like counting county splits is an easy thing fr a user to see and understand. The math of counting links gets only a little more complicated when comparing plans with different region counts, and is still less complicated than understanding a purely geometrical concept.

Eve if the math for perimeters were easy to understand, it leaves the problems associated with perimeter-based formulas as I noted in point 1.

Relative compactness measures are used in an attempt to restrict abuse by gerrymanders that use block assignment.  But the true abuse is block assignment.  Total perimeter length is just as effective (more effective in actuality) as county links in measuring compactness of county-based plans.

Perimeter is a grade-school math concept.   Given a simplified outline anyone could measure it on a map.  Not they wouldn't have to, since any reasonable software would be calculating it as they clicked, particularly it it were a contest criteria.

The map of Washington was not necessarily what a contest entrant would see.  It was a presentation of the data that was being used for the contest.  Both the actual borders, and the simplified borders would be available, as well as information about whether the border was passable or not.

It is certainly easier to measure the length of the Okanogan-Lincoln border or the Columbia-Franklin border than to try to figure out why I think one is "1" and the other is "0", and you think the opposite.