[EM] A 48% Group elects 60% of the Droop Members

[DEMOREP1 at aol.com]

The Droop ratio approaches the Hare ratio with an increasing number of 
seats-----





S= Number of Seats, D/H = Droop/Hare ratio (as a percentage)

S    D/H (each is little more depending on the total number of votes)

1    50.00
2    66.67
3    75.00
4    80.00
...
9    90.00
...
19    95.00
...
49    98.00
...
99    99.00
...
999    99.90

Can a bare majority of Droop quotas in a single at large district produce 
indirect minority rule ?

S= Number of seats, T = Total votes

Odd number of seats-

(S+1)/2  x   (T/(S+1) +1) = (T + S + 1)/ 2  votes

Since the result is greater than T/2 the answer is NO.  
Note that with a large T and a small S, the result will barely be over 50 
percent of T.

Even number of seats-

(S/2 +1) x   (T/(S+1) +1) =  (S+2)/2 x  (T + S + 1)/(S+1)  votes

Not obvious.

S       Result (votes)

2       2T/3 + 3
4       3T/5 + 5
6       4T/7 + 7
...
10     6T/11 +11
...
100   51T/101 +101 
...
1000    501T/1001 +1001

With a large S the result nears T/2.

Since the result is greater than T/2 the answer is NO.

Note that with a large S, the result will barely be over 50 percent of T.

To avoid borderline apportionment problems in multi-party elections, there 
would seem to be a need for a majority requirement (i.e. any 2 or more 
parties with a majority of the votes must get a majority of the seats -- but 
this may not be possible due to conflicting overlapping partial majorities 
regarding the marginal seat(s) -- especially with 4 or more parties.)

Again I note that the various ratios of party votes/party seats for the 
different parties will almost never be the same (especially with a low number 
of seats in the legislative bodies of many private groups and local 
governments).

I note again -- proxy p.r. avoids most, if not all, of the above math 
problems.