Our elections are free but are they fair?


After the September voting gave NZ First 7% everyone must wonder why this group has the power to create the next Government. It seems that MMP is the reason and because a 1993 referendum chose it over FPP we also wonder why. The first thing to realise is that democracy is unfair – according to Ian Stewart writing for the New Scientist.

In an ideal world, elections should be two things: free and fair. Every adult, with a few sensible exceptions, should be able to vote for a candidate of their choice, and each single vote should be worth the same.
Ensuring a free vote is a matter for the law. Making elections fair is more a matter for mathematicians. They have been studying voting systems for hundreds of years, looking for sources of bias that distort the value of individual votes, and ways to avoid them. Along the way, they have turned up many paradoxes and surprises. What they have not done is come up with the answer. With good reason: it probably doesn’t exist.

First Past the Post (FPP)

In first-past-the-post or “plurality” systems, borders matter. To ensure that each vote has roughly the same weight, each constituency should have roughly the same number of voters. Threading boundaries between and through centres of population on the pretext of ensuring fairness is also a great way to cheat for your own benefit a practice known as gerrymandering, after a 19th-century governor of Massachusetts, Elbridge Gerry, who created an electoral division whose shape reminded a local newspaper editor of a salamander.

Maths is used to illustrate how a minority can win using the careful drawing of boundaries which is possible in a big city like Auckland. Here is another case where ranking is used.

The anomalies of a plurality voting system can be more subtle, though, as mathematician Donald Saari at the University of California, Irvine, showed. Suppose 15 people are asked to rank their liking for milk (M), beer (B), or wine (W). Six rank them M-W-B, five B-W-M, and four W-B-M. In a plurality system where only first preferences count, the outcome is simple: milk wins with 40 per cent of the vote, followed by beer, with wine trailing in last.

So do voters actually prefer milk? Not a bit of it. Nine voters prefer beer to milk, and nine prefer wine to milk – clear majorities in both cases. Meanwhile, 10 people prefer wine to beer. By pairing off all these preferences, we see the truly preferred order to be W-B-M  the exact reverse of what the voting system produced. In fact Saari showed that given a set of voter preferences you can design a system that produces any result you desire.

What about an ideal system?

The American economist Kenneth Arrow listed in 1963 the general attributes of an idealised fair voting system. He suggested that voters should be able to express a complete set of their preferences; no single voter should be allowed to dictate the outcome of the election; if every voter prefers one candidate to another, the final ranking should reflect that; and if a voter prefers one candidate to a second, introducing a third candidate should not reverse that preference.

All very sensible. there’s just one problem: Arrow and others went on to prove that no conceivable voting system could satisfy all four conditions.

Proportional representation (including MMP).

NZ is not the only country that has had trouble with proportional representation, Germany has a similar system with similar woes.

One criticism of proportional voting systems is that they make it less likely that one party wins a majority of the seats available, thus increasing the power of smaller parties as “king-makers” who can swing the balance between rival parties as they see fit. The same can happen in a plurality system if the electoral arithmetic delivers a hung parliament, in which no party has an overall majority – as might happen in the UK after its election next week.

Where does the power reside in such situations? One way to quantify that question is the Banzhaf power index. First, list all combinations of parties that could form a majority coalition, and in all of those coalitions count how many times a party is a “swing” partner that could destroy the majority if it dropped out. Dividing this number by the total number of swing partners in all possible majority coalitions gives a party’s power index.

Better minds than mine can grapple with these problems. Simple democratic solutions seem to be impossible. Some more complex ones also have problems. The Banzhaf power index may get closer to democracy but will then cause a problem because no one will understand it. In the US people still cannot understand why Trump won with 3mill less votes but with an outstanding electoral college advantage.

Locally, why cannot Bill and Jacinda talk it over?


This post was written by Intern Staff

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