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Transactions . 



Art. XLV. — Preferential Voting in Single- member Constituencies, with 

 Special Reference to the Counting of Votes. 



By G. Hogben, M.A., F.G.S. 

 [Bead before the Wellington Philosophical Society, 24th September, 1913.] 



This subject was so fully discussed in October, 1882, by Professor B. J. 

 Nanson, of Melbourne University, in a paper on " Methods of Election," 

 read before the Royal Society of Victoria, that some apology seems to be 

 necessary for discussing it again. But if an interval of thirty years is not 

 in itself a sufficient reason for a restatement of the arguments then used, 

 I may be pardoned for saying that, absolutely conclusive as Professor 

 Nanson's arguments were, the method of counting the votes proposed by 

 him was so cumbrous as to deter the ordinary politician from giving the 

 case set forth in the paper the attention it deserved. 



In this paper I have described a method of counting that appears to 

 me much simpler than Nanson's, although in reality based on the same 

 general principles as his ; and I have added the outline of a proof which 

 is, to some extent at all events, independent of his proof . 



I shall begin by taking an example. Let us suppose that there are five 

 candidates, of whom one is to be elected, and that there are 400 voting- 

 papers, the votes being distributed as follows : — 



Table I. — Order of Preference shown. 



A B C D E on 100 papers. 

 B A C E D on 80 

 D E C A B on 100 

 E D C B A on 60 

 AC BED on 10 

 C B A D E on 40 

 BC ADE on 10 



(I assume, for this example, that these are the only preferences shown 

 out of all the possible permutations.) According to Nanson, the score- 

 sheet would be then exhibited as follows : — 



Table II. 



In order to ascertain which should be rejected, Nanson constructs, after 

 the method of Borda, the trial table below, which is obtained from Table II 

 by multiplying the first choices by 4 (one less than the number of candi- 

 dates), the second choices by 3, the third by 2, the fourth by 1, and the 

 fifth by 0. 



