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Voting matters - Issue 10, March 1999

A problem for Andrae and Hare

I D Hill

David Hill's great-great-great-grandfather is the Thomas Wright Hill mentioned in this article.

With any form of STV there is a question about the best way to transfer surpluses when they arise. Some people seem to think that provided the right number are taken, and no vote is specifically misused, it does not much matter how it is done. Others claim that such conditions are not sufficient, and that methods should be used that correctly interpret the wishes of the relevant voters as a whole.

The argument turns up interestingly in a fascinating book, to be found in the McDougall Library Andrae and Proportional Representation by Poul Andrae, son of Carl Andrae who introduced STV to Danish elections in 1855. The book is partly a complaint that Thomas Hare gets all the credit for the invention of STV and his father very little. Hare first suggested STV in 1857, whereas Andrae actually introduced it in 1855. The complaint appears to be justified and it looks as though perhaps Hare himself did not really want to know about Andrae, but it is always dangerous to judge something like this after hearing argument from one side only. The author of the book is evidently totally unaware of what Thomas Wright Hill did in 1819.

Andrae's system was simply to shuffle the voting papers and then count them just once, allocating each to its earliest preference who had not already attained a quota, and finally elect all those with a quota, plus the highest of those with less, to give the right number to fill the seats. There was no system of exclusions, with redistribution of those votes. Hare's earliest versions were somewhat similar to this.

On the question of how to redistribute a surplus, there is in the book a problem that was put to Andrae, of a case where it was said that his system could give an absurd answer. Andrae, in reply, points out that one of the rules of his system is that the voting papers are to be thoroughly shuffled before counting and, if that rule is obeyed, the probability that they are counted in the particular order on which the absurd result depends is so small that it can be ignored. In this he is correct (and he calculates the probability correctly too).

However the problem was also put to Hare, and Hare's reply is to try to justify the absurd answer as reasonable. I wonder whether any STV supporter nowadays would agree with Hare.

The problem concerns 5 candidates for 3 seats, and votes:

299 ABD
200 ACB
101 ACE
Hare and Andrae agree that the quota should be 600/3 = 200 and for present purposes let us not dispute that, even though we think that Droop's quota is preferable. The problem says: suppose the votes are counted in the order as given, using Andrae's system. Then, of the first 299, 200 go to A and 99 to B, the next 200 all go to C (leap-frogging A) and the final 101 to E (leap-frogging both A and C). As the system does not use exclusions, the final seat is awarded to E, because 101 exceeds 99 even though nowhere near a quota.

Andrae's correct reply is that, even in the unlikely event of such votes being made, the probability merely that all the 299 come out before any of the 200 is 1/q where q is a number of 117 digits, without even taking account of the fact that all those have to come before the final 101. This is certainly a remote enough probability to be ignored.

He does not mention that a similarly silly answer could result from

where the probability is as high as 1/90, but I feel sure that he would have said that his system was designed for big elections, not such tiny ones, though to my mind a good system ought to work sensibly for any size of election.

Hare, however, according to the book, wrote

I am willing ... to adopt the result, which I believe is perfectly reconcilable with the principle that is at the foundation of this method of voting, and also reconcilable with justice. The object is to give the elector the means of voting for the candidate who most perfectly attains his ideal of what a legislator should be, but it does not contemplate giving him the choice of more than one ...

The primary purpose of giving the voter the opportunity of adding to his paper the second, third, or other names for one of whom his vote is to be taken on the contingency of the name at the head not requiring it, is not to add greater weight to his vote, but to prevent it from being thrown away or lost owing to a greater number of voters than is necessary placing the same popular candidate at the head of their papers ...

Thus the first 200 voters, whose voting papers are appropriated to A, have no ground of complaint (because of the non-election of B), for their votes have been attended with entire success ... Still less have the second 200 voters, whose votes were appropriated to C, any reason to complain, for they also have not only elected a favourite candidate of their own, but, equally with the first 200, they are gratified by the triumphant success of A. The 99 voters for B have also the latter satisfaction, and if they failed to return their next favourite candidate, it is simply because 101 are more than 99.

I should have to change my mind about supporting STV, if that were good STV reasoning, but I do not accept that it is. I agree that it is right to allow each voter just the one vote, but if 299 say AB whereas 301 say AC, to pass A's surplus as 301 to C and only 99 to B, instead of dividing it out in proportion to the voters' wishes, is grotesque.

It is extraordinary that Hare thinks it just and reasonable to elect E even though the total number of voters mentioning E at any level of preference is far less than a Droop quota. Any modern STV system would take the quota as 600/4 = 150, elect A with a surplus of 450 to be divided almost equally between B and C, who then each have more than a quota and all seats are filled.

Even if the votes had been merely

200 ACB
101 ACE
to elect ACE rather than ACB would be obviously absurd. With the additional 299 ABD votes it becomes even more so. Does any reader think that Hare was talking sense?
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