BY E. L. PIESSE, B.SC, LL.B. 33 



1913. 



74. The following is an investigation of the conditions 

 in which the larger of two nearly equal parties may get 

 only a minority of seats in a country divided into single- 

 member constituencies. It is assumed that in each dis- 

 trict there are only two parties each with one 

 candidate, or that if there are more than two candidates a 

 system of preferential voting is used to secure that the 

 candidate returned has received votes from an absolute 

 majority of the voters; and that the constituencies are 

 equal in the number of voters. 



Let .S^ be the party which has less than half of the votes 

 polled throughout the country, and let L be the party 

 which has more than half; and let the strengths of S, L 

 throughout the country be (50 - a) % and (50 + a) %. 

 Let there be 100 constituencies and v voters in each. 



Consider the constituencies in each of which S has a 

 majority, and consequently wins the seat; let there be x 

 of these, and let the average strength of *S^ in them be 

 (50 + s)%. Consider also the constituencies in which L 

 has a majority ; let there be y of these, and let the aver- 

 age strength of I in them be (50 + I) %. 



Then, considering the total number of votes obtained 

 by S, we have : 



50 + .S , 50-/ ^O-rr ,^^ 



100 100 100 



or 



.^0 + , ^ 50-/ 



+ ~T7^. • y = ^0-rt (1) 



100 100 



Similarly, from L's votes we get — 

 5()-« .'0 + / 



Also — 



X + y = 100. (3) 



Subtracting (1) from (2), we get — 



ly - sx ^ 100a (4) 



The condition for equal representation of the larger and 

 smaller parties [x = y — 50) is 



I - s = 2a. (5) 



As an example, let the average strength of ^S* in the dis- 

 tricts in which it is in a majority be 51 %, and let the aver- 

 age strength of L in the districts in which it is in a 

 majority be 57 %, and let the average strengths of the two 

 parties throughout the country be 47 %, 53 %. The values 

 of s, /, a are then s = 1, / = 7, o = 3, and I - s = 2a. 



