222 Methods of Election. 



For, in the demonstration of the last proposition, the 

 essential condition is that a + c — 26 should be positive. 

 Now, if we suppose as before that the accepted result is A 

 better than B, and B better than C, we must have a, b, c all 

 positive and b the smallest of the three, so that it is clear 

 that a + c — 26 is positive. 



Comparing then Ware's method with the Venetian method, 

 we see that both may be right, or one wrong and one right, 

 but both cannot be wrong ; so that, if these two methods 

 agree, the result cannot be shown to be wrong. If, however, 

 they do not agree, we cannot tell which is right without in 

 effect having recourse to the proposed method. 



(vi.) If a = b = c, single and double vote methods give 

 different results. 



For A's scores on the two methods will be respectively 

 N — a and N + a. Thus, if y > /? > a, the candidates are 

 in the order A, B ; C on the single vote method, and in the 

 order C, B, A on the double vote method. In this case 

 Borda's method leads to a tie, and consequently the proposed 

 method also. Ware elects A or B as c is positive or negative, 

 and Venetian method elects C or B as a is negative or posi- 

 tive. Thus, in this case, Ware and Venetian method give 

 different results. 



(vii.) If a =.p = y, double vote method, and therefore also 

 Venetian method, gives a tie ; single vote method and Borda 

 lead to same result ; but Ware and proposed method will not 

 necessarily lead to same result. If one only of the three, 

 b — c, c — a, a — b, be negative, Ware and proposed method 

 will lead to same result ; but if two be negative the results 

 may or may not agree. 



(viii.) If AB = AC, BC = BA, CA = CB, all the 

 methods will give the same result, and that result will be 

 demonstrably correct. 



This is the case in which the strong supporters of each 

 candidate are equally divided as to the merits of the remain- 

 ing candidates. In this case we have 



a = fi — y, b = y — a, c = a — /?, 

 and A's scores on the single, double, and Borda's method are 

 respectively 2a, N + a, N + 3a. Thus, if a > j3 > y } it is 

 obvious that each of these methods will put A first, B second, 

 and C third, and it is clear that this result is correct, for a, c 

 are positive and b negative. It is at once seen that all the 

 methods which have been discussed will lead to the same 

 result in this case. 





