Counting

I am considering some issues relating to the conduct of elections in the UK. In Lewisham Labour we are selecting a candidate for Mayor and the election will be conducted using a simple eliminating ballot, sometimes referred to as an Alternative Voting scheme. This, it can be argued is a special case of a more proportionate voting system; the special nature being that there is only one winner and thus the result is not so proportionate. The rest of this article is a technical description of an alternative vote election, a single transferable vote election and a closed list proportional representation election using the D’Hondt counting system.

Alternate Vote

An AV ballot result might look like this.

The ballot paper requires voters to mark the candidates in order of preference where 1 is the first choice, and 2 the second etc. It is not mandatory to mark all candidates in order.

In my example, we have four candidates. The votes cast are as state in the “Round 1 Votes” column. In this example, because Dave’s votes are less than the difference between the next two candidates, only he is eliminated and his votes redistributed to his voter’s second choice; two voters chose not to express a second preference. The votes are then recounted including the second preferences. In our example, Anne comes last and her votes are redistributed in the final round and Charlotte wins. In all rounds, the transferred votes count as a single vote.

Multi-member Single Transferable Vote

Proportional counting systems are more complicated, either based on a slight tweak of the system above or based on list systems. The first variant is called the Single Transferable Vote. I have made another example below.

In this example, we have six candidates for three places. Voters state their preferences as before. The system requires that a winning number of votes is calculated, this is called the quota. This is the number of votes cast divided by the number of seats/winners plus 1.

Q = Σ Votes Cast / ( Seats + 1)

In our example, we have three seats; the quota is 25% of the votes cast and the formula ensures that minorities can win, although in this case, as we’ll see, the victory threshold is quite large. The winning quota is lower if the constituency has more seats. The more seats/winners within an electorate, the more proportional the result. This system is used for electing the Northern Ireland Assembly, local elections in Scotland and Northern Ireland and European Parliament elections in Northern Ireland.

In the example above, there are 1432 votes cast, and the quota is 359. In the first round, Charlotte’s votes exceed the quota, her surplus is calculated and redistributed. Surpluses are redistributed first. Each ballot cast for the winning candidate is examined and their second preference counted and the transferred. Each transfer is valued as the proportion of the surplus to the votes cast. In this case, 111/470, the fact they are illustrated as transferred as cardinal numbers is a feature of the example.

TV = Surplus / (Votes Cast)

After the transfer of Charlotte’s surplus, no further candidate has achieved a quota, and so the bottom candidate, Frank, is eliminated. His votes are redistributed. Frank’s votes need to be examined and his first preferences redistributed as 1 vote and his transfers from Charlotte are transferred as .24 of a vote. Since no-one has achieved the winning quota, Dave is also eliminated in the next round. Dave’s votes are redistributed, his first preferences redistributed as 1 vote and his transfers from Charlotte are transferred as .24 of a vote and his transfers from Frank valued depending on whether they are 2^nd preferences (from Frank, 1 vote) or 3^rd preferences (from Charlotte, .24 vote). It takes two rounds because Frank’s vote count was greater than the difference between Dave & Edwina.

The redistribution of votes in counting round 3, mean that Edwina and Anne overtake Bertie who is eliminated in the final round of counting.  If Dave’s votes had split more decisively in favour of Edwina or Bertie such that one of them exceeded the quota, then their surplus may have needed to be redistributed to see if it made a difference for Anne.

In our example Bertie & Edwina are declared elected to seats No. 2 & 3.

Closed List (D’Hondt counting)

For the European Parliament election in Great Britain  and Greater London Authority City wide list a further form of PR is undertaken. Parties register lists of candidates, and voters vote for their preferred party.

The counting is done using a method called the Dhondt method.

A quota is calculated. This varies during the count. The quota is calculated as follows;

Quota = Votes Cast / ( Seats won + 1 )

I have an example, again there are three seats.

Parties are ranked in order of the votes cast, and the first winner is declared, and their quota is adjusted. In my example, the Red Party won and the win the first seat. Their quota is adjusted to 235, which is lower than the Blue score. The Blues get the second seat and their quota is adjusted (124.5) The Red’s second round quota is higher than the remaining party’s first round scores and so it gets the 3^rd seat.

Since this method requires an iterative sort and knowledge of the iteration count, this is exceedingly hard to program in SQL, ( I have had a second thought about how to do it in Excel and documented how to do it here …) but it does not require a second or further examination of the ballot papers and thus is much cheaper to count. It does give the party machines much more power since they can make a take it or leave it offer, and in the case of D’Hondt, they control the order in which candidates are elected. In STV, the electorate make that choice by expressing a preference between candidates in the same party.