226 Methods of Election. 



whichever of the candidates of the first group wins, the 

 result cannot be shown to be erroneous. If the division 

 into groups can be made in more than one way it is clear 

 that the last statement applies only to the smallest group of 

 the first kind. Now in the proposed method the successful 

 candidate must belong to the smallest group of the first 

 kind. Hence then it is clear that the result of the proposed 

 method cannot be shown to be erroneous in any case. 



It is clear that no candidate can have more than 

 N (2n — 2) votes on any scrutiny, n being a,s before the 

 number of unexcluded candidates at the commencement of 

 that scrutiny. For a candidate could only have this number 

 by obtaining the first place on each voting paper. 



Again, if any candidate obtain N (2n — 3) votes on any 

 scrutiny, there is an absolute majority in his favour, so that 

 we can at once elect him. For if a candidate were not put 

 first on half the papers, he could not have so many as 

 (n — 1) N + (n — 2) N votes, this being the number he 

 would have if he were put first on one half of the papers 

 and second on the other half. It is clear, too, that if any 

 candidate has less than N votes there is an absolute majority 

 against him ; for if a candidate has less than N votes, he 

 must be last on at least half of the papers. These results 

 are not of much use except in the case of three candidates ; 

 for if there be more than three candidates, it is only in cases 

 of remarkable unanimity that a candidate can have so many 

 as N (2n — 3), or so few as N votes. If, however, there be 

 three candidates only, the above results may be stated as 

 follows : — The average is 2N ; the largest number of votes 

 any one candidate can have is 4N ; if any candidate has 3N 

 votes, or more, there is an absolute majority for him, and 

 we can elect him at once, no matter whether the second 

 candidate is above the average or not ; if any candidate 

 has less than N votes, there is an absolute majority against 

 him, so that the result of the proposed method is demon- 

 strably correct. 



In the case of any number of candidates it will some- 

 times save a great deal of trouble if we first examine if 

 there be an absolute majority for or against any candidate. 

 This is easily done, and the results arrived at in the inquiry 

 will be of use in carrying out the proposed method, if such 

 be found necessary. For let A b A 2 . . . A M denote the 

 numbers of papers on which A occupies the first, the 

 second . . . the last or nth place, and let similar meanings 



