[EM] Hare, Droop and MEP

[DEMOREP1 at aol.com]

Changing Tom Round's `yesterday' post is lost to Hare, Droop and MEP----

I mention again that the Hare versus Droop math causes problems due to having 
1 vote per legislator seat and that the party votes/ party seats ratios will 
almost never be equal.

Also- for a small number of seats the Droop versus Hare ratio causes problems.

Divisors
H = Hare (and Seats), D = Droop
H     D     Droop as a percentage of Hare (plus 1 vote in all cases)
1     2     50.00
2     3     66.67
3     4     75.00
4     5     80.00
5     6     83.33
6     7     85.71
7     8     87.50
8     9     88.89
9    10     90.00
***
19   20     95.00
***
49   50     98.00
***
99  100     99.00
***
Etc.        Approachs 100.00

I suggest that BOTH Hare and Droop have problems since they both are based on 
integer units (i.e. to get a seat, a full Hare or a full Droop quota is 
needed (for all but the last seat) --- example 5 seats-  Hare quota = All 
votes/ 5,  Droop quota = (All votes / 6) +1 vote).

The U.S. currently apportions 435 U.S. Representative seats among the 50 
States based on their populations (See U.S. Constitution, 14th Amendment, 
Sec. 2) by the statutory method of equal proportions (MEP) used since the 
1940 U.S. population census.

MEP is based on the theory that the next seat should be given such that the 
percentage differences between any two population(P)/seat(S) ratios is 
reduced.

A, B, etc. = States, P=  Population, S = Number of Seats (before giving out 
the next seat),  R = P/S Ratio

PA/SA   = RA
PB/SB   = RB
PC/SC  = RC
etc.
The next seat goes to the State which lowers its relative inequality with the 
other States.

That is, for any 2 States, X and Y, one of the ratios will be nearer to one - 
RX (+1 seat)/RY or RX/RY (+1 seat).

In practice, the math involves dividing the State populations by a series of 
divisors (the square root of 1 x 2, 2 x 3, 3 x 4, etc.) with the State having 
the highest ratio at a given time getting the next seat.

With a large number (N) such square roots get very close to N + 0.5.

Example - Square root of 50 x 51 = 50.4975+

See-- Methods of Apportionment in Congress by Edward V. Huntington, 76th 
Congress, 3d Session, Senate Document No. 304 (1940) (41 pages) (in the U.S. 
Congressional Serial Set Vol. 10469).     The Huntington study resulted in 
the Nov. 1941 Act of Congress using MEP to apportion the U.S. Rep. seats 
among the States. 





 The U.S. Supreme Court upheld the constitutionality of the MEP law around 
1996.

The first approximation is obviously 435 x State Population / Total 
Population of all States (= some multiple of the Hare Total Population/ 435 
ratio) .  With an normal population distribution a State will get the integer 
number of seats in such multiple or the integer plus 1.

With MEP the ranges then become
N   Square root of N x (N+1)
1    1.41
2    2.45
3    3.46
4    4.47
5    5.48
6    6.48
7    7.48
8    8.49
9    9.49
10   10.49
11   11.49
12   12.49
13   13.49
14   14.49
Etc.

Example-- a State having between 1.41 and 2.45 [Hare] population ratios will 
get 2 seats (most of the time but with possible changes near the 1.41 and 
2.45 outer boundaries depending on the other borderline State ratios).

Change population to votes and States to parties for apportioning p.r. seats 
(with each seat having 1 vote in a legislative body).   

If a party has over a half ratio fraction, then it will get the additional 
seat in almost all cases.  Example - 5 seats. A Party gets 52 percent of the 
total votes = 5 x 0.52 = 2.6 Hare ratios.  It should get 3 seats.  

Another repair is to have combinational votes of all the parties.
Example -
A, B, C, D, E,  etc. = Parties

Each voter votes YES or NO for his/her choices and number ranks them.

Sample-
A NO  4
B YES 3
C  YES 1
D  NO 5
E  YES 2
F  NO  6

One Party Choice
C  (assume that no party gets a majority of the votes)

Two Parties Choice
CE (other voters may vote EC)

Three Parties Choice
CEB  (other voters may vote CBE, EBC, ECB, BCE, BEC)

Etc.

At some level, there may be a majority YES vote for one or more party groups.

If 2 or more groups get majority YES votes, then they would go head to head 
using the number votes (with the winning group getting a majority of the 
seats - to be apportioned among each other in like manner).

Proxy p.r. avoids ALL of the above math complexities.  Each legislator would 
have a voting power in a legislative body equal to the number of first choice 
and transferred votes that he/she finally receives.  A legislator who gets 
transferred votes would obviously notice the first choice (losing party) 
votes of such transferred votes.