Cube Root Rule Legislative Districts

[muon2]

Under the cube root rule IL should have about 235 legislators for 2010, compared to the actual number of total reps and sens of 177 so it isn't too far off. Interestingly IL had 236 legislators up to 1980. That year a constitutional amendment passed to cut back the number of house members from 177 to 118 based on the argument that it would save money by having fewer reps.

[muon2]

Of course, city councils are usually way off, for example, San Francisco has only 11 for a population of 800,000.

While Frankfurt has 93 for a population of 650,000.

Similarly Chicago has 50 aldermen, one per ward, and the rule would predict 142 for 2.85 M people.

For smaller cities divided into wards IL state law provides a statutory number of aldermen (two per ward) on the city council, though that number can be changed in certain cases (and generally reduced) by referendum or ordinance. IL law seems to use just under half the number of aldermen predicted by the cube root rule.

Population Range	Aldermen	Cube Root Prediction
up to 3 K	6	0 to 14
3 K to 15 K	8	15 to 24
15 K to 20 K	10	25 to 27
20 K to 50 K	14	28 to 36
50 K to 70 K	16	37 to 41
70 K to 90 K	18	42 to 44
90 K to 500 K	20	45 to 79

[muon2]

Of course, city councils are usually way off, for example, San Francisco has only 11 for a population of 800,000.

While Frankfurt has 93 for a population of 650,000.

Similarly Chicago has 50 aldermen, one per ward, and the rule would predict 142 for 2.85 M people.

For smaller cities divided into wards IL state law provides a statutory number of aldermen (two per ward) on the city council, though that number can be changed in certain cases (and generally reduced) by referendum or ordinance. IL law seems to use just under half the number of aldermen predicted by the cube root rule.

Population Range	Aldermen	Cube Root Prediction
up to 3 K	6	0 to 14
3 K to 15 K	8	15 to 24
15 K to 20 K	10	25 to 27
20 K to 50 K	14	28 to 36
50 K to 70 K	16	37 to 41
70 K to 90 K	18	42 to 44
90 K to 500 K	20	45 to 79

Those numbers in a city council I think would generate chaos myself.

Not at all. With a strong committee system and a mayor who understands procedure, the full council meetings can be to the point. The council  where I live has 14 and rarely has a chaotic meeting. The same statement applies to the town next door which also has 14 on the council.

Granted, many large cities reduce the size through referendum. For example, Aurora has 188 K population but uses only 12 aldermen. They reduced the 10 wards to one alderman per ward and added two at-large aldermen.

[muon2]

The People at www.RangeVoting.com have an explanation for the cube root rule (in addition to an interesting voting method idea):

Suppose some constant fraction of the Constituency probably wants to communicate with the Legislator, which is c·P/L communication with Him where c is a constant, P is the population of the country, and L is the cardinality of the legislature.

Meanwhile, each Legislator needs to communicate with all the Others (or anyhow a constant fraction of them) to get things done (e.g. convince Them to do something He wants). That's about k·L communication for each Legislator per thing He wants to do (where k is another constant).

If We now minimize c·P/L + k·L, by choice of L, We get the square-root law, L = (P · c/k)^(1/2),    i.e.    L ∝ √P which is the "optimum" legislature size which minimizes total communication to make something that Legislator wants, get done.  This formula is "optimum", if We assume each Legislator aims for some constant number of goals per (fixed length) term.

Suppose the communication with the Constituents is by mail or email or telephone; but the communication with fellow Legislators is face-to-face 1-on-1 meetings in random order. Further, all the Legislators are along one long corridor. Then each Legislator typically must walk a distance proportionate to L to reach a random target Legislator. So the difficulty of communication with the L-1 others is then not proportional to L, but rather to its square. In that case, We need instead to optimize by minimizing c·P/L + k·(L²), by choice of L, now getting the cube-root law, L ∝ P^(1/3).

If instead of one corridor, They sit in a 2-dimensional grid, then the typical walk-distance is proportional to √L.  In that case, optimizing is instead to minimize c·P/L + k·L^(1.5), by choice of L, now getting the two-fifths power law, L ∝ P^(2/5).

So it seems as though some power law is the "right answer" – although perhaps it is now not so clear what the correct power is!  (I do not see any good argument for L∝log(P).)

Interesting theory. Do they try to fit the different power laws to see which best matches existing bodies? I ask, since the link isn't working for me.

[muon2]

Nothing New Under The Sun

The Ohio Constitution of 1851 (which is the base for the current constitution) has a provision for apportioning fractional representatives to counties.  Counties entitled to more than one representative could have additional representation for each 1/5 of a quota, which would result in the additional member being elected to between 1 and 4 sessions of a decade,

The constitution was amended in 1903, supposedly at the behest of Mark Hanna, to guarantee each county one representation - but not to remove the fractional representation.  I can't find any evidence of any change until 1960's when the apportionment scheme was challenged in court.  But this appears to have been based on one man, one vote rulings based on the guarantee of one representative per county, and the at-large election of representatives (one plaintiff was the Cincinnati NAACP, and Hamilton County would have had around 10 representatives at the time).  The entire apportionment article was replaced in 1967.

The schedule of representation in the 1851 Constitution is still part of the Constitution.  At the time the quota would have been 19,803.  

There were 35 counties with between 0.5 and 1.2 quotas who had one representative for the entire period.  It is not clear how rounding was done.  Clark would have been entitled to 1.120 representatives and got 1, but Mahoning was entitled to 1.198 and had one representative for each session, plus one for the fifth.  So it appears that something like D'Hondt was used.  They may have had different numbers than modern census figures, or perhaps they fudged.  Seneca was entitled to 1.369 and got 1.2 members, while Brown was entitled to 1.380 but got 1.4.

There were 11 counties with 1.2 representatives; and 13 with 1.4; 4 with 1.6; and 4 with 2.0.  Muskingum had 2.0, Cuyahoga 2.4; and Hamilton 7.8.

There were 7 multi-county districts, mainly in the northwest.  There were 3 counties that could have had a district on its own merits (only 50% was needed), but it appears that they had a smaller neighbor or two attached.

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Even though OH no longer conducts an apportionment between the counties in the truest sense of their original constitution, the state still refers to the redistricting body as the apportionment board.