Hogben. — Preferential Voting in Single-member Constituencies. 305 



A 

 B 

 C 

 D 

 E 



Table III (First Trial Table). 



440 + 240 + 100 + 100 + = 880 points, x « 

 360 + 420 + 20 + 60 + = 860 „ | §• 

 160+ 60 + 680+ + = 900 „ 

 400 + 180+ + 150 + = 730 „ 

 240 + 300 + + 90 + = 630 





Nanson then rejects all those whose trial totals are not greater than 

 800, the average of all the trial totals. (He shows by a rigid process the 

 soundness of this step.) A new score-sheet similar to II is then made for 

 the remaining candidates, A, B, C, disregarding votes given for D and E. 

 It will be as follows : — 



Table IV. 



Multiplying the first choices by 2, the second by 1 , and the third by 0, 

 we get a trial table similar to Table III, thus : — 



A 

 B 



C 



Table V (Second Trial Table). 



.. 220 + 180 + = 400 points.' 

 .. 180 + 200 + = 380 „ 

 . . 400 + 20 + = 420 „ 



Shi 



The average of the totals is 400 ; he cuts out A and B, who have totals 

 not greater than the average, and C is accordingly elected. Assuming that 

 the preferences are truly and rightly given, and that each voter exercises 

 all his preferences, this method is infallible. 



Nanson proposes to neglect the cases in which no preference is shown 

 for more than a few candidates ; but his method could be made complete 

 by distributing the points that would thus be unassigned in Table III 

 equally among the unmarked candidates. It is generally agreed, however, 

 that Nanson's method, although mathematically sound, is too cumbrous for 

 actual use. 



But it can be shown that a trial table constructed as Table VI below 

 gives the same totals as Nanson's Trial Table III above, and that the other 

 trial tables required can be derived directly from it without any fresh count 

 or calculation, except mere addition of figures already obtained. 



The Trial Table VI is thus constructed : We make as many lines and as 

 many columns as there are candidates ; we count the number of ballot- 

 papers on which the first candidate on the list is preferred to the second, 

 all other names being disregarded, and papers on which no preference is 

 shown for either of these candidates being divided equally between them. 

 The number so found is inserted in the second line of the first column — that 

 is, total preferences of A over B = 210. (The same count gives us the 

 number for the first line of the second column — that is, the number of 

 papers (190) on which the second candidate is preferred to the first.) Each 

 of the other spaces in the table is similarly filled up. 



