AF»Dix, APPENDICES.
APPENDIX I.
Extract from a paper read by Professor E.
J.
Nanson before the Royal Society of Yictoria,
October 12th, 1882, on methods of election to fill one seat:— If there be several candidates for an office of any kind and the appointment rests in the hands of several persons, an election is held to decide who is to receive the appointment.
The object of such an election is to select, if possible, some candidate who shall, in the opinion of a majority of the electors, be most fit for the post.
Accordingly, the fundamental condition which must be attended to in choosing a method of election is that the method adopted must not be capable of bringing about a result which is contrary to the wishes of the majority.
There are several methods in use, and none of them satisfy this condition.
The object of this paper is to prove this statement, and to suggest a method of election which satisfies the above condition.
Let us suppose, then, that several persons have to select one out of three or more candidates for an office.
The methods which are in use, or have been put forward at various times, may be divided into three classes.
The first class includes those methods in which the result of an election is arrived at by means of a single scrutiny.
The second class includes those in which the electors have to vote more than once.
The third class includes those in which more than one scratiny may be necessary, but in which the electors have only to vote once.
In describing these methods, the number of candidates will in some cases be supposed to be any whatever, but in other cases it will be assumed, for the sake of simplicity, that there are only three candidates.
The case in which there are only three candidates is the simplest, and it is of frequent occurrence.
I propose, therefore, to examine, for the case of three candidates, the results of the methods which have been proposed, and to show that they are erroneous in this case.
This will be sufficient for my purpose, for it will be easily seen that the methods will be still more liable to error if the number of candidates be greater than three.
I shall then discuss at some length the proposed method in the case of three candidates, and afterwards consider more briefly the case of any number of candidates.
Methods of the First Class.
In the first class three methods may be placed, viz.,
the single vote method, the double vote method, and the method of Borda.
In these methods the electors have only to vote once, and the resxilt is arrived at by means of a single scrutiny.
The Single Vote Method.
This is the simplest of all methods, and is the one adopted for Parliamentary elections in all English-speaking communities in the case in which there is only one vacancy to be filled.
As is well known, each elector has one vote, which he gives to some one candidate, and the candidate who obtains the greatest number of votes is elected.
This method is used for any number of candidates; but in general the larger number of candidates the more unsatisfactory is the result.
In this method, unless some candidate obtains an absolute majority of the votes polled, the result may be contrary to the wishes of the majority.
Eor, suppose that there are twelve electors and three candidates, A, B, C, who receive respectively five, four, and three votes.
Then A, having the largest number of votes, is elected.
This result, however, may be quite wrong; for it is quite possible that the four electors who vote for B may prefer C to A, and the three electors who vote for 0 may prefer B to A, If this were the case, and the question.
That A is to be preferred to B
were put to the whole body of electors, it would be negatived by a majority of two, and the question
That A is to be preferred to C would also be negatived by a majority of two.
Thus the single vote method places at the head of the poll a candidate who is declared by a majority of the electors to be inferior to each of the other candidates.
In fact if A and B were the only candidates B would win; or if A and C were the only candidates C would win; thus B and C can each beat A, and yet neither of them wins.
A wins simply because he is opposed by two men, each better than himself.
Thus the single vote method does not satisfy the fundamental condition.
It appears also not only that the best man may not be elected, but also that we are not even sure of getting in the second best man.
It is clear that if any candidate obtain an absolute majority of the votes polled this error cannot occur.
All we can say, then, about the single vote method is that if any candidate obtain an absolute majority the method is correct, but if no one obtains such a majority the result may be quite erroneous.
These results are well known, and consequently in elections under this plan great efforts are generally made to reduce the number of candidates as much as possible before the polling day, in order to avoid the return of a candidate who is acceptable to a small section only of the electors.
This reduction can, in practice, be made only by a small number of the electors, so that the choice of a candidate is taken out of the hands of the electors themselves, who are merely permitted to say which of two or more selected candi¬ dates is least objectionable to them.
The Double Vote Method.
In this method each elector votes for two candidates, and the candidate who obtains the largest number of votes is elected.
This method is erroneous, for it may lead to the rejection of a candidate who has an absolute majority of votes in his favour, as against all comers.
Eor suppose that there are twelve electors, and that the votes polled are, for A, nine; for B, eight; for C, seven, then A is elected.
]Sow, in order to show that this result may be erroneous it is merely necessary to observe that it is possible that each of the seven electors who voted for C may consider C better than A and B; that is to say, an absolute majority of the electors may consider C to be the best man, and yet the mode of election is such that not only does C fail to win, but in addition he is at the bottom of the poll This is an important result; we shall see presently the effect it has on other methods of election.
In the case in which there are only three candidates this method is, in fact, equivalent to requiring each elector to vote against one candidate, and then electing the candidate who has the smallest number of votes recorded against him.
Borda's Method.
This method was proposed by Borda in 1770, but the first published description of it is in the volume for 1781 of the " Memoirs of the Royal Academy of Sciences."
For some remarks on the method see Todhunter's "History of Probability," p.
433, where the method is described.
In the case of three candi¬ dates, in is as follows: Each elector has three votes, two of which must be given to one candidate, and the third vote to another candidate.
The candidate who obtains the greatest number of votes is elected.
In order to show that this method may lead to an erroneous result, suppose that there are twelve electors, of whom five prefer A to B and B to C, whilst two prefer A to C and 0 to B, and five prefer B to Q and F 2
