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Alternative voting systems for choosing a voting system

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Martin Rosenbaum | 10:29 UK time, Tuesday, 2 February 2010

Gordon Brown is announcing today his plans for a referendum on electoral reform for UK general elections.

I therefore thought I should revive a post of mine from last year (with some corrections to its flaws in terminology, for which thanks to commenters). And apologies again for it being a little off-topic (although it could at least be argued in my defence here that freedom of information and electoral systems are both aspects of constitutional reform).

Ballot box So if there is to be a UK referendum on voting systems, what should be the actual voting system for that referendum? Or to put it another way, what decision-making system is it right to use when what you're doing is choosing between different decision-making systems?

Suppose there were four options on offer for a voting system - the current first past the post system (FPTP), the supplementary vote (SV) used in mayoral elections in London and elsewhere, the alternative vote (AV) now backed by Gordon Brown and some other Labour electoral reformers, and Borda, a points-based system advocated by those who regard it as a more consensual approach. (A full explanation of each system is given below.)

Suppose also that there are 100 voters and they happen to be divided into four opinion groups, with their order of preference for the voting systems as follows:

Voting table

Now what would happen if there was a referendum with all four options on the ballot paper with the votes counted under the various systems themselves (assuming no tactical voting)?

With such a distribution of public opinion, each system would win according to its own system and lose under all the others. Thus, if FPTP was used to count the result of the referendum, it would win; if SV was used, it would win; if AV was used, it would win; and if Borda was used, it would win.

So that leaves the puzzling question: what would be the fairest system for a referendum on electoral reform? Maybe what we need is a referendum to decide which voting system should be used in a referendum...?

To work through this example in detail:

Under FPTP, only the first preferences matter. FPTP has the most with 28 and it wins.

Under SV, the bottom two on first preferences (AV and Borda) are eliminated; FPTP gets no second preferences while SV gets 24 (from those who put AV first) to add to its 27 first preferences, so it scores 51, overtakes FPTP's 28 and wins.

Under AV, just Borda is eliminated on the first round. Since the 21 second preferences of those who put Borda first go to AV, on the second round AV has 45 votes and SV is now bottom as it still has 27. So SV is now eliminated, AV picks up another 27 votes in second preferences, so it now has 72 votes and beats FPTP, still on 28, in the final round.

Under Borda, FPTP scores 205 points, SV scores 257, AV scores 268, while it is Borda which finally seizes victory with 270.

So this analysis confirms that under this distribution of public opinion, each system wins if and only if the votes are counted in line with that system.

How different systems work

First-past-the-post What happened in 2010 In the current system, people get a single vote for who they want to represent their constituency and whichever candidate gets the most votes wins.

UK use: Election to Westminster and local government in England and Wales.

Single transferable vote Applied to 2010 result Several constituencies are combined and voters rank the candidates. Members are elected once they pass a certain number of votes, known as a quota.

UK use: Used in Northern Ireland for elections to Assembly, European Parliament and local government. Also used for local elections in Scotland.
Alternative vote Applied to 2010 result Voters rank the candidates. If no candidate has 50% of first preferences then second preferences are counted and so on until someone has a majority.

UK use: By-elections to Northern Ireland Assembly.

Alternative vote plus Applied to 2010 result The same as AV to elect most of the Commons but with a second element - the "plus" part - which would be used to elect 100 MPs in a more directly proportional system.
AV+ has yet to be put into practice anywhere in the world.
Proportional representation Applied to 2010 result The crudest version of proportional representation would give all parties seats in parliament based directly on their share of the vote. In practice, countries which employ PR have thresholds in place to screen out the smallest parties.
Simple PR is not in use in the UK.
Source: Electoral Reform Society

Notes on each voting system:

• First past the-post (FPTP), the system currently used for UK general elections. Voters have one vote and the candidate with the most votes wins.

• Supplementary vote (SV), as used in mayoral elections. Voters mark a first and a second preference. The first preferences are counted and all except the top two candidates are eliminated. The second preferences of the votes cast for the eliminated candidates are examined and where they are for either of the top two they are added to that candidate's total. The candidate from the initial top two with the highest total of first preferences plus second preferences transferred from eliminated candidates wins.

• Alternative vote (AV), which is also equivalent to applying the single transferable vote (favoured by the LibDems for multi-member constituencies) to a single-member constituency; and since a referendum would only be selecting one voting system, there's no point in my example here in having STV as a separate method. Voters list candidates in order of preference. After the first preference count, the bottom candidate only is eliminated, and the second preferences from his/her votes are added to the totals for the other candidates in line with those preferences. After this the candidate who is now bottom is eliminated, and more second/third preferences are transferred. Then the bottom candidate is eliminated and this process continues until only two are left and the candidate with the higher total of votes including transferred preferences is elected.

• Borda count - as with AV, voters list candidates in order of preference, but unlike AV Borda is a points-based system. If there are four candidates and the voter puts them all in order of preference, a first preference is worth four points, a second is worth three points, a third is worth two, and a fourth is worth one. The candidate with the highest points total wins.

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