UNSW academics have cracked a problem in proportional representation posed by leading researchers – the latest chapter in a mathematical debate more than 200 years old.
In proportional representation, allocating parliamentary seats is not as simple as dividing them according to each party’s share of the vote.
For example, if Party A receives 34.5% of the votes in an election for a 100-seat parliament its exact entitlement would be 34.5, but there is no way of allocating 0.5 of a seat.
So how do you decide whether they get 34 seats or 35?
That might seem like a small decision, but in certain circumstances such choices can have significant consequences, especially when several parties need to join together to form a coalition to govern.
The problem is that in some existing methods there remains the possibility that when a coalition increases their share of the vote in a proportional representation system their chances of gaining a crucial extra seat can actually decrease.
The Sampford method
Now, a new paper in Spektrum by UNSW researchers has proven that one particular method known as ‘Sampford apportionment’ does not have the same counterintuitive issue, settling a recently posed conjecture in a mathematical debate more than two centuries old.
“This proof rules out a particularly troubling kind of tactical dilemma,” says Professor Haris Aziz, one of the co-authors of a working paper along with Dr Simon Mackenzie and Dr Mashbat Suzuki.
“We have shown that whatever party you vote for, and whatever coalition they might be in, that giving your vote cannot decrease the chance of reaching a certain number or a certain threshold.
“It gives that reassurance that is not actually true of other methods.”
The problem of how to fairly divide a fixed number of parliamentary seats in proportion to parties’ votes has been debated for more than 200 years, with Thomas Jefferson – a future US President – tackling the equivalent problem of dividing congressional seats among US states as far back as 1792.
Researchers have found that no single mathematical method can guarantee every kind of fairness they might want from a voting system.
More recently, they have explored methods that use an element of chance to distribute the final seats, while still ensuring that, on average, each party receives exactly its fair share.
But until now, no such method had been proved to provide this additional guarantee for coalitions.
“Politics is not about one party,” Prof. Aziz says. “Coalitions may care about whether they can reach a majority, block legislation, qualify for funding or cross some other politically important threshold.
“A system can therefore look fair for each individual party while still behaving strangely for a group of parties taken together.”
How the maths works
The paper uses a simple example involving three parties competing for four seats. Parties A and B are allies. Initially, their proportional entitlements are 0.9 seats each, while Party C is entitled to 2.2 seats.
Under a system that keeps each party within its fair rounding range and gives it its exact share on average, A and B have an 80% chance of receiving two seats between them.
Now imagine support shifts towards the alliance. A’s entitlement rises to 1.03 seats and B’s to 0.92, while C’s falls to 2.05.
The alliance’s chance of winning at least two seats now rises from 80% to 95%, while C’s chance of receiving a third seat falls from 20% to just 5%.
There is an apparent oddity: the chance that A and B are both technically “rounded up” falls from 80% to zero. But that is only because A is now guaranteed one seat rather than needing to be rounded up to it.
This illustrates why the researchers focus not on who gets rounded up, but on the politically meaningful question: does gaining voter support improve a coalition’s chance of reaching a particular number of seats?
In 2024, an international team of researchers – José Correa, Paul Gölz, Ulrike Schmidt-Kraepelin, Jamie Tucker-Foltz and Victor Verdugo – formally conjectured that the Sampford method – a sampling technique devised in 1967 by a University of Edinburgh statistician – would provide this guarantee for coalitions, but a proof had remained elusive.
The UNSW team has now provided that proof, establishing Sampford apportionment as the first method proven to combine three guarantees: every party receives one of the two whole-seat totals immediately around its exact entitlement; each party receives its exact proportional share on average; and a coalition that gains support never becomes less likely to reach any seat target.
The central result was derived with the help of extensive and interactive use of ChatGPT (Sol 5.6), and subsequently simplified and verified by the researchers.
Around 130 countries around the world, including New Zealand, Switzerland, Germany, Sweden and the Netherlands, use some form of proportional representation to elect their lower chamber – although in Australia preferential voting is used to elect federal MPs to each individual seat.
Amending voting systems
Despite the new proof regarding Sampford, Prof. Aziz acknowledges that changing long-established electoral systems is an unlikely effect.
“I don’t think just by presenting this mathematical proof that suddenly everybody is going to run their election in a new way,” he says.
“Most systems, whether they use proportional representation or even First-Past-The-Post systems like in UK general elections which have their own known problems and issues, have been in place for maybe hundreds of years and they are very hard to change.
“But people are intellectually curious about this, including all the way back to Thomas Jefferson in the United States in 1792 and it has been discussed for a very long time.
“This proof is not just about coalitions reaching a certain threshold, let’s say securing a majority in parliament. There might be a lower threshold where they receive certain funding, or get some kind of other benefit even without a majority.
“This work shows that for every threshold your chance of reaching that threshold improves when your votes increase, which bizarrely is sometimes not true with other methods. So it is giving reassurance to all coalitions that this method is responsive to increased support from voters.”
