224 Methods of Election. 



= 2N. Now suppose that at the commencement of any 

 scrutiny the unexcluded candidates are A, B, C, . . . . P, 

 then the score of A on that scrutiny will be 



ab + ac + ad + . . . . + ap. 

 For suppose that there are n unexcluded candidates, and 

 consider a voting paper on which A now occupies the rth 

 place. For this A gets n — r votes. Now on this paper A 

 stands before n — r other candidates. Thus the n — r votes 

 which A receives may be considered each as due to the fact 

 that A stands before one of the following n — r candidates. 

 Thus we see that on any one voting paper A receives one 

 vote for every candidate placed after him. Summing up for 

 all the voting papers, we see that A receives one vote for 

 each candidate placed after him on each paper. Now 

 ab denotes the number of times B is placed after A on all 

 the papers, and similarly for ac, ad, &c. Thus it is clear 

 that A's score is 



ab + ac + ad + . . . . + ap. 

 This result was stated by Borda,* but proved only for the 

 case of three candidates. 



The whole number of votes polled is 



2N (1 + 2 + 3 + 4 ... + n—l) 

 or N% (n — 1). Thus the average polled by all the candidates 

 is "N(n — 1). Now let us suppose that there is a majority for 

 A as against each of the other candidates, then each of the 

 n — 1 numbers ab, ac, ad, . . . . a/pis greater than N; 

 thus the sum of these numbers, which is equal to A's score, 

 is necessarily greater than (n — 1) N, that is, greater than 

 the average score. Thus A will be above the average on 

 every scrutiny, so that he must win on the proposed method. 



Next, let us suppose that there is a majority for each of 

 the other candidates against A. Then each of the numbers 

 ab, ac, . . . ap is less than N, and therefore their sum, which 

 is equal to A's score, is less than (n — 1) N, that is, less than 

 the average score. Thus A is below the average, and will, 

 therefore, be excluded at the first scrutiny. 



The results which have just been proved are particular 

 cases of a more general theorem, which may be enunciated 

 as follows : — 



If the candidates can be divided into two groups, such 

 that each candidate in the first group is, in the opinion of a 



* Mdmoires de V Academic Royal des Sciences, 1781, p. 663. 



