Methods of Election. 223 



(ix.) If we suppose that 



N N N 



« = -o + P ( b — c )> P = ~o + P (° — a )>7= -o + P («— V), 

 o o o 



then A's scores on the single, double, and Borda methods 

 will be respectively 



2N 4N 



— - (p + 1) (b-c),— + p (K-c), ^ - V-c). 



Hence we see that 



If p < o and > — 1, the results of all three methods will 

 be the same. 



If p < — 1, double and Borda methods will give the same 

 result, which will be opposite to that of single method. 



If p > o, single and Borda methods will give the 

 same result, which will be opposite to that of double method. 



Thus, if p > o or < — 1, single and double methods will 

 give different results. If we suppose that b, c are positive 

 and a negative, and also that 26 < c + a, then it may be 

 shown that these different results will both be wrong. 



Cases of More than Three Candidates. 



It remains now to state and examine the method pro- 

 posed for the case in which there are more than three can- 

 didates. 



A series of scrutinies are held on Borcla's system of voting, 

 and all candidates who on any scrutiny have not more than 

 the average number of votes polled on that scrutiny are ex- 

 cluded. As many scrutinies are held as may be necessary 

 to exclude all but one of the candidates, and the candidate 

 who remains uneliminated is elected. 



The method proposed cannot lead to the rejection of any 

 candidate who is in the opinion of a majority of the electors 

 better than each of the other candidates, nor can it lead to 

 the election of a candidate who is in the opinion of a 

 majority worse than each of the other candidates. These 

 results are an extension of those alreacty proved for the case 

 of three candidates, and they may be proved as follows : — 

 As before, let 2N be the number of electors, and let the can- 

 didates be denoted by A, B, C, D, &c. Let the compound 

 symbol ab denote the number of electors who consider A 

 better than B, and let corresponding meanings be given to 

 ac, ad, ba, &c, so that ba will denote the number of electors 

 who prefer B to A, and we shall, therefore, have ab -f ba 



