Vote counting in the Australian Senate

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Recently we have seen senators behaving in ways that seem stupid, or contrary to accepted public opinion. And then people will start jumping up and down and complaining that such a senator only got a tiny number of first preference votes. One commentator said that one senator, with 19 first preference votes, "couldn’t muster more than 19 members of his extended family to vote for him". This displays an ignorance of how senate counting works. In fact first preference votes are almost completely irrelevant; or at least, far less relevant than they are in the lower house.

Senate counting works on a proportional system, where multiple candidates are elected from the same group of ballots. This is different from the lower house (the House of Representatives federally) where only one person is elected. For the lower house, first preference votes are indeed much more important. As for the lower house, senate voting is preferential: voters number their preferred candidates starting with 1 for their most preferred, and so on (but see below).

A full explanation is given by the Australian Electoral Commission on their Senate Counting page; this blog post will run through a very simple example to demonstrate how a senator can be elected with a tiny number of first preference votes.

An aside on micro parties and voting

One problem in Australia is the proliferation of micro parties, many of which hold racist, anti-immigration, or hard-line religious views, or who in some other ways represent only a tiny minority of the electorate. The problem is just as bad at State level; in my own state of Victoria we have the Shooters, Fishers and Farmers Party, the Aussie Battlers Party, and the Transport Matters Party (who represent taxi drivers) to name but three. This has the affect that the number of candidates standing for senate election has become huge, and the senate ballot papers absurdly unwieldy:

Initially the law required voters to number every box starting from 1: on a large paper this would mean numbering carefully from 1 up to at least 96 in one recent election. To save this trouble (and most Australian voters are nothing if not lazy), "above the line voting" was introduced. This gave voters the option to put just a single "1" in the box representing the party of choice: you will see from the image above that the ballot paper is divided: the columns represent all the parties; the boxes below the line represent all the candidates from that party, and the single box above just the party name. Here is a close up of a NSW senate ballot:

Almost all voters willingly took advantage of that and voted above the line. The trouble is then that voters have no control over where their preferences go: that is handled by the parties themselves. By law, all parties must make their preferences available before the election, and they are published on the site of the relevant Electoral Commission. But the only people who carefully check this site and the party's preferences are the sort of people who would carefully number each box below the line anyway. Most people don't care enough to be bothered.

This enables all the micro-parties to make "preference deals"; in effect they act as one large bloc, ensuring that at least some of them get a senate seat. This has been handled by a so-called "preference whisperer".

The current system in the state of Victoria has been to encourage voting below the line by allowing, instead of all boxes to be numbered, at least six. And there are strong calls for voting above the line to be abolished.

A simple example

To show how senate counting works, we suppose an electorate of 100 people, and three senators to be elected from five candidates. We also suppose that every ballot paper has been numbered from 1 to 5 indicating each voter's preferences.

Before the counting begins we need to determine the number of votes each candidate must amass to be elected: this is chosen as the smallest number of votes for which no more candidates can be elected. If there are \(n\) formal votes cast, and \(k\) senators to be elected, that number is clearly

\[\left\lfloor\frac{n}{k+1}\right\rfloor + 1.\]

This value is known as the Droop quota. In our example, this quota is

\[ \left\lfloor\frac{100}{3+1}\right\rfloor +1 = 26. \]

You can see that it is not possible for four candidates to obtain this value.

Suppose that the ballots are distributed as follows, where the numbers under the candidates indicate the preferences cast:

Number of votes A B C D E
20 1 2 3 4 5
20 1 5 4 3 2
40 2 1 5 4 3
5 2 3 5 1 4
4 4 3 1 2 5
1 2 3 4 5 1

Counting first preferences produces:

Candidate First Prefs
A 40
B 40
C 4
D 5
E 1

The first step in the counting is to determine if any candidate has amassed first preference votes equal to or greater than the Droop quota. In the example just given, both A and B have 40 first preferences each, so they are both elected.

Since only 26 votes are needed for election, for each of A and B there are 14 votes remaining which can be passed on to other candidates according to the voting preferences. Which votes are passed on? For B it doesn't matter, but which votes do we deem surplus for A? The Australian choice is to pass on all votes, but at a reduced value known as the transfer value. This value is simply the fraction of surplus votes over total votes; in our case it is


for each of A and B.

Looking at the first line of votes: the next highest preference from A to a non-elected candidate is C, so C gets 0.35 of those 20 votes. From the second line, E gets 0.35 of those 20 votes. From the third line, E gets 0.35 of all 40 votes.

The votes now allocated to the remaining candidates are as follows:

C: \(4 + 0.35\times 20 = 11\)

D: 5

E: \(1 + 0.35\times 20 + 0.35\times 40 = 22\)

At this stage no candidate has amassed a quota, so the lowest ranked candidate in the counting is eliminated - in this case D - and all of those votes are passed on to the highest candidate (of those that are left, which is now only C and E) in those preferences, which is E. This produces:

C: 11

E: \(22 + 5 = 27\)

which means E has achieved the quota and thus is elected.

This is of course a very artificial example, but it shows two things:

  1. How a candidate with a very small number of first preference votes can still be elected: in this case E had the lowest number of first preference votes.
  2. The importance of preferences.

So let's have no more complaining about the low number of first preference votes in a senate count. In a lower house count, sure, the candidate with the least number of first preference votes is eliminated, but in a senate count such a candidate might amass votes (or reduced values of votes) in such a way as to achieve the quota.