As the election gets closer, and Labor are concerned about losing votes to the Greens, the old myth that voting Green might help the Liberals has arisen once again. This is effective because of a mathematical sleight of hand that tricks you into thinking there’s something in it, I’m going to explain why it’s not right.

Imagine a lower house electorate, where the count is

Labor 45%
Liberals 40%
Greens 6%
Family First 5%
Independent 4%

As nobody has over 50% the count will go to preferences. As Labor are only 5% short you would expect they should be the first to get to 50%, especially with the Greens having enough to get them there. Let’s say the independent splits 1% to each other candidate

Labor 46%
Liberals 41%
Greens 7%
Family First 6%

Next Family First, and their voters faithfully follow the how to vote and all preference the Liberals (actually this also includes the independent voter who preferences FF, lets suppose their next preference is also Liberal).

Liberals 47%
Labor 46%
Greens 7%

Now Labor are behind, but the Greens are eliminated next and their preferences go to Labor (including the Independent voter who chose Greens, they still control their own next preference, it doesn’t have to go along with the Greens voters).

Labor 53%
Liberals 47%

This is typical of many lower house counts (though note that a very large number are decided without any preferences even being distributed).
Now suppose that the Greens vote grows at the expense of Labor only Let’s say 15% change from Labor to Green, so the primary votes are

Liberals 40%
Labor 30%
Greens 21%
Family First 5%
Independent 4%

The sleight of hand is to say look, now the Liberals are in front, they are much more likely to win, they only need 10% whereas Labor need a whole 20%. This argument may be superficially appealing but it is misleading.
Let’s see that actually, nothing has changed in the two party preferred count. After distributing preferences for the independent and Family First we get

Liberals 47%
Labor 31%
Greens 22%

Now the Greens preferences transfer to Labor giving the same result as before.

Labor 53%
Liberals 47%

On the other hand, suppose at some point in the count the Liberals get over 50% in a race where Labor and Liberal are the two leading candidates. Then at this point over 50% of voters have expressed a preference of Liberals over Labor. If every voter who voted Green before preferencing Labor changes their 1 vote to Labor then it makes no difference to this, because none of them are in this majority. For example suppose we have

Liberals 40%
Labor 40%
Others (preference Lib) 11%
Others (preference Lab) 9%

After preferences Liberals win 51% to 49%. It’s really closed and if you voted Green and then Labor you might think “if only I put Labor first”, but there is no way it could make a difference. Suppose all of the 9% who voted for a smaller party but put Labor ahead of Liberal put Labor 1 instead.
Labor 49%
Liberals 40%
Others (preference Lib) 11%

It’s still not enough, after preferences Liberals win, that’s because the only way of changing it is shifting someone from preferring Liberal over Labor to the other way around. Otherwise we are just shifting votes around within the losing 49%, as long as 51% have Liberal ahead of Labor, they win. You, the Greens voter who prefers Labor, are not part of the 51% so changing your vote will not reduce it.

All of this assume as two horse race, which applies in the majority of electorates around the country, but not all of them. Suppose, starting from our original preferences, that enough people change Labor to Green so that the Green candidate overtakes Labor.

Liberals 40%
Greens 26%
Labor 25%
Family First 5%
Independent 4%

After first two eliminations we have

Liberals 47%
Greens 27%
Labor 26%

At this point Labor is eliminated, and the outcome now depends on the preferences of Labor voters instead of Green voters, it is up to them to stop the Liberals, in which case a Green would win.

In the earlier examples you might wonder why vote for a party who isn’t going to win anyway. There are a few reasons, and remember all of this assumes that you are considering voting Green in the first place, so you have decided they are your preferred party:

• You don’t know how everyone else will vote, it may be that your vote does help elect your favoured candidate
• You might not help them win this time, but you can make the seat winnable next time.
• If Labor wins on Green preferences they know they will know that voters in that electorate care about the issues the Greens are advocating.
• Your number  1 vote determines who gets public funding, given the preferential voting system there is no reason to direct your money to anyone other than your favoured candidate.