Winning the Chalkdust coin game

Go is a two-player strategy game. Players score points for surrounding territory and capturing opponents’ pieces. In 2016, Google challenged the world’s top Go player, Lee Sedol, to a five-game match against their AlphaGo program and won 4-1. The program was successful because it learned to play Go through a machine learning algorithm (specifically deep learning) trained on 30 million moves from games played by human experts (you can read more about AlphaGo here).

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Changing the status quo

In issue 3 of Chalkdust, Alex Bolton wrote an article about the maths of voting systems. In this week’s blog post, Alex looks at some interesting referenda relating to changing the status quo, and some difficulties in designing the system used.

In 2011, New Zealand was asked whether it would like to change its voting system. Voters were asked two questions:

  1. Should New Zealand keep the current voting system?
  2. If New Zealand were to change to another voting system, which voting system would you choose? (4 options were given)

If a majority were in favour of changing the voting system then the new system would be chosen using the alternative vote on the alternatives. This system has a clear problem: if both

  • 49.9% of the electorate think the status quo is the best system, and
  • the status quo would beat any alternative in a one-on-one contest,

A sample ballot paper for the 2003 California recall election.

A sample ballot paper for the 2003 California recall election.

the status quo can still lose because it requires a majority to think it is the best system in order to win. Knowing this, voters who would like to change the status quo are put in a difficult position. A voter who prefers alternative A to the status quo but prefers the status quo to alternative B may well have to vote unwillingly for the status quo or else risk the system changing to an alternative that they don’t like. In the end, 58% voted to keep the current system, but we do not know which system would have won the election if all the systems had been put together and people were asked to choose their favourite.

A similar system was used in the 2003 California governor recall election. Voters were simultaneously asked whether they wanted to recall the current governor Gray Davies, and who they would like to replace Davies. 55% of voters chose to remove Davies on the first ballot, and he did not appear on the second ballot. The Republican Arnold Schwarzenegger won the second ballot. It was very easy to be nominated as a candidate for the second ballot, and Arnold Schwarzenegger was one of 136 candidates, including former child actor Gary Coleman and 46 other Republicans.






flag6Candidates in the New Zealand flag referendum.

A different system was used for the 2015-16 New Zealand referendums on changing the country’s flag. Many alternative flags were submitted and five were chosen for people to vote on. An alternative vote contest was held to choose which of the five alternatives would compete against the status quo. This system is preferable to the first system because people vote on the status quo with knowledge about what the alternative would be, and people have no extra incentive to vote against the status quo when it is put on the ballot. Of course the disadvantage of this system is that the votes have to be held separately, increasing the cost of running the ballot for both the government and the electorate. Indeed, after the first vote to choose an alternative, 57% of New Zealanders chose to keep the original flag.

The design of a referendum on the status quo is critical to ensuring that the wishes of the electorate are understood. Care needs to be taken to ensure that the electorate have a good idea of what the alternative to the status quo is in order for voters to articulate their true preferences without unintended incentives to do the contrary.


Header image: Flag referendum candidates flying above Albatross Fountain on Wellington Waterfront, Flickr user glasnevinz, CC BY-SA 2.0.
Flags: (1) Public domain. (2) Kyle Lockwood, CC BY 3.0. (3) Kyle Lockwood, CC BY 3.0. (4) Public domain. (5) Alofi Kanter, CC By 3.0. (6) Andrew Fyfe CC BY 3.0


A mathematical view of voting systems

Soon after I began my undergraduate degree, Barack Obama won the 56th United States presidential election. The next day, my pure maths tutor asked me if I had followed it closely, saying that elections really interested him. I was surprised to hear this: surely the only maths involved was adding up the number of votes? In reality, voting systems hold considerable interest for mathematicians, and there are several mathematical results and theorems concerning electoral processes. The main thing that I like about the language of mathematics is that it allows us to make extremely precise statements without ambiguity and, as you’ll see, we can make precise mathematical statements about voting systems—with some surprising results.

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