214 Methods of Election. 



When, however, we have to deal with a body of men, this 

 result may easily occur, and no one of the candidates can be 

 elected without contradicting some one of the propositions 

 stated above. If this result does occur, then, no matter 

 what result any method of election may give, it cannot be 

 Remonstrated to be erroneous. We have examined several 

 methods, and all but the one now proposed have been shown 

 to lead to erroneous results in certain cases. It may fairly 

 be urged, then, that that method which cannot be shown to 

 be erroneous in any case has a greater claim to our considera- 

 tion than any of the other methods which can be shown to 

 erroneous. On this ground alone I think the method pro- 

 posed ought to be adopted for all cases. 



We can, however, give other reasons in favour of the 

 method proposed. We have seen that it gives effect to the 

 views of the majority in all cases except that in which the 

 three results (1), (2), (3) are arrived at. In this case there is 

 no real majority, and we cannot arrive at any result without 

 abandoning some one of the three propositions (1), (2), (3). 

 It seems most reasonable that that one should be abandoned 

 which is affirmed by the smallest majority. Now, if this be 

 conceded, it may be shown that the proposed method will 

 give the correct result in all cases. For it is easily seen that 

 the majorities in favour of the three propositions (1), (2), (3) 

 are respectively 2a, 2b, 2c. Hence, then, in the case under 

 consideration, a, b, c, must be all positive. Let us suppose 

 that a is the smallest of the three. Then we abandon the 

 proposition (1), and consequently C ought to be elected. 

 Now let us see what the proposed method leads to in this 

 case. B's score at the first scrutiny is 2N — c + a, and this is 

 necessarily less than 2N, because c is greater than a, and 

 each is positive. Again, C's score is 2N — a + b, and this is 

 necessarily greater than 2N, because b is greater than a, and 

 each is positive. Thus B is below the average, and C is 

 above the average. Therefore, at the first scrutiny B goes 

 out and C remains in. If A goes out also, C wins at the 

 first scrutiny. But if A does not go out, C will beat A at 

 the second scrutiny. Thus C wins in either case, and, there- 

 fore, the proposed method leads to the result which is 

 obtained by abandoning that one of the propositions (1), (2), 

 (3) which is affirmed by the smallest majority. We have 

 already seen that in the case in which the numbers a, b, c 

 are not all of the same sign, the proposed method leads to 

 the correct result. Hence, then, if it be admitted that when 



