City of Hudson's weighed voting system under scrutiny

[jimrtex]

Livingston County

Livingston County has a board of supervisors using weighted voting to balance voting power with population.   Livingston has no cities, and the largest town, Geneseo, only has 16.45% of the population.

Using voting weights proportional to population would give Geneseo 7.91% relative extra power, and every other town a negative error.  With adjusted weights, the match of power to population is quite good, with a standard deviation of 0.53%.  With 17 towns and no huge concentrations of population, this is about as good as weighted voting gets.

The town of Groveland apparently has a prison which contains over half of its population.

Wayne County

Wayne County adjusts voting weights so that voting power matches population.  The largest town, Arcadia has 52.4% of the population.  Splitting it in two would eliminate the need to use anything but simple weights.

Saint Lawrence

Saint Lawrence County has put its legislative history since the 1960s online.  It scanned its old typewritten minutes into PDFs, and created an index.  It is not at all clear which local laws went into effect, or were turned down in a referendum, or overturned by a court, because a decade later, the votes don't appear to be based on the law previously passed.

In 1966, weighted votes for the board of supervisors were established.  St.Lawrence has 32 towns and one city, Ogdenburg.  Ogdenburg had 4 wards, so under a traditional configuration had 36 supervisors.  The smallest town, Clare, has just over 100 persons, and the 5 largests towns (city), Potsdam, Massena, Canton, Ogdenburg city, and Gouverneur contain a majority of the population.  Clare had one vote and the larger towns dozens of votes.

By the 1980s, the laws provided for weighting of the votes of 22 legislators, so somewhere in between the board of supervisors was abandoned and replaced by a county legislature.  The weights were approximately equal, so it appears that perhaps equal-population districts had been created, but it was difficult to maintain equality, particularly when splitting was restricted to the largest towns.  In one proposal, the weights were mostly 10, with a few ranging to as much as 8 or 12, indicating that the weights were based on populations rounded to 10% of a quota.  In another proposal, the weights were numbers such as 4.7 and 5.1.   Since they added to 100.0, they must have been the share of the county population rounded to 0.1%.  It appears a goal was to provide smaller numbers that could be added up by hand.

In the 1990s, the debate was over switching to 11, 13, or 15 districts.   It must have been decided that a large board did not provide that much representation to the smaller towns, but rather multiple members to the largest towns, as well as added expense to pay that many legislators.  The votes on the various proposals did not appear to be weighted, so it appears that weighting never occurred.

The 2000s discussion included much more extensive minutes.  One member mentioned the one-year term, and limits set by a judge on the amount of permitted deviation, so the previous districts must have been overturned in court, and new elections ordered.

The current legislature has 15 members.

[jimrtex]

Iannucci vs. Board of Supervisors of Washington County and Saratogian, Inc. vs. Board of Supervisors of Saratoga County, 20 N.Y. 2d 244, 299 NE 2d 195, 282 NYS 2d 502, 1967 were a pair of cases decided in 1967 by the New York Court of Appeals, the state's highest court.

The court overturned lower court decisions that said weighted voting for county boards of supervisors was unconstitutional, but at the same time said that simple weighting based on population was not valid, and that instead that it was voting power that should be proportional to population.  It didn't really say that the counties schemes were invalid, but rather that the counties had not offered any evidence that their apportionment scheme would produce valid results, and that it would likely be expensive to prove it.

John Banzhaf III (re)creator of the Banzhaf Power Index was an amicus curiae and they cited his  law review article 'Weighted Voting Doesn't Work: A Mathematical Analysis'.  That article had included the example of the Nassau County Board of Supervisors, whose weighted voting scheme very badly did not work, since the votes of the supervisors from the two cities were totally irrelevant (Nassau County had (has) only three towns and two cities, and over half the population was in the Town of Hempstead.

It turns out that the counties schemes of simple weighting were valid, or were close to it.

In Washington County, one vote was apportioned for every 279 persons.  In addition, towns with more than 15 votes were given additional supervisors with the vote divided between them.  For example, Kingsbury, the largest town was apportioned 39 votes, and given 3 supervisors, each casting 13 votes.  The smallest towns had 2 votes.  The magnitude of votes was as if they were apportioning whole numbers of members to each town, and wanted to make sure every town had at least two.

If the populations of the towns had been used as the voting weights, we see the familiar pattern of the largest entity, in this case Kingsbury, being overpowering, while every other town is about equally underpowered.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Kingsbury      11012   41480  22.72%  27.27%  20.03%
Fort Edward     6523   19274  13.46%  12.67%  -5.85%
Granville       5015   14930  10.35%   9.81%  -5.14%
Whitehall       4757   14144   9.81%   9.30%  -5.26%
Greenwich       3969   11752   8.19%   7.72%  -5.65%
Fort Ann        3124    9266   6.44%   6.09%  -5.49%
White Creek     2365    6950   4.88%   4.57%  -6.36%
Salem           2258    6642   4.66%   4.37%  -6.27%
Argyle          1898    5564   3.92%   3.66%  -6.59%
Easton          1681    4918   3.47%   3.23%  -6.78%
Cambridge       1610    4712   3.32%   3.10%  -6.74%
Hartford        1058    3084   2.18%   2.03%  -7.12%
Hebron          1026    2984   2.12%   1.96%  -7.33%
Jackson          795    2342   1.64%   1.54%  -6.13%
Putnam           490    1460   1.01%   0.96%  -5.06%
Hampton          469    1378   0.97%   0.91%  -6.38%
Dresden          426    1254   0.88%   0.82%  -6.20%

But if we divide the votes for the largest towns among more than one supervisor, the results are much better.  The largest errors are for the smallest towns.  But that is not due to weighted voting, but rather apportionment error.  It appears that the quota of 279 was chosen so that Dresden would be entitled to slightly more than 1.5 votes, which would be rounded to 2.  White Creek does very poorly because its entitlement of 8.47 is rounded down to 8.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Kingsbury         13  536040  22.72%  22.71%  -0.03%
                  13  536040
                  13  536040
Fort Edward       12  492064  13.46%  13.29%  -1.25%
                  11  448836
Granville          9  364212  10.35%  10.29%  -0.56%
                   9  364212
Whitehall          9  364212   9.81%   9.70%  -1.15%
                   8  322640
Greenwich         14  580902   8.19%   8.20%   0.20%
Fort Ann          11  448836   6.44%   6.34%  -1.64%
White Creek        8  322640   4.88%   4.56%  -6.60%
Salem              8  322640   4.66%   4.56%  -2.18%
Argyle             7  281498   3.92%   3.98%   1.54%
Easton             6  240646   3.47%   3.40%  -1.99%
Cambridge          6  240646   3.32%   3.40%   2.33%
Hartford           4  159884   2.18%   2.26%   3.46%
Hebron             4  159884   2.12%   2.26%   6.68%
Jackson            3  119762   1.64%   1.69%   3.13%
Putnam             2   79758   1.01%   1.13%  11.43%
Hampton            2   79758   0.97%   1.13%  16.42%
Dresden            2   79758   0.88%   1.13%  28.17%

If we instead apportion 1000 votes, in effect giving each town one vote for each 0.1% of the population (48.476 persons) we get very good conformance.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Kingsbury         76  540471  22.72%  22.99%   1.22%
                  76  540471
                  75  532843
Fort Edward       68  479915  13.46%  13.57%   0.86%
                  67  472539
Granville         52  362635  10.35%  10.23%  -1.09%
                  51  355475
Whitehall         49  341167   9.81%   9.72%  -0.92%
                  49  341167
Greenwich         82  587125   8.19%   8.37%   2.18%
Fort Ann          64  450279   6.44%   6.42%  -0.44%
White Creek       49  341167   4.88%   4.86%  -0.36%
Salem             47  326897   4.66%   4.66%   0.00%
Argyle            39  270193   3.92%   3.85%  -1.67%
Easton            35  241983   3.47%   3.45%  -0.57%
Cambridge         33  227937   3.32%   3.25%  -2.21%
Hartford          22  151599   2.18%   2.16%  -1.03%
Hebron            21  144959   2.12%   2.07%  -2.41%
Jackson           16  110069   1.64%   1.57%  -4.37%
Putnam            10   68705   1.01%   0.98%  -3.15%
Hampton           10   68705   0.97%   0.98%   1.19%
Dresden            9   61825   0.88%   0.88%   0.24%

The standard deviation for the error is 1.63% (population weighted 1.38%).

So had Washington County used a finer apportionment, they likely could have used population-weighted voting for their board of supervisors.

[jimrtex]

The plan in Saratoga County was similar.  One vote was apportioned for each 600 persons.  Towns with more than 20 votes were given an extra supervisor, with the vote for the town split between the two supervisors.  Using the population of each town as its voting weight, we repeat the pattern of the largest town having too much power.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Saratoga Spr.  16630  624945  18.67%  22.15%  18.68%
Moreau          8406  256993   9.43%   9.11%  -3.45%
Waterford       7231  220395   8.12%   7.81%  -3.74%
Milton          7114  216695   7.98%   7.68%  -3.80%
Mechanicsville  6831  207885   7.67%   7.37%  -3.89%
Ballston        5752  174445   6.46%   6.18%  -4.22%
Corinth         5167  156505   5.80%   5.55%  -4.34%
Clifton Park    4512  136317   5.06%   4.83%  -4.58%
Stillwater      4416  133433   4.96%   4.73%  -4.57%
Halfmoon        4120  124361   4.62%   4.41%  -4.67%
Saratoga        3515  105983   3.95%   3.76%  -4.77%
Charlton        3024   91069   3.39%   3.23%  -4.89%
Greenfield      2548   76669   2.86%   2.72%  -4.97%
Malta           2223   66831   2.50%   2.37%  -5.05%
Wilton          1902   57239   2.13%   2.03%  -4.96%
Galway          1746   52479   1.96%   1.86%  -5.07%
Northumberland  1353   40619   1.52%   1.44%  -5.19%
Hadley           982   29533   1.10%   1.05%  -5.02%
Edinburg         602   18057   0.68%   0.64%  -5.27%
Providence       556   16645   0.62%   0.59%  -5.45%
Day              466   13983   0.52%   0.50%  -5.23%

Splitting Saratoga Springs between two supervisors considerably improves the situation, but there is a problem with the coarseness of the apportionment.  Northumberland has 38% more population than Hadley, but each has 2 votes.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Saratoga Spr.     14  626781  18.67%  19.07%   2.17%
                  14  626781
Moreau            14  626781   9.43%   9.53%   1.06%
Waterford         12  529801   8.12%   8.06%  -0.70%
Milton            12  529801   7.98%   8.06%   0.94%
Mechanicsville    11  482867   7.67%   7.35%  -4.19%
Ballston          10  436753   6.46%   6.64%   2.91%
Corinth            9  391329   5.80%   5.95%   2.65%
Clifton Park       8  346515   5.06%   5.27%   4.09%
Stillwater         7  302187   4.96%   4.60%  -7.25%
Halfmoon           7  302187   4.62%   4.60%  -0.59%
Saratoga           6  258227   3.95%   3.93%  -0.43%
Charlton           5  214745   3.39%   3.27%  -3.75%
Greenfield         4  171459   2.86%   2.61%  -8.80%
Malta              4  171459   2.50%   2.61%   4.54%
Wilton             3  128403   2.13%   1.95%  -8.50%
Galway             3  128403   1.96%   1.95%  -0.33%
Northumberland     2   85513   1.52%   1.30% -14.34%
Hadley             2   85513   1.10%   1.30%  18.02%
Edinburg           1   42729   0.68%   0.65%  -3.80%
Providence         1   42729   0.62%   0.65%   4.16%
Day                1   42729   0.52%   0.65%  24.28%

Increasing the total voting weight to approximately 1000, so that each vote represents 0.1% of the county's population (89.096 persons) improves matters considerably.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Saratoga Spr.     94  629230  18.67%  19.08%   2.21%
                  93  621766
Moreau            94  629230   9.43%   9.60%   1.70%
Waterford         81  534826   8.12%   8.16%   0.49%
Milton            80  527746   7.98%   8.05%   0.79%
Mechanicsville    77  506706   7.67%   7.73%   0.78%
Ballston          65  423730   6.46%   6.46%   0.09%
Corinth           58  376442   5.80%   5.74%  -1.01%
Clifton Park      51  329706   5.06%   5.03%  -0.72%
Stillwater        50  323062   4.96%   4.93%  -0.60%
Halfmoon          46  296698   4.62%   4.52%  -2.16%
Saratoga          39  250762   3.95%   3.82%  -3.07%
Charlton          34  218310   3.39%   3.33%  -1.92%
Greenfield        29  185890   2.86%   2.83%  -0.88%
Malta             25  160058   2.50%   2.44%  -2.18%
Wilton            21  134394   2.13%   2.05%  -4.00%
Galway            20  127926   1.96%   1.95%  -0.45%
Northumberland    15   95946   1.52%   1.46%  -3.65%
Hadley            11   70290   1.10%   1.07%  -2.75%
Edinburg           7   44718   0.68%   0.68%   0.92%
Providence         6   38310   0.62%   0.58%  -6.38%
Day                5   31902   0.52%   0.49%  -6.99%

Even better is to simply give two supervisors to Saratoga Springs and make the voting weight the same as the population.  The standard deviation for the error is 1.49%, with a weighted value of 1.65%.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Saratoga Spr.   8315  624945  18.67%  19.06%   2.10%
                8315  624945
Moreau          8406  632555   9.43%   9.65%   2.24%
Waterford       7231  536669   8.12%   8.18%   0.84%
Milton          7114  527237   7.98%   8.04%   0.69%
Mechanicsville  6831  504975   7.67%   7.70%   0.44%
Ballston        5752  421231   6.46%   6.42%  -0.50%
Corinth         5167  376703   5.80%   5.74%  -0.95%
Clifton Park    4512  327739   5.06%   5.00%  -1.31%
Stillwater      4416  320623   4.96%   4.89%  -1.35%
Halfmoon        4120  298719   4.62%   4.56%  -1.49%
Saratoga        3515  254049   3.95%   3.87%  -1.80%
Charlton        3024  218155   3.39%   3.33%  -1.98%
Greenfield      2548  183523   2.86%   2.80%  -2.14%
Malta           2223  159929   2.50%   2.44%  -2.25%
Wilton          1902  137069   2.13%   2.09%  -2.09%
Galway          1746  125517   1.96%   1.91%  -2.33%
Northumberland  1353   97189   1.52%   1.48%  -2.40%
Hadley           982   70555   1.10%   1.08%  -2.38%
Edinburg         602   43255   0.68%   0.66%  -2.38%
Providence       556   39795   0.62%   0.61%  -2.76%
Day              466   33501   0.52%   0.51%  -2.33%

[jimrtex]

Dobish v. State of N. Y. 53 Misc.2d 732 (1967) was decided by the Supreme Court of Wayne County.  Wayne County had proposed weighting the votes of the town supervisors, by weighing the vote to the nearest 1000.  The judge had a computer analysis done "by the New York Scientific Center, a division of International Business Machines Corporation, on an IBM 360 Computer, Model 40.".  50 years later, this is easily in range of a spreadsheet on a PC.

The analysis had found that for Wayne County, the voting power was not proportional to the population.  In a quite predictable by now result, the largest town of Arcadia had 11.9% additional power.  The tables in the opinion are remarkably similar to those that I have been producing, including using a fixed pitch typeface.  Because of the error, the judge ruled the plan unconstitutional.

But the judge had also gone further and had IBM adjust the weights for the largest towns downward, anticipating the adjusted voting weights now used in Hudson and Columbia County.  He did have a small error, in that he compared the voting power to the original weights rather than to the population.

The plan proposed by Wayne County, and ruled unconstitutional.

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Arcadia          125    8454  17.61%  19.61%  11.36%
Sodus             79    4758  11.02%  11.04%   0.13%
Palmyra           69    4130   9.62%   9.58%  -0.41%
Williamson        61    3602   8.54%   8.36%  -2.14%
Lyons             59    3466   8.24%   8.04%  -2.41%
Ontario           54    3194   7.53%   7.41%  -1.65%
Macedon           48    2814   6.70%   6.53%  -2.53%
Galen             45    2626   6.32%   6.09%  -3.58%
Walworth          35    2046   4.96%   4.75%  -4.34%
Wolcott           35    2046   4.90%   4.75%  -3.14%
Marion            32    1858   4.53%   4.31%  -4.93%
Rose              22    1286   3.13%   2.98%  -4.72%
Savannah          17     970   2.45%   2.25%  -8.10%
Butler            16     926   2.23%   2.15%  -3.59%
Huron             16     926   2.20%   2.15%  -2.55%

By reducing the voting weight for the larger town - each iteration I reduced the weight of the town with the largest positive error by one - I was able to reduce the maximum error for the largest towns to less than 1%.  The errors for some smaller towns is due to lack of resolution in the original weights.   One could start with the population as the original weights, but then you end up with adjusted voting weights, that look like adjusted population.  The standard deviation for the error reduced from 4.09% (5.50% weighted), to 1.61% (1.29% weighted).

Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Arcadia          113    7820  17.61%  17.77%   0.86%
Sodus             77    4896  11.02%  11.12%   0.90%
Palmyra           68    4264   9.62%   9.69%   0.68%
Williamson        60    3768   8.54%   8.56%   0.23%
Lyons             58    3624   8.24%   8.23%  -0.08%
Ontario           54    3348   7.53%   7.61%   0.95%
Macedon           48    2960   6.70%   6.72%   0.40%
Galen             45    2798   6.32%   6.36%   0.60%
Walworth          35    2136   4.96%   4.85%  -2.21%
Wolcott           35    2136   4.90%   4.85%  -0.98%
Marion            32    1956   4.53%   4.44%  -1.99%
Rose              22    1334   3.13%   3.03%  -3.22%
Savannah          17    1030   2.45%   2.34%  -4.45%
Butler            16     974   2.23%   2.21%  -0.70%
Huron             16     974   2.20%   2.21%   0.37%

[jimrtex]

Dobish v State of New York 54 Misc.2d 367 (1967) was the follow on case to Dobish v. State of N. Y. 53 Misc.2d 732 (1967), which had been decided only a couple of months earlier.

In the prior case, the judge had declared the weighted-voting plan proposed for the Wayne County Board of Supervisors unconstitutional because if failed to produce voting power proportional to population, and had used computerize analysis to demonstrate this.

In this case, the judge relying on the expert testimony of Lee Papayanopoulos who had produced two alternatives, ordered Wayne County to adopt one of them.   Lee Papayanopoulos is the consultant who produced the the 1975, 2004, and 2013 plans for Hudson, and is currently a professor at Rutgers Business School.  In 1967 he was an IBM Systems Engineer and mathematics and special research specialist.  He is also author of 'Computerized weighted voting reapportionment; ACM AFIPS '81 Proceedings of the May 4-7, 1981, national computer conference.

I have not found a direct link between Papayanopoulos and Banzhaf, though Banzhaf has a BSEE from MIT.  At the time he wrote the article that weighted voting does not work, Banzhaf was editor of the Columbia Law Review.  Now he is more known for his anti-tobacco litigation.

I did come across this BBC broadcast that includes an interview with Banzhaf about 6 minutes in.
BBC broadcast on electoral methods

[jimrtex]

Slater v. Cortland County Board of Supervisors 66 Misc.2d 108 (1971) found that Cortland County violated OMOV.  At the time, Cortland County continued to use the traditional form with one supervisor for each of the 15 towns and 6 wards of the city of Cortland.  The 11 smallest towns with about 1/5 of the total population could control a majority on the board.

It is interesting that the configuration had lasted until after the 1970 census.

The judge ordered that weighted voting go into effect immediately for the board of supervisors, with one vote per 100 persons, and ordered the county to implement a constitutional apportionment within 3 months.

Slater v. Cortland County Board of Supervisors 42 A.D.2d 795 (1973) was a follow-up where the judge approved a 19-district legislature with districts comprised of groups of smaller towns, or parts of larger towns and cities.  There was about a 25% difference between the largest and smallest districts, so that there was also weighting applied.

By 1990, Cortland County had 19 legislative districts with the voting weight of each identical to their population.  The city of Cortland had 8 districts, while the towns of Cortlandville and Homer had 3 each.  The other 13 towns were combined in 5 multi-town districts.  The division of the city of Cortland was quite balanced, with districts ranging in population from 2451 to 2496 (the largest only 1.8% larger than the smallest).  The results were quite interesting.

District        Vote   Swing  R.Pop.   R.Pow    Dev. 
17              3244   52076   6.63%   5.66% -14.59%
19              3058   51964   6.25%   5.65%  -9.60%
16              2922   51268   5.97%   5.57%  -6.65%
13              2853   50672   5.83%   5.51%  -5.51%
15              2744   49896   5.60%   5.42%  -3.26%
18              2632   48924   5.38%   5.32%  -1.11%
12              2614   48832   5.34%   5.31%  -0.61%
14              2587   48652   5.28%   5.29%   0.05%
2               2496   47728   5.10%   5.19%   1.73%
4               2491   47676   5.09%   5.18%   1.82%
7               2480   47636   5.07%   5.18%   2.19%
1               2479   47628   5.06%   5.18%   2.21%
6               2475   47596   5.05%   5.17%   2.31%
5               2474   47592   5.05%   5.17%   2.34%
8               2455   47484   5.01%   5.16%   2.90%
3               2451   47444   5.01%   5.16%   2.98%
9               2315   46344   4.73%   5.04%   6.50%
11              2172   45648   4.44%   4.96%  11.81%
10              2021   45264   4.13%   4.92%  19.16%

Unlike the other cases we've looked at, where the largest district was overpowered, it is the smallest districts that have the extra power.  I suspect what is happening is that the populations are similar enough that the distribution of the total vote for the various combinations is quite close to a normal distribution, with the effect that the combinations where the largest districts are critical are starting to become less probably.  In addition, a simple majority in a 19-member equal weight body is 10/19 (52.6%).  The extra 2.6% means that most 10-member combinations can still achieve a majority, give the relatively overall variation in Cortland.   For example, the combination of the 3rd through 12th smallest districts can produce a majority.

In 2010, the legislature was reduced to 17 members.  One district in the city of Cortland was eliminated; and Homer was combined with Preble and Scott, then divided into 3 districts.  Previously Homer had 3 districts of its own, and Preble and Scott had been its own district.  Population equality within divided towns was quite severe.  For example. the range for the 7 districts in the city of Cortland was from 2740 to 2747; for the three districts formed from parts of Homer, 2989 to 2995; and for the three districts in Cortlandville, 2819 to 2853.  The districts continued to be weighted by their population, but this has no effect.

District        Vote   Swing  R.Pop.   R.Pow    Dev. 
17              3379   12870   6.85%   5.88% -14.11%
14              3344   12870   6.78%   5.88% -13.21%
16              3192   12870   6.47%   5.88%  -9.08%
10              2995   12870   6.07%   5.88%  -3.10%
8               2990   12870   6.06%   5.88%  -2.94%
9               2989   12870   6.06%   5.88%  -2.91%
13              2853   12870   5.78%   5.88%   1.72%
11              2837   12870   5.75%   5.88%   2.30%
12              2819   12870   5.71%   5.88%   2.95%
2               2747   12870   5.57%   5.88%   5.65%
1               2744   12870   5.56%   5.88%   5.76%
5               2744   12870   5.56%   5.88%   5.76%
3               2743   12870   5.56%   5.88%   5.80%
4               2743   12870   5.56%   5.88%   5.80%
7               2743   12870   5.56%   5.88%   5.80%
6               2740   12870   5.55%   5.88%   5.92%
15              2734   12870   5.54%   5.88%   6.15%

Because of the high level of equality, and the reduced size of the legislature, there are no 8-member combinations that command a majority, with the largest falling short at 49.8%.  And as a corollary, all 9-member combinations form a majority.

So the result is the same as if the districts were treated as having equal population.  The range in population from 16% above to 6% below the ideal is pushing the OMOV limits, but may be acceptable given the emphasis in New York of recognizing the significance of towns within the overall government structure.

[jimrtex]

It appears that Nassau County's weighted voting for the Board of Supervisors was introduced in 1936, much earlier than I had thought.  After a constitutional amendment was passed in 1935, county charters were granted to four large counties: Monroe, Westchester, Nassau, and Suffolk, one containing Rochester and the other three in the New York City suburbs.

Unlike modern county charters which are written and approved by the county citizens, these were written by the legislature.  Nassau County only had three towns and two cities, and it appears to have tried to compensate for this.

The largest town, Hempstead had 60% of the population and was given two supervisors on the board of supervisors.  The other two towns and two cities were given one supervisor each.  The voting weight of each entity was its census population divided by 10,000.  The charter specified the whole number of the quotient - it is unclear if that means truncation, by dropping the fractional remainder; or rounding to the nearest whole number.  Each entity was guaranteed one vote.  The votes for Hempstead were to be divided evenly between the two supervisors, which suggests that an even number total.  

And finally, no town could have a majority on the board.  In 1937, the county attorney interpreted this to mean that the Hempstead representation would be reduced so that it was less than a majority, without increasing the representation of the other units.  In addition a majority was considered to be a majority of the originally apportioned totals.  In effect, the votes were taken from Hempstead - and converted to permanent Noes.

As might be expected the result was not very good.  Hempstead, based on population would be entitled to 19 votes on a 31-vote council.  But the charter required its representation to be reduced to an even number less than a majority, or 14, which was divided between the two supervisors.  But the majority threshold was kept at 14.  This increased the power of North Hempstead and Oyster Bay since they became critical members of more coalitions.  Since the two Hempstead members could not form a majority, they needed another member to join them.  And since the majority threshold remained at 16, the four smaller entities could still not win.

Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead          7      13  61.62%  52.00% -15.61%
                   7      13
North Hempstead    6      11  20.53%  22.00%   7.19%
Oyster Bay         4      11  12.17%  22.00%  80.83%
Glen Cove          1       1   3.77%   2.00% -46.97%
Long Beach         1       1   1.92%   2.00%   4.20%

With only 6 members, there are only 64 voting combinations, with a total of 192 members voting No.  But since two members have the same vote, there are only 48 unique combinations of voting weights (for every combination where Hempstead 1 votes Aye, and Hempstead 2 votes Nay; there is an equivalent combination where the two votes are reversed).  And in this particular case, the number of votes for the two cities was also the same, reducing the combinations of voting weights to 36.  There is simply not enough material to work with, even when manipulating weights.

Historical tidbits: Queens County used to include the area that is now Nassau County.  First the western part of the county was made a borough of New York City, and then the eastern part of the county was detached to form Nassau County.  It was named after the Dutch name for Long Island, Isle of Nassau.

The original Queens County had 6 towns, Flushing, Jamaica, and Newtown were dissolved when Queens became part of New York City.  Flushing and Jamaica are still recognized areas of Queens.  Newtown was in the northern part of Queens, to the west of Flushing (opposite Manhattan).  Long Island City was an incorporated city in the western part of Newtown, where it also the name of a neighborhood.

The other three towns of Hempstead, North Hempstead, and Oyster Bay became Nassau County.  The Rockaway peninsula was detached from Hempstead and was also added to Queens Borough.   Glen Cove and Long Beach were created after the division.  In New York, cities can only be created by the state legislature, and the last, Rye, was chartered in 1942.  Villages may be created by the citizens, which is why the suburbs are covered with villages.

In New York, a village remains part of its town, while a city is independent.  In effect, it is similar to an independent city in Virginia, with the independence being from a different level of government.

Kings and Queens, Dukes, and Dutchess counties were created at the same time.  Dukes still exists, but as a Massachusetts county.  The original Dukes County included Martha's Vineyard, Nantucket, and the Elizabeth Islands.  The county was transferred to Massachusetts Bay Colony in 1691, where it is the only county with a royal name.

By 1940, Hempstead's share of the population continued to grow, and instead of 26 of 41 votes was restricted to 20 of 41, with 6 imputed Noes.  The restriction helps Hempstead's power match its population, and it also provides an opportunity for the two small cities to be the critical vote.  Since Hempstead is limited to just shy of a majority, either city can join with the two Hempstead supervisors to form a bare majority.  But Oyster Bay and North Hempstead are badly out of balance.

Majority = 21
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead         10      15  63.75%  60.00%  -5.89%
                  10      15
North Hempstead    8       9  20.50%  18.00% -12.20%
Oyster Bay         4       9  10.47%  18.00%  71.89%
Glen Cove          1       1   3.05%   2.00% -34.47%
Long Beach         1       1   2.22%   2.00%  -9.97%

In 1950, Hempstead reached its peak share of population, but the weightings still badly mismatched.

Majority = 35
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead         17      15  64.29%  55.56% -13.58%
                  17      15
North Hempstead   14      11  21.20%  20.37%  -3.90%
Oyster Bay         7       7   9.95%  12.96%  30.30%
Glen Cove          2       3   2.25%   5.56% 147.03%
Long Beach         2       3   2.32%   5.56% 139.80%

In 1960, Oyster Bay had passed North Hempstead as the second largest town.  While Oyster Bay proper is on Long Island Sound, the town spans the whole eastern edge of the county.  Its population quadrupled in the decade, as developed surged through Nassau County which nearly doubled to 1.3 million during the decade, and on into Suffolk County that increased 141%

Majority = 66
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead         32      15  56.97%  55.56%  -2.49%
                  32      15
Oyster Bay        29      11  22.31%  20.37%  -8.69%
North Hempstead   22       7  16.85%  12.96% -23.07%
Glen Cove          2       3   1.83%   5.56% 203.28%
Long Beach         3       3   2.04%   5.56% 172.85%

The 1960 data was that which Banzhaf based his 'Weighted Voting Doesn't Work: A Mathematical Analysis' on.  His tables indicate that the apportionment was based on citizen population (excluding aliens), and that the weights were determine by dividing the (citizen) population by 10,000 then truncating the fraction.   It makes no difference in the result, and in terms of error is slightly worse.

Majority = 63
Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Hempstead         31      15  57.11%  55.56%  -2.72%
                  31      15
Oyster Bay        28      11  22.38%  20.37%  -8.99%
North Hempstead   21       7  16.71%  12.96% -22.44%
Glen Cove          2       3   1.78%   5.56% 211.52%
Long Beach         2       3   2.01%   5.56% 176.28%

Banzhaf had apparently assumed that a "majority" was a majority of the votes cast, and not a
majority of the votes initially apportioned and produced this table that showed that not only did the two small cities have no effect on the result, but that the supervisor from North Hempstead who represented 1/5 of the population.

Majority = 58
Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Hempstead         31      16  57.11%  66.67%  16.73%
                  31      16
Oyster Bay        28      16  22.38%  33.33%  48.93%
North Hempstead   21       0  16.71%   0.00%-100.00%
Glen Cove          2       0   1.78%   0.00%-100.00%
Long Beach         2       0   2.01%   0.00%-100.00%

[jimrtex]

John Banzhaf showed in his article 'Weighted Voting Doesn't Work: A Mathematical Analysis' that the voting weights used in Nassau County produced an anomalous result, where the three supervisors from the two largest towns (2 from Hempstead and one from Oyster Bay had all the voting power, while the other 3 supervisors had no voting power.  This result is not surprising for the two small cities, since they each had about 2% of the population.  They would not have there own legislator in a equal-district population unless the legislature had 50 members, rather than a 6-member board.  But more remarkable is that the supervisor from North Hempstead had no voting power, even though he had 16.8% of the votes on the board, which matched the town's 16.7% share of population.

As it turns out, Banzhaf misunderstood the system used in Nassau County.  While it reduced the number of votes for the Hempstead supervisors, it maintained the number of votes needed to pass a measure.  125 votes were initially apportioned, with a majority of 63 required.  But then 5 votes were deducted from the Hempstead supervisors' voting weight, the number of votes required to pass a measure remained at 63.

In effect, it was as if the Hempstead football team had received a 5-yard penalty, making it less likely that their field goal attempt would be made.   Banzhaf's version had the penalty, but had also moved the goalpost's forward.

Apportionment based on population, with majority required.

Majority = 63
Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Hempstead         36      16  57.11%  66.67%  16.73%
                  36      16
Oyster Bay        28      16  22.38%  33.33%  48.93%
North Hempstead   21       0  16.71%   0.00%-100.00%
Glen Cove          2       0   1.78%   0.00%-100.00%
Long Beach         2       0   2.01%   0.00%-100.00%

Now the system used in Nassau County where the majority threshold was maintained, even while 10 votes were deducted from Hempstead.

Majority = 63
Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Hempstead         31      15  57.11%  55.56%  -2.72%
                  31      15
Oyster Bay        28      11  22.38%  20.37%  -8.99%
North Hempstead   21       7  16.71%  12.96% -22.44%
Glen Cove          2       3   1.78%   5.56% 211.52%
Long Beach         2       3   2.01%   5.56% 176.28%

And finally, the system assumed by Banzhaf with a 10-voted deduction for Hempstead, paired with a reduction in the majority condition.

Majority = 58
Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Hempstead         31      16  57.11%  66.67%  16.73%
                  31      16
Oyster Bay        28      16  22.38%  33.33%  48.93%
North Hempstead   21       0  16.71%   0.00%-100.00%
Glen Cove          2       0   1.78%   0.00%-100.00%
Long Beach         2       0   2.01%   0.00%-100.00%

The first and last outcomes are the same.  If we penalize Hempstead, but also move the goalposts, the result is the same.

But what if we did not penalize Hempstead, but simply moved the goalposts, would it change the results?  Yes.  For example, if we increase the winning threshold to 65, the results change, so that the supervisors from the 3 smallest towns/cities are no longer powerless.

Threshold = 65
Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Hempstead         36      16  57.11%  61.54%   7.75%
                  36      16
Oyster Bay        28      14  22.38%  26.92%  20.29%
North Hempstead   21       2  16.71%   3.85% -76.99%
Glen Cove          2       2   1.78%   3.85% 115.67%
Long Beach         2       2   2.01%   3.85%  91.27%

And if we set an even higher threshold?  The results are even better.  If we increase the threshold to 73, the results are better.

Threshold = 73
Town           Vote    Swing  R.Pop.  R.Pow    Dev. 
Hempstead         36      15  57.11%  60.00%   5.06%
                  36      15
Oyster Bay        28       9  22.38%  18.00% -19.58%
North Hempstead   21       9  16.71%  18.00%   7.70%
Glen Cove          2       1   1.78%   2.00%  12.15%
Long Beach         2       1   2.01%   2.00%  -0.54%

The population-weighted standard deviation is 10.63%, compared to the 52.41% when a simple majority of the vote was required, or the 39.11% with the actual system used with its vote penalty for Hempstead.

But is if fair if a supermajority of the vote is required?  Can we even refer to the threshold as a simple majority, when it might as well be called a magic number?

But consider that if there were 6 members elected from equal-population districts.  A simple majority of 6 is 4/6 or a 2/3 supermajority.   That is, for a measure to pass it would require the support of members who represent 2/3 of the population, not 50% + 1 person of the population.

73/125 = 58.4%, or 3.504/6.  That is a threshold of 73 corresponds to 3.5+ members of a 6-member body favoring the motion.  Round 3.504 to 4 and a motion requires the support of  representatives of approximately 2/3 of the population to pass.

It should be noted that this occurrence may just be happenstance.   In this particular combination of voting weights, 73 is one more than the combined vote of the two Hempstead supervisors, which means that either of the two city supervisors may join them to pass a motion.  Meanwhile any three of the four heaviest supervisors may pass a motion, with each being critical to passage.

[jimrtex]

Looking at Nassau County, we find that superior results may be obtained by simply adjusting the threshold to pass a motion while keeping voting weights proportional to population.

But what happens when the alternate strategy of adjusting voting weights, away from population proportionality is used?

When population proportionality is used, we are assured that when a motion passes, that the members who vote for the motion represent a majority of the population:

That is for coalition C, a motion passes iff P_C / P_T > 0.5

As a corollary, if the representatives of coalition D represent as large or larger population than coalition C and coalition C was successful, then coalition D will be successful.  That is:

if P_D >= P_C and W(C) = TRUE, then W(D) = TRUE

Similarly, if the representatives of coalition D represent as small or smaller population than coalition C and coalition C was unsuccessful, then coalition D will be unsuccessful.  That is:

if P_D <= P_C and W(C) = FALSE, then W(D) = FALSE

While these properties may seem trivial, they do not necessarily hold if we adjust the voting weights of the representatives.

For each non-member i of each voting coalition C, there is a voting coalition C+i that represents i switching his vote from No to Aye.   For an n-member body there are

n ^. 2ⁿ

such coalition pairs.

The addition of i to voting coalition C can have three possible effects:

W(C) is FALSE and W(C+i) is FALSE, the addition of i was not enough to change the result.
W(C) is TRUE and W(C+i) is TRUE, the addition of i was superfluous.
W(C) is FALSE and W(C+i) is TRUE.   i swung the result and was critical to doing so.

The relative Banzhaf Power Index is the share of critical switches for each member divided by the total number of critical switches for all members.

If we adjust the voting weights such that the Banzhaf Power index is different, then we have necessarily changed the criticality of some switches.   That means some voting coalitions that were unsuccessful will become successful under the new voting scheme or (inclusive) some voting schemes that were successful , will become unsuccessful.  If there is no change in the criticality, then the change in voting weights had no effect on the voting power.

Let W(C) equal success under the original voting scheme, and W'(C) under an alternative scheme.  Then it is possible that:

if W(C) is FALSE and W(C+i) is FALSE, then

W'(C) is FALSE and W'(C+i) is FALSE the change to the voting weights had no effect on success;
W'(C) is FALSE and W'(C+i) is TRUE the change made the switch critical.  This can either be due to an increase in the weights for the original coalition making the threshold within reach for i, or the voting weight of i increasing so he can reach the threshold, or a combination of both; or
W'(C) is TRUE and W'(C+i) is TRUE the vote of i is superfluous.   This is somewhat unlikely, since originally even the inclusion of i in the  coalition was not enough.

if W(C) is TRUE and W(C+i) is TRUE, then

W'(C) is FALSE and W'(C+i) is FALSE, this is unlikely, but would have no effect on criticality;
W'(C) is FALSE and W'(C+i) is TRUE the change made the switch critical, since the initial coalition became unsuccessful, while with i added it remained successful; or
W'(C) is TRUE and W'(C+i) is TRUE the change in voting weights had no effect on success.

if W(C) is FALSE and W(C+i) is TRUE, then

W'(C) is FALSE and W'(C+i) is FALSE, the switch is no longer critical.
W'(C) is FALSE and W'(C+i) is TRUE, the switch remains critical; or
W'(C) is TRUE and W'(C+i) is TRUE the switch is no longer critical.

The bolded conditions reflect a change in success under the new voting weights that changed the criticality of the switch.  It is impossible to change criticality, without the voting weights having caused a change in success - making losers winners, or winners losers.

If the original voting weight was simply the population of each member's district, then any set of voting weights that changes the voting power will necessarily result in some coalitions that represent less than 50% of the population being successful or (inclusive) some coalitions that represent greater than 50% of the population being unsuccessful.

There is also a risk of the total vote not increasing as the population representation represented by the coalition increases.

That is for coalitions C and D (neither of which is a subset of the other), then

P_C > P_D does not necessarily imply that V_C > V_D

If the relation does hold, then we could have achieved the same results by adjusting the threshold of success, rather than changing the voting weights.  This would be consistent with the observation that in a small body of unweighted members, a small supermajority is required to pass a motion.  For example, with 11 members, a majority is 6/11 or 54.5%.

[jimrtex]

Franklin v Mandeville 57 Misc.2d 1072 (1968) by the Nassau County Supreme Court found that Nassau County's scheme of weighted voting for its Board of Supervisor was unconstitutional because it apportioned less than a majority of the vote to the town of Hempstead, even though it had a majority of the population.

In Nassau County, the voting weight of the three towns and two (small) cities was apportioned on the basis of population; but then the apportionment for a town with over half of the voting weight (Hempstead) was reduce to less than half of the (original) total voting weight.  The truncated votes were not reallocated, but simply treated as always voting No.

Because Hempstead had a majority of the population in 1936 when the New York legislature had provided the charter for the county, the court ruled that the reduction in representation for the largest town was not severable.  That is, it is unlikely that the other towns would have accepted a scheme where Hempstead had a majority vote on the board, so that the reduction was an integral part of the adopted scheme.

In Franklin v Mandeville 32 A.D.2d 549 (1969), the appellate division of the supreme court affirmed the lower court decision.  They further declined to consider whether the plan complied with Iannucci v. Board of Supervisors (20 N.Y.2d 244), that required the voting power of members to approximately proportional to the population they represent.

There is a contradiction here that they were either unaware of, ignored, or deliberately avoided recognizing.  As was shown in a previous post, any adjustment to voting weights away from a strictly population-based scheme that changes the relative voting power of any member (his likelihood of being a critical vote) necessarily will produce some successful voting coalitions that represent less than 50% of the population or unsuccessful voting coalitions that represent more than 50% of the population.  That is, the adjustment that Iannucci appeared to require, would cause the problem that that occurred in Nassau County - representatives representing a majority of the population unable to to pass a motion.

Franklin v Mandeville 26 N.Y.2d 65 (1970) by the Court of Appeals (NYS Supreme Court) upheld the lower court decision.   By the time of this decision in early 1970, so much time had passed that the court ordered any remedy to be delayed until after the results of the 1970 census were announced.  It also noted that referendums to change the voting scheme had been defeated in 1965 and 1967.

In 1972, after the census, Nassau County prepared a new voting scheme.  It defined the "voting power" of a Supervisor as "the mathematical possibility of ... casting a decisive vote on a particular matter", and said that the voting-power share should approximate the population of the town or city, and that every entity will have a non-zero voting power.  It also required an "an independent computerized mathematical analysis"

Nassau County is a very bad case for weighted voting.  It only has 3 towns, one of which had 56% of the population, and two relatively small cities.  If population-based weights are used (here the population is in 1000s) the results are very bad.

Threshold = 715
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead        401      16  56.10%  66.67%  18.84%
                 401      16
Oyster Bay       333      16  23.32%  33.33%  42.91%
North Hempstead  235       0  16.45%   0.00%-100.00%
Long Beach        33       0   2.32%   0.00%-100.00%
Glen Cove         26       0   1.80%   0.00%-100.00%

The four weightier supervisors are so comparable, that the weight of any three is always greater than any two, which in turn is greater than the largest individual.  Moreover, adding in the small weights of the smaller cities has no effect on the relationship.  Five bucks and the votes from the two cities buys a chai latte at Starbucks.  They are not relevant.

Any two of three the supervisors from Hempstead and Oyster Bay form a majority.  North Hempstead can not form a majority with one of the three bigs, and the smaller cities don't matter.   Hempstead and Oyster Bay have all the power because they are necessary to every majority.

We can adjust the threshold so that Hempstead+Oyster Bay is no longer successful, and so that the addition of North Hempstead or both of the two small cities is critical.

Threshold = 768
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead        401      16  56.10%  61.54%   9.70%
                 401      16
Oyster Bay       333      10  23.32%  19.23% -17.55%
North Hempstead  235       6  16.45%  11.54% -29.88%
Long Beach        33       2   2.32%   3.85%  65.80%
Glen Cove         26       2   1.80%   3.85% 113.14%

We can then reduce the weight of the two Hempstead supervisors, and set the threshold such that the two Hempstead supervisors can not combine for majority.

Threshold = 733
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead        366      15  56.10%  55.56%  -0.97%
                 366      15
Oyster Bay       333      11  23.32%  20.37% -12.66%
North Hempstead  235       7  16.45%  12.96% -21.22%
Long Beach        33       3   2.32%   5.56% 139.50%
Glen Cove         26       3   1.80%   5.56% 207.87%

If we wanted to, we could use the actual populations of Oyster Bay (333,089), North Hempstead (234,984), Long Beach (33,127), and Glen Cove (25,770), and then set the weight of Hempstead to that of Oyster Bay + Long Beach (366,216), and set the threshold to one greater than twice that (732,433).  Of course this would make the manipulation more obvious.

Or we could increase the weight of Oyster Bay, and not drop that of Hempstead as much.

Nassau County in 1972 actually used the following (initial weights were based on population divided by 10,000; but you will note that Hempstead = Oyster Bay + Long Island and the threshold is one greater than twice that of Hempstead.

Threshold = 71
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead         35      15  56.10%  55.56%  -0.97%
                  35      15
Oyster Bay        32      11  23.32%  20.37% -12.66%
North Hempstead   23       7  16.45%  12.96% -21.22%
Long Beach         3       3   2.32%   5.56% 139.50%
Glen Cove          2       3   1.80%   5.56% 207.87%

Modifying the resolution does not change the fact that there are at most only 48 different vote totals when there are 6 members, and two equal to each other.

Note that the second example where we used the actual population (divided by 1000) and raised the threshold to 769 (53.7%) has less relative error (power/population - 1), than the final result.  It appears that they gave more emphasis to the difference between population share and power share, so that Glen Cove only had 3.76% extra voting power, rather than 3 times as much voting power as its population.

The adjusting of the weights also caused an inversion such that a motion supported by both Hempstead supervisors who represented 801,110 persons would fail, while a motion supported by one Hempstead supervisor along with those from Oyster Bay, Long Beach, and Glen Cove who collectively represented 792,541 persons would succeed.

In Franklin v Krause 72 Misc.2d 104 (1972), the Nassau Supreme Court considered the Nassau County plan and found it unconstitutional.  The judge noted that Hempstead with 56.3% of the population was given 54.0% of the vote, but the threshold had been set at 54.6%, so as to prevent Hempstead from exercising its majority.

The judge also ruled that it could not be said that the share of voting power was approximately proportional to the share of the population.  This is not truly correct.  It would be like saying that the apportionment of representatives among the states does not approximate their population distribution, just because it is not totally proportional.

The judge also suggested that weighted voting was only constitutional on a temporary basis, for example to adjust voting in a deliberative body, without requiring new elections.

The Court of Appeals reversed the supreme court decision in Frankin v. Krause 32 N.Y.2d 234 (1973).

The court found that a computer analysis had generated 2000 sets of voting weights, and about 100 had been given to the board attorney, who had passed on a half dozen or so to the board, and therefore the plan was in compliance with Ianucci.

But Suffolk County is quite amenable to pen-and-paper analysis.  With only 6 members, there are only 64 voting combinations, and 32 of those have a split vote by the Hempstead supervisors, so that there are only 48 vote totals.

The court also observed that the deviation for Glen Cove was only 3.7% above their population, overlooking that this would be equivalent to giving them 3 seats in a 55-member body when they were only entitled to one.  They said the total deviation was 7.3%.  They compared this to the US Supreme Court decision in Mahan v Howell where a 16.4% deviation was accepted. But in that case, it was the relative deviation from the ideal (the smallest district had a population of 6.8% below the ideal, and the largest 9.6% greater.

In June 1974, the voters of Nassau County disapproved of the adjusted weighted voting plan.  So it was back to court in Franklin v Krause 81 Misc.2d 52 (1975).  Nassau County was still operating under the 1960s plan, since when the final decision on its constitutionality was rendered the 1970  census was eminent.

The Board submitted two plans which would divide the county into 15 legislative districts.  One plan had a deviation of 22%, which the court rejected since it was greater than that accepted by the SCOTUS in Mahan v Howell.  The second plan had a deviation of 8.6%, and that was within the 10% safe harbor limits of White v Regester.   The court refused to consider issues such as political fairness, division of unincorporated communities, and whether the board had deliberately chosen a plan that would be rejected by the voters.

The 15-district plan was rejected by the voters of Nassau County in June 1975.

In Franklin v Krause 83 Misc.2d 42 (1975) the court created a judicial commission to devise a plan.  That commission apparently chose the adjusted voting weight plan proposed by the board of supervisors and accepted by the New York Court of Appeals, but rejected by the voters.  I didn't find any actual decision, but the weighted voting scheme was at issue in the 1980s litigation.

[jimrtex]

The 1970s saw a 7.5% drop in the population of Nassau County, as baby boomers left home for Nassau County where they could afford to live, or NYC where they couldn't afford to live, but wanted to, and higher density infill development had not begun.  Nonetheless the relative proportions of population had not changed much.

If population had been used directly for weights, the result would have been the same as the previous decades, with all power held by the three supervisors from Hempstead and Oyster Bay, any two of which formed a majority.

Threshold = 661
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead        369      16  55.88%  66.67%  19.30%
                 369
Oyster Bay       306      16  23.14%  33.33%  44.08%
North Hempstead  219       0  16.54%   0.00%-100.00%
Long Beach        34       0   2.58%   0.00%-100.00%
Glen Cove         25       0   1.86%   0.00%-100.00%

If the threshold was increased slightly to 710 (53.7% of the total vote) the results are much better.

Threshold = 710
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead        369      16  55.88%  61.54%  10.12%
                 369      16
Oyster Bay       306      10  23.14%  19.23% -16.88%
North Hempstead  219       6  16.54%  11.54% -30.25%
Long Beach        34       2   2.58%   3.85%  49.18%
Glen Cove         25       2   1.86%   3.85% 106.48%

Instead the voting weights were wildly manipulated, and the "majority" threshold set to 65 (60.2% of the total vote of 108).

Threshold = 65
Town           Vote   Swing  R.Pop.  R.Pow   Dev.  
Hempstead         30      15  55.88%  53.85%  -3.64%
                  28      13
Oyster Bay        22      11  23.14%  21.15%  -8.56%
North Hempstead   15       9  16.54%  17.31%   4.62%
Long Beach         7       3   2.58%   5.77% 123.77%
Glen Cove          6       1   1.86%   1.92%   3.24%

Coalitions that represented as much as 57.7% of the population would fail to win a vote.

Legal challenges moved to the federal courts after the 1980 reapportionment.   In League of Women Voters v Nassau County Bd. 737 F.2d 155 (1984), the 2nd Circuit upheld the apportionment scheme.  In doing so, they essentially said that the issues were the same as that in the 1970s, when the SCOTUS had summarily rejected an appeal from the New York courts.

In particular, the court found that the use of deviation between the power share and population share could be measured as the arithmetic difference, rather than using the relative deviation.

For example, Long Beach had 2.58% of the population, and its supervisor 5.77% of the voting power.  The courts had measured this as a 3.19% deviation (5.77% - 2.58%) rather than 123.77% relative deviation (5.77% - 2.58%)/2.58%.

The court warbled on about how this was how it had been presented to the SCOTUS in the 1970s, and that the SCOTUS had suggested flexibility and there can be no single mathematical method.

But it is the voter that is denied equal protection, and you simply can't say that a voter in Long Beach has only 3.19% additional voting power.

You might as well say that each person in Long Beach represents 0.000076% of the county population and has 0.000169% of the total voting power, and therefore is only overrepresented by 0.000094%, or less than 1 part in a million.

[Torie]

I just came across this entry (from 2011), in the blog that started this thread.

Some History of the Weighted Vote

I haven't come across the specifics of how the weights are calculated in Hudson.  My assumption was that the President should have a voting strength based on 1/10 of the city's population, and each alderman 1/2 of the population of his ward.  This would give a total represented population of 1.1 times the actual population, with the president representing 0.1/1.1 or 1/11 of the total - the same as he would if there were equal population districts with no weighting.

In the 2000s plan, there was one instance where the two aldermen from a single ward had voting weights that differed by one.  Aldermen have different voting weights, depending on whether the vote is a simple majority; 2/3 supermajority; or 3/4 supermajority.  That the weight of a member varies slightly on the type of vote is likely to be viewed with incredulity.  That two members have different weights also would be treated skeptically.  In the 2000s, one alderman had one additional vote on a 2/3 vote (the alderman with the greater popular vote when elected, received the bonus vote).

There was an alternative to the current plan that would have applied a similar split, but would be used on simple majority votes.  There were some small adjustments to other voting weights.  These weights were said to produce results slightly more comparable to population. I think the votes opposed to the current plan were based on a preference for the alternative.

I then searched the blog for "weighted" and came across some interesting entries.

Forty Years of Weighted Votes

Note the pictures.  These demonstrate that a picture is worth a 1000 words

"where the "old Hudson" folks, who grew up there still control matters in Hudson, but perhaps for not much longer, holding the power and most of the very much sought after government jobs.

Or alternatively, that the Appalachians do extend through New York.

The State of the Weighted Vote

Eureka!

This entry has a link to Lee Papayanopoulos's study of voting weights.

He cites a number of court cases from around 1970 upholding weighted voting schemes in several New York counties.

I think that the blogger might not understand the term 'egality' which is the number of wards where the two alderman have the same weight.  An egality of 5, means the weights are identical for all 5 wards.

The Papanopoulos study has different ward populations than we have assumed:

Ward 1  770  v 828
Ward 2  1,281 v 1204
Ward 3  1,142 v 1142
Ward 4   725 v 744
Ward 5   2,485 v 2,485

He says: "Populations based on the decennial census, adjusted to exclude institutional inmates and to reconcile overlapping election districts."

These would require a change in the block splits:

Front Street block: Ward 1:2  131:234 v 73:292
Great Northern block: Ward 2:4  51:238 v 32:257

If we use Ward 2: 6 buildings x 8 units + 2 buildings x 12 units = 72 units; and Ward 1: 5 buildings x 8 units = 40.

365 x 72/(72 + 40) = 234.6

The block split was reversed.   Someone from Ward 2 should sue.

The flip seems to be the census numbers, not the numbers used for voting weights, if I follow your mathematics. Can you walk through this more carefully for me?

It does seem that the weighted voting system is legal, since the variations between population and voting power are far less than the 10% variance max that NY courts allow. I don't know where that 46% voting power number came from for the 5th ward that was bandied about.

[Torie]

The issue of weighted voting was discussed by the Legal Committee of the Common Council of Hudson. Yours truly had a cameo role. Jimtex's incredible spadework on this, is going to have the effect of changing the system, and consigning it to history I think. Kudos to him.

[jimrtex]

The issue of weighted voting was discussed by the Legal Committee of the Common Council of Hudson. Yours truly had a cameo role. Jimtex's incredible spadework on this, is going to have the effect of changing the system, and consigning it to history I think. Kudos to him.

The article implied that the committee's attorney (Tuczinski) was going to investigate the issue of whether the block along Front Street had been properly allocated.  I'll present my analysis below.

The article says that Hudson was divided into wards in 1921.  Hudson had two wards in 1850, and 4 wards since before 1860, the 5th ward was added by 1890.   Based on the racial composition, I suspect that the split in 1850 was a east/west split, and I would guess along 3rd Street.

In 1860, the 2nd and 4th wards had a notably larger black population share, just as they still do.  Presumably this is related to proximity to the port.  By 1880, the 4th ward had 35% of the population, that was recognized by creation of the 5th ward.  The split in 1890 was reasonable for the era: (1) 17.6%. (2) 24.0%. (3) 22.9%, (4) 17.0%, (5) 18.5%.

Hudson fell below the threshold used by the Census Bureau for reporting population by ward, and this may have contributed to the lack of updating between 1970 and 2000.  In 1990, the Census Bureau began producing data for census blocks, which permitted a city to define arbitrary configurations of population for election districts.   It is possible that the Census Bureau did generate data, but simply did not publish it.  Hudson may simply have been lackadaisical.  The fact that they did not bother to switch to weighted voting until 1974 suggests some disregard for the issue.

The article says that the fact that Hudson is the only city in the US to use weighted voting raises constitutional questions.  That may not be true, or is over-simplistic.   New York states grants cities a great level of home rule authority, including determining how they are governed.  Home rule would be meaningless if a city could not determine a different method of conducting elections.  They are of course subject to broader standards such as equal protection; and in New York, since 2010, the counting of prison populations.

There are SCOTUS opinions that ruled (not un-)constitutional the weighting of representation for towns within a county, on their board of supervisors.  In those instances, each town had multiple representatives, apportioned on the basis of their population.  But it does not follow that having one representative casting five votes is necessarily different in an constitutional sense from five representatives elected at-large each casting one vote.

The SCOTUS said that the nature of the relationship between towns and counties in New York, and the dual service of town supervisors on town boards and county boards of supervisors provided justification for a greater population deviation, than might be legitimate with equal-population single-member districts.  Rockland County had a very fortuitous population distribution such that the population of its towns were roughly integer multiples of the smallest.

The SCOTUS does not make rulings as to what is constitutional, but rather what is unconstitutional or not unconstitutional.  It appears that those who are suggesting that Hudson's system is unconstitutional because it is a town rather than a county, and the units of representation are wards rather than towns, are making the erroneous conclusion that because wards are simply lines on a map, and not themselves functioning units of government, that there is not, and can not be, any justification for keeping the wards fixed and using weighted voting.  But this is a hypothetical that the SCOTUS simply has not addressed.

A challenger would still have the burden of showing that the Hudson system violated equal protection.  This would be particularly hard since the deviation between voting weight and population is not very large (unlike systems where effective voting weights are restricted to small integers).

You must be logged in to read this quote.

In the 1960s, Dr. Papayonopoulos was working for IBM in New York.   An early court decision on weighted voting (simple population-based weighting), said that weighting should be based on the voting power, such that the voting power was proportional to population, but that it was much too complicated to calculate.  In a subsequent case in a different county, Papayonopoulos demonstrate that the voting power for a given set of voting weights could be calculated.  At the time, it required use of an IBM 360, and it was apparently part of Papayonopoulos's job to find applications for which the computer was suitable.  At least for a body as small a Hudson's council, it can easily be computed with a spreadsheet on a garden variety PC.  It is simply not a complicated or complex calculation.

In the same case, Papayonopoulos then generated a set of voting weights that the court ordered to be implemented.  I suspect he understands the legal background of weighted voting as much as anyone.

A fault in the method is that it appears that there is no systematic method of generating voting weights.  In the case of Hudson, he generated nearly a million plans, and selected those which had the least deviation between voting power and population.

Note that underlying the whole discussion is that Hudson has twice considered equal population districts, which have been rejected by the voters.  Though Hudson may exercise home rule in how it is governed, it is subject to referendum.

Those who want equal-population districts would prefer to go to the electorate claiming that they were only trying to comply with the 14th Amendment and SCOTUS rulings; rather than we know you've rejected this change twice before, but we are going to keep forcing it down your throat until you vote for it.  Alternatively, they are hoping for a lawsuit, that Hudson would decide not defend against, claiming that it was not constitutionally defensible.

The problem is that weighted voting is clearly not unconstitutional per se.

[Torie]

Tbe law, unfortunately, is far more complex than the above. The memo I will write, may be the most complex one I have ever written about the law. The courts have great difficultly understanding math, and are inconsistent in applying it (mostly when it comes to calculating deviations from effective population, decisive combination equality). The deeper I dig into the cases, the more I realize this. And one assumption is that if a Ward or Town has two alderman, or supervisors, vis a vis each other, they will vote randomly, and that is assumed in the the Monte Carlo voting power calculations. That is subject to attack. There is probably a path for a municipal weighted voting system to survive legal attack, but a very narrow one, with a lot of constraints to meet to get there. 

[jimrtex]

Dr. Papayonopoulos Analysis For Hudson Voting Weights 2013

Papayonopoulos gives the following voting weights:

Ward 1  770
Ward 2  1,281
Ward 3  1,142
Ward 4   725
Ward 5   2,485

Along with the footnote: "Populations based on the decennial census, adjusted to exclude institutional inmates and to reconcile overlapping election districts."

Since the total population of 6403 clearly reflects the exclusion of the prison population less  prisoners allocated to census blocks based on their residence prior to imprisonment, it must be those used in calculating the voting weights.  The spreadsheet from the Hofstra report does not show such a prison adjustment.

Two census blocks are split by ward boundaries:

New York, Columbia County, Tract 13, Block 1002
GEOID10=360210013001002

This is the long block to the west of Front Street.  The boundary between Wards 1 and 2 is Warren Street extended to the west, along the walkway to the river overlook.  There are apartment units to both the north and south of this line.  The population is 365.

New York, Columbia County, Tract 12, Block 1000
GEOID10=360210012001000

This is a large block (275 acres) in the northern part of the city between 2nd Street and Harry Howard, north of the defined street grid.  The boundary between Wards 2 and 4 is 3rd Street extended to the northern boundary of the city.  Most of the population is along the edges of  the block.  It includes the Firemen's Home.  The population is 289.

Wards 3 and 5 do not split any census blocks, and their populations match the total of their constituent blocks.  Note: For the moment I'll ignore that the current electoral boundaries do not appear to conform with those specified in the city charter.

Since there us only one split census block in Ward 1, we can determine that Block 13/1002 was allocated Ward 1:307, Ward 2: 58.   And similarly we can determine that Block 12/1000 was allocated Ward 2: 19, Ward 4: 270.

HELP!!!

Can someone independently check the block populations for 2010?   Is it conceivable that there was a 38% decline in the population of Ward 1, outside the Front Street block.

How did Dr. Papayonopoulos determine the ward populations on which the voting weights were calculated?

Was it someone for the city?   Someone from the county board of elections?  Papayonopoulos?

Something is totally messed up.  It would appear that the actual population of Ward 1 is closer to 600 than 800?

2000 block populations:



2010 block populations (ignore divisions of census block).




IGNORE THE FOLLOWING FOR NOW

There are relatively few houses in Ward 2 within Block 12/100: a few northwest of 2nd Street and on the portion of Mill Street east of 2nd Street (since Mill Street dead-ends it does not divide the census block).   This is the area west of Third Street extended.  Within Ward 4, there are houses on the east side of Third St north of Rope Alley; on the north side of Rope Alley east of 3rd Street; on the north side of State Street east of its intersection with Rope Alley; on the north(west) side of Carroll Street (some of these may be commercial, but most have back yards); on the west side of Short Street north of its intersection with Carroll Street; and on the west of Harry Hopkins, including the house along Lucille Drive, and the eastern segment of Mill Street; and the Firemen's Home.  Given the large differences in number of houses between the two wards, the 51:238 split is quite plausible.   Note, the Census Bureau does provide census counts within blocks for "group quarters".  From this, it should be possible to determine the census population for Firemen's Home.  If the allocation did not take this in to account, it may be somewhat off, but probably by no more than a dozen or so persons.

[Torie]

By independently check the populations for Ward 1, you mean just go over the census block numbers that you entered? Would you post those please? Carole Osterlink who lives in the most depopulated census block in the First Ward tells me a lot of triplexes (and more) on her block were converted into single family homes between 2000 and 2010. It was one of the most premier Hudson "hubs" of gentrification, and is now a very desirable hood. It seems also some residences were torn down to make a parking area across the tracks from the train station as well.

Where did you get the 292-73 split for the Front Street census block, and the 257-32 split for the Howard-Mill Street census block in your prior map, that you are now varying slightly?  Papayonopoulos seemed to have also had a different figure for the Mill-Howard Street as well. In any event, my analysis of the numbers is reflected below, based off of yours. If you see an error somewhere, please let me know.

Oh, and how does the City Charter map description vary from what the map shows, as to the ward boundaries, to which you allude above?

[jimrtex]

By independently check the populations for Ward 1, you mean just go over the census block numbers that you entered? Would you post those please?

Someone needs to get the LATFOR numbers

2010 Amended Population
(Prisoner Adjustment)

On that page there is a link to a zip file that contains several csv files which can be read into a  spreadsheet, along with a link to documentation.

The relevant file is pl_adjusted_DOJ_block.CSV which has one line for each census block in the state (State_ID 36).   You want the blocks that are in Columbia County (County_ID 21), Census Tracts 12 (Tract_ID 1200) and Tract_ID 1300).   These two tracts cover Hudson, and together are coincident with the Hudson city limits.

I created the map of population numbers transferring from the spreadsheet to the map.  I (just now) recalculated the populations of the wards using both the map, and the spreadsheet.  While I match the numbers that Papayonopoulos used for Wards 3 and 5, the splits of the two census blocks to match the Papayonopoulos ward totals would appear to require almost all of the Front Street block in Ward 1.

When I had first come across the Papayonopoulos numbers, I had concluded that it required a split of the Front Street block almost identical to your first spreadsheet, but with wards reversed.

But now I can't figure out how I came to such a conclusion.  I'm either doing something stupid or careless OR Hudson is way off on their base population numbers.  That's why I need someone to check independently not trusting my map.

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The census gives a population of 362 for the Front Street block.  The LATFOR numbers increase this to 365.  The prison adjustment includes both (a) deleting prisoners from the location of their prisons; and (b) adding them back to the location where they lived prior to incarceration (prisoners who were resident out of state, or whose prior address is unknown, or were in federal prisons) disappear from the counts for legislative and local redistricting.

The spreadsheet you had posted from the Hofstra report implied a 290:72 split.  I simply added the three prisoners maintaining the proportionality of that split.   290+2:72+1.  There were no prisoners added to the Great Northern block, so I was using the maps implied by the Hofstra report.  Since the Hofstra spreadsheet didn't have the prison adjustment, I would totally disregard their block splits - and thus should be disregarded from my map.

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Hudson City Charter/Code Book

Ward boundaries are in Section C1-4.

Ward 1: Warren Street and 3rd Street (extended to west boundary, and south boundary)

It is the extension of Warren Street that splits the Front Street Block.

Ward 2: Warren Street and 3rd Street (extended to west boundary and north boundary)

It is the extension of 3rd Street that splits the Mill St-Howard Great Northern block.

Ward 3: From the southern boundary and 3rd Street Extended to Warren Street to 7th Street, diagonally across Public Square (now 7th Street Park) to the intersection of Gifford Place (now Park Place?) and Columbia Street, thence along Columbia Street and Columbia Turnpike to the eastern limits.

It appears that elections are now conducted using Columbia Street to the eastern boundary, shifting the triangle between Columbia Street and Columbia Turnpike from the 5th Ward to the 3rd Ward.   Columbia Turnpike was chartered in 1799.  It would not surprise me if it at one time included the portion of Columbia Street beginning at 7th street, where Columbia Street angles off from the main street grid, and that Columbia Street was later extended to the east.

Following the 2000 Census, the Common Council passed a resolution declaring the ward populations.  Hudson had not bothered to update the voting weights since they were first introduced in 1974.  It was some time after the resolution was passed that the weights were actually implemented.  Based on the numbers in the resolution, and the 2000 census block populations, the triangle between Columbia Street and Columbia Turnpike was included in Ward 5 in 2000, but in Ward 3 in 2010.

It is quite possible that the Common Council (or the clerk who prepared the numbers) used the charter, even though elections were conducted based on the use of Columbia Street.

Ward 4: From the northern boundary and 3rd street extended, to Warren Street to 5th Street, extended to the northern boundary.

The apparent current electoral boundary jogs west from 5th Street on Prospect to Short Street
and thence on out Harry Howard.  This has the effect of moving the population in the Short-Washington-5th-Prospect block; the Short-(Clinton)-5th-Washington block; and on the south side of Harry Howard north of Underhill Pond from the 4th Ward to the 5th Ward.  Note that Clinton does not actually go through between 5th and Short, but appears to exist as a utility easement, foot path, and possible jeep trail-short cut. 

It appears that the council resolution followed the charter, but the 2010 numbers are based on the jog.

I'll comment on your spreadsheets in a separate post.

[jimrtex]

Carole Osterlink who lives in the most depopulated census block in the First Ward tells me a lot of triplexes (and more) on her block were converted into single family homes between 2000 and 2010. It was one of the most premier Hudson "hubs" of gentrification, and is now a very desirable hood. It seems also some residences were torn down to make a parking area across the tracks from the train station as well.

Do many NYC people have weekend houses in Hudson?  The train schedules are such that commuting to NYC would be possible, but grueling, especially when making allowance to make sure that you don't miss a train when the next train is an hour or two later.   There is a very early southbound train and a fairly late northbound train.

But someone might have an efficiency in NYC, where they sleep during the week and a house in Hudson mainly for weekends.   Commuting on Monday morning and Friday afternoon is plausible.  Since they live most of the time in Manhattan, that is likely where they were counted for the census, even though they might have considered their house in Hudson as a main residence.  This would make the house in Hudson, unoccupied for census purposes.   They might be able to use the house in Hudson for tax and voting purposes.

[Torie]

Carole Osterlink who lives in the most depopulated census block in the First Ward tells me a lot of triplexes (and more) on her block were converted into single family homes between 2000 and 2010. It was one of the most premier Hudson "hubs" of gentrification, and is now a very desirable hood. It seems also some residences were torn down to make a parking area across the tracks from the train station as well.

Do many NYC people have weekend houses in Hudson?  The train schedules are such that commuting to NYC would be possible, but grueling, especially when making allowance to make sure that you don't miss a train when the next train is an hour or two later.   There is a very early southbound train and a fairly late northbound train.

But someone might have an efficiency in NYC, where they sleep during the week and a house in Hudson mainly for weekends.   Commuting on Monday morning and Friday afternoon is plausible.  Since they live most of the time in Manhattan, that is likely where they were counted for the census, even though they might have considered their house in Hudson as a main residence.  This would make the house in Hudson, unoccupied for census purposes.   They might be able to use the house in Hudson for tax and voting purposes.

Yes, many do, particularly on that side of town. The trains run just about every hour, until about 11 pm. My cousin's partner is a literary agent in NYC, and stays 3 nights a week or so down there in a hotel, other than the summers, when she repairs to Maine. The house next door to them is a second home for a radiologist in NYC who is rarely there.

[Torie]

I added the block census numbers on your map sheet and found that they tied to the numbers for the 3rd Ward and 5th Ward, and as my spreadsheet shows, close for the 4th Ward (a different split was used for what you call the great northern block). So if there is an error, it must be in the blocks for the second and first wards. Are you worried that you copied and pasted the wrong census blocks or something? Is there a way that you could email to me the applicable portion of the spreadsheets that you downloaded for those two wards? swdunn1@gmail.com

[jimrtex]

I added the block census numbers on your map sheet and found that they tied to the numbers for the 3rd Ward and 5th Ward, and as my spreadsheet shows, close for the 4th Ward (a different split was used for what you call the great northern block). So if there is an error, it must be in the blocks for the second and first wards. Are you worried that you copied and pasted the wrong census blocks or something? Is there a way that you could email to me the applicable portion of the spreadsheets that you downloaded for those two wards? swdunn1@gmail.com

Done.

I get 455 for Ward 4, exclusive of the Great Northern block (Tract 12, Census Block 1000).  To get to the total of 725, this requires a a 270:19 split of the block.

[jimrtex]

Where do the number of units for the split block come from.  There are in identical proportion to what I had guessed from the satellite view, but 50% greater.

I had guess 40 and 72, while you have 60 and 108.   I think the allocation makes a lot of sense.

The problem is that if we take the 463 which I count in Ward 1, exclusive of the split block, then the population of Ward 1 is 593, which is way below 770.

I would rearrange the 2nd spreadsheet.  I would start out with the version that would exclude adjustments for the council president.  There are two reasons for this.   (1) That is the basis on  which Dr.Papa+ determined the voting weights.  His calculation of voting power share excludes the combinations where the council president is a critical vote; (2) In general, courts have separated district members from at-large members in determining equal protection violations.  This avoids the need to develop a mathematical formula to combine the contributions of the two types of members (insert quote about mathematical quagmire here); and recognizes that a voter has an equal protection right to elect different types of officers.  An alderman might be expected to be more parochial with regard to their ward; while the council president may have a broader citywide interest, along with management and organization of the council.   That is, they are more than mere votes on council motions.

An exception to this is in Hawaii, where the senate and house members are apportioned among the four island units, which results in rather large deviations between districts.  (eg Maui might be entitled to 1.3 senators and 2.6 representatives; they would be given one oversized senate district; and 3 undersized house districts).   There is a formula that in effect determines a voters "legislative" influence.

I would include the actual voting weights.  In Roxbury Taxpayers vs Delaware County Board of Supervisors, the equal protection test was based on the voting weight share vs. population, rather than power share; even though the voting weights had been calculated so as making voting power and population proportional.

My order of rows would be:

1) Ward populations;
2) Ward population share;
3) Voting weights (eg 256, 456, 392, 256, 864)
4) Voting weight share.
5) Difference between voting weight share and population share.
6) Discrepancy between voting weight share and population share.
7) Critical combinations.
8) Voting power share (excluding council president)
9) Difference between voting power share and population share.
10) Discrepancy between voting power share and population share.

I don't understand what Line 4 (Error Factor From Line 12 Below ...) represents.

I'm not sure how you are calculating discrepancy (line 7 and line 19), the numbers that I calculate are similar but not exact.

11+) I would then make the calculations that include an adjustment for the council president.

The way I calculated the discrepancy for the council president was to assume he represented 640.3 persons (the average for the 10 alderman 6403/10).  In effect, each person received a 0.1 bonus that was used to support the president.   The president would thus represent 1/11 of the adjusted population and expected to have 1/11 of the voting power, just as he would have if the 10 aldermen were elected from equal-population districts.

I can't say that your approach is better or worse.   Your approach will make the proportionality between ward voting power and ward population slightly better because you are distributing the president's voting power in precise proportion.

Neither may be necessary.  Dr.Papa+ didn't have an apparent constraint on the voting weight or voting power of the president.  However, the voting power of the president varied between 9.26% and 10.53% in the plans presented to the council.   9.09% is 1/11.   But there would be no legal distinction between a mixed-member council that had 10 district members and one at-large member, or 10 district members and three at-large members, or five-at large, etc, at least as far as equal protection for district voters (there might be issues with the number of at-large members if it interfered with representation of racial or political minorities).

Arguably, a role of the council president is to regulate council business.  Giving him a bit more or less voting power in order that the political power of the alderman is (more) proportionate to the population of their wards, is consistent with that role.

I don't understand where the numbers on Lines 21,22 are coming from.

[jimrtex]

Edit Picture: Corrected Ward 5 Population

This is the first of several maps intended to check the population for Hudson wards.  The first digit of a 4-digit census block number, is the Block Group (eg census block 1004, is formally census block 004 of block group 1.

Census Geography has the following proper nesting:

State -> County -> Census Tract -> Block Group -> Census Block

[jimrtex]

Tbe law, unfortunately, is far more complex than the above. The memo I will write, may be the most complex one I have ever written about the law. The courts have great difficultly understanding math, and are inconsistent in applying it (mostly when it comes to calculating deviations from effective population, decisive combination equality). The deeper I dig into the cases, the more I realize this. And one assumption is that if a Ward or Town has two alderman, or supervisors, vis a vis each other, they will vote randomly, and that is assumed in the the Monte Carlo voting power calculations. That is subject to attack. There is probably a path for a municipal weighted voting system to survive legal attack, but a very narrow one, with a lot of constraints to meet to get there. 

I would not characterize this as a Monte Carlo method.   Banzhaf power is based on an exhaustive compilation of all voting combinations.   Monte Carlo methods are used to estimate probabilities based on a random selection of potential events (in this case, voting combinations).

In Hudson, with 11 voting members, there are only 211 (2048) voting combinations.  This is quite amenable to exhaustive analysis using a spreadsheet with 2048 rows, one per possible voting combination.  In larger cases with many more members, Monte Carlo methods may be suitable.  For example, for a 21-member body such as the 1960s New Jersey Senate, there are 221 (2.097 million) voting combinations, which is twice the number of rows that a typical spreadsheet can handle.  But even with that many, it would feasible with a bit of programming to loop over all voting combinations.

Dr. Papa+ generated nearly a million sets of voting weights, and tested each for proportionality between population and voting power, then examining the closer examples.  There does not appear to be a systematic way of determining a best set of voting weights, even if there were an established criteria for best.  If you start using population-based weights, calculate voting power and adjust the weights for the districts with the greatest difference between population and voting power, this tends to improve proportionality, but iteration does not always converge.  Improving one ward may harm another.

He may have used random plans, or he may have done an exhaustive search within certain constraints.  For example, his report says he attempted to make the total weighted vote around certain totals such as 500, 1000, 2000.  There are also many sets of weights which produce the same voting power.  With the two members from each ward having the same voting weight, there are only 3⁵*2 = 486 possible vote totals at most.  Presumably, he also constrained voting weights such that a ward with more population could not have a lesser weight.