METHODS FOR COUNTING IN ELECTIONS. 66l 



than one-third of the votes to secure election, and two less than 

 oaie-third of the votes are necessarily non-effective. So if four 

 members are to be chosen the quota is one more than one-fifth 

 of the votes ; the necessarily non-effective votes are four less 

 than one-fifth of the votes. 



If N represents the number of voters, and n the number of 



N 

 members to be elected, the quota under this system is [- i. 



"■ + ' 

 This is the absolute majority quota. 



It is generally known as Droop's quota by those interested 

 in proportional representation. 



Droop's quota can be used to find the members most pre- 

 ferred when the votes are all of one grade of choice ; where 

 only one choice has been marked by the voter, as in the case of 

 the single non-transferable vote; or in cases with the trans- 

 ferable vote where the election is over on first choices only. In 

 those two cases it correctly shows which are the members most 

 preferred by the voters, for no transfers are made. 

 Droop's quota is the smallest quota that can be used 

 with a majority system of counting; for, if we diminish it by 



A' 



even one unit, we get the sum , a sum which it is evident one 



n -f- I 

 more than n members could each get ; for N is divided in w -|- i 



N 



equal parts. Thus it is evident that [- i is the smallest pos- 



n -f I 

 sible njajority quota. It is " the absolute majority "' quota, and 

 by it the smallest possible number of first choice votes and of 

 second choice votes, etc., would be counted to the member. But 

 to find out the members most preferred by all the voters, when 

 they have the transferable vote, the largest possible number 

 of first and other high-grade choices must be counted to them, 

 as is only done when we use Hare's quota, A'' divided by ;/, which 

 we have seen must be used to secure Proportional Representation. 



4. The Single Transferable Vote. — In England in 1857 

 Hare published a system of voting, which had been introduced 

 in Denmark two years before by Andrae, a famous mathema- 

 tician. This system was afterwards very strongly advocated by 

 John Stuart Alill. It is the system of the single transferable vote. 

 Among its merits it secures : — 



(a) That the largest possible number of votes, and of 



first and other high-choice votes, is allotted to and made 

 effective for the selection of the members ; 



(b) That every voter has the opportunity of giving a vote 



which secures for him representation by the member he 

 most prefers. 



(c) That every member represents the largest possible equal 



number of voters ; and 



