As a scientist people often ask me what's the difference between FPTP, AV and Deal or no Deal.  The answer is nothing - All three involve guessing what's inside a box before it is opened so as to make a seemingly important tactical choice on no real statistical information using a very poor grasp of elementary mathematics.

However, as both the Yes and No to AV campaigns seem to have totally avoided any real statistical analysis here is some elementary mathematic analysis of AV what I carried out as part of a software beta test.

AV is a form of runoff voting.  Runoff voting has a long history and was invented by the political class as a way of getting on with each other just enough for mutual survival because voters dont tend to vote for parties that are split. 


Runoff voting is where instead of one ballot you have a series.  Eliminating the lowest polling candidates one at a time till there are two left.  Sometimes the balloting must continue until not only is there a winner but they have passed a set threashold.  Classic examples of runoff voting are the selection of Popes ....


....and of leaders of the Conservative party or any other situation where the electorate cannot be open about their real opinions and the candidates cannot be seen to be openly canvassing for support.  Either because it might be seen as disloyalty or because it's supposed to be down to the Holy Spirit and not a political process at all...


In the latter quarter of the 20th century voting system problems plagued the Parliamentary Conservative Party when they decided to remove Ted Heath and later Margaret Thatcher.  Their challengers could not be seen to openly canvass for support so rules were invented that in the first round ballot the winning candidate must get more than 50%.  Such runoff ballot systems are traditionally utilised for leadership elections.  Another reason that runoff ballots are utilised in such elections...



...is that often such elections are not secret ballots so the leader has the advantages of intimidation to secure their 50% support.  Thus it used to be felt that if the leader couldn't rely on the open support of at least 50% of the parliamentary party they were ...erm... not a leader as by definition the party had split.  Thus a new leader was required.  As FPTP creates a two and a half party system for reasons explained below the main parties are regularly split over a large number of issues ; Europe, Immigration, whether or not to support the AV referendum, the Iraq War, the 10p Tax rate and all the other issues that never make it into any manifesto but seem to become very important once party in power has been elected.

Also it is VERY important for the leaders of the main parties to visibly have over 50% party support for Constitutional reasons.  Technically the Queen appoints the Prime Minister.  Of course this is a bit of a fairy tale in the 21st century (no disrespect to the Queen she's a very good compere) so to keep it alive we are now supplied with photographs of the appointment process...


Which look remarkably like the title card of an old episode of Minder.  However, the meaning of this odd ritual is that the PM now has the right to he appoint the hugely well paid government jobs - so the LEADER is required to have enough support in the House of Commons to avoid his government being terminated by a vote of no confidence resulting in an immediate General Election.  During the Coalition talks David Cameron ( who spent the previous 3 years calling for an "immediate general election") suddenly became extremely worried about how wrong it was for parliaments to be disolved early.  Not having a full majority Mr Cameron worried about the viability of a 5 year coalition government with the Liberal Democrats attempted to fix this constitutional problem with a proposed increse in the majority needed to disolve parliament from 50% + 1 to 55% + 1 to avoid a general election being triggered by a vote of no confidence but abandoned the scheme when it was fiercely ridiculed even within his own party. To quote Christopher Chope the Secretary of the 1922 committee:




Indeed in the 1990 election for leadership of the Conservative Party the barrier the PM needed to break to win the first round was actually 50% and “a lead over the 2nd place candidate of 15%" - so Mrs Thatcher was effectively eliminated with 53.8% because she was 1.3% off the lead over Michael Heseltine needed for a decisive first round win.  This was a bit ironic really as she'd actually only needed 53.3% to win the job of Leader of the Party back in 1974 ....


...but the rules of the Conservative Party leadership election process were, of course, designed to disadvantage the incumbent.


Under the somewhat abstract rules the 1st round runoff led to a second in which candidates who were not in the 1st round could enter.  In the 2nd round John Major polled only 49.7% - not enough to win.  This was supposed to lead to a third and final ballot to be decided via AV but Michael Heseltine and Douglas Hurd withdrew with only 35.2% and 15.1% as by this time everyone was slightly bored.


Interestingly when Edward Heath stood for party leader in 1965 he didn’t achieve a 15% lead either but Reginald Maudling dropped out and no other candidates came forward and there was no incumbent to beat so the 15% margin did not apply.  For some reason there tend to be more candidates entering in the 2nd round of the runoff system when the party is in Government.


In the 1997 election for leadership of the Conservative Party William Hague who verhmently condemns AV for allowing the person who comes 2nd or 3rd to come 1st was, of course, in 2nd place during the 1st and 2nd ballots until John Redwood's elimination from the competition allowed him to push Kenneth Clarke into second place.


In the 2001 election for leadership of the Conservative Party the rules were changed again so that the final two candidates had to be voted on by the entire party.  Although Micheal Portillo was the leader in the 1st ballot (there were actually 2 first ballots as the two bottom ranking candidates tied for last place and the Chairman of the 1922 Committee couldn’t decide who to eliminate first which matters as the runoff is not instant but on sequential days) when David Davies and Michael Ancram were eliminated most of their backers migrated to Kenneth Clarke and Iain Duncan Smith so that Micheal Portillo was eliminated in the penultimate round by 1 vote.  The story of his life.  Many People blamed tactical 3rd round FPTP voting for Portillo's disqualification.  However, it is just possible that actually no one likes him.


Iain Duncan Smith then won the membership vote despite being 4 points behind Kenneth Clarke in the MPs vote.  Iain Duncan Smith’s narrow margin in the MPs ballot meant he was in a very weak political position and Michael Howard managed to usurp him 2 years later ....




The Labour party uses, of course, a very complicated weighted form of AV to elect its leaders split into an electoral college of Affiliates, Members & MPs.   An MP's vote is worth 0.12 per cent of the total electorate, a party member's vote is worth 0.0002 per cent and an affiliated member's vote is worth 0.00000943 per cent (although these figures assume all affiliated members bother to vote and one can be affiliated to up to about 10 other organisations meaning the total amount of votes one can "buy" is 0.0003%).  This makes Labour leadership elections extremely expensive (about £2 million a go) which maybe why the party seems to manage to avoid them. 



Gordon Brown was famously "elected" unopposed as any challenger would have needed to be nominated by 12.5% of Labour MPs - the process favours the incumbent.  No AV ballot before the most recent has gone beyond 1 round since 1980 due to either a limited number of candidates entering (it's personally very expensive to run) or one candidate getting an absolute 1st round majority.  Unlike the Conservative Party, the Labour Party likes to hang on to bad leaders for grim death.  The last election for leader was highly unusal in having 5 candidates and progressing through all 5 rounds of the AV process.



As we can see it is the reallocation of losing candidate Ed Balls, Andy Burnham & Diane Abbot's votes that make the differece between Ed and David Miliband winning or losing.   On a slight tangent one could also analyse which elements of the electoral college the two finalists derived most support from which is interesting.


There is no doubt that politicians themselves think that runoff voting is a better way of arriving at political stability than a simple FPTP election.  However, the reason that General Elections do not utilise runoff voting is twofold. 

Firstly cost.  It would be extremely expensive to perform multiple ballots. 

And secondly since there are 650 consituencies all electing MPs simultaneously - if those where the result was close enough to go to more than one round the voters in the key marginals would have detailed knowledge of the results of other constituencies to inform their choice for the next round which would not be fair to other voters. 

In order to solve the problem of how to run a runoff type election without the voters knowing the results of the previous ballot at the next round ...in 1871 American architect William Robert Ware invented "instant runoff voting" or "AV" - a system whereby the voters rank their candidates in order of preference 1,2,3,4 etc so that all the "runoff rounds" can be contained and calculated from the numbered preferences of one ballot paper.


The problem is that although the permutations through which every voters choices are passed are equal (all votes are treated equally) because they are expressed on one piece of paper instead of 3 it is possible to claim that the votes are not of equal value because physical recounting of every single ballot paper is not required once they have been sorted at the end of each round.  This actually ended in a court case in the US (which uses AV for the selection of Mayors) known as...



in 1975 at which Circuit Judge James G. Fleming decided that

Under the 'A.V. System', however, no one person or voter has more than one effective vote for one office.
No voter's vote can be counted more than once for the same candidate.

In the final analysis, no voter is given greater weight in his or her vote over the vote of another voter, although to understand this does require a conceptual understanding of how the effect of a 'A.V. System' is like that of a run-off election.

The form of majority preferential voting employed in the City of Ann Arbor's election of its Mayor does not violate the one-man, one-vote mandate nor does it deprive anyone of equal protection rights under the Michigan or United States Constitutions.


AV is often criticised as a highly complex mathematical model.  Critics such as Baroness Warsi claim that AV puts power in the hands of extremists.  Leaving aside the obvious point that FPTP would seem to put extreme power in the hands of tiny parties... here's a graph showing Andrew Pelling's 2005 Conservative majority for Croydon Central of 75 votes displayed in proportion to the minor parties votes in that election to give some indictation how sought after their votes are in a marginal...


It is easy to write of Lady Warsi as simply the person given the dirty work of the Conservative party because she's the one who cant lose her seat but the reality is a tad more complictated.  Lady Warsi's argument for the dangers of AV (as explained to the readers of "the Sun") was that in Dewsbury where she stood in 2005 the BNP polled more than the difference between the two major candidates (as we shall see below this doesn't actually give their voters the ballence of power due to threashold effect) but her arguments are worth considering as Dewsbury in which both the Conservatives and Labour fielded asian muslim candidates (the Labour candidate was Shaid Malik) in 2005 polled the highest BNP vote in the country. 

However, if we look at the data from 2001 to 2010 (there were some minor boundary changes in 2010) we see a few interesting things.   Firstly...


The swing to the BNP is massive ...and out of line with the National trend and seems to almost completely correct its self in 2010...? The only common factor not present in 2001 and 2010 but present in 2005 that I can find is ...erm Lady Warsi.  Perhaps voters held a racial prejudice against the first ever Tory muslim candidate.  Or was it something she said.  The battle for Dewsbury seems to have been extremely hard fought in 2005 and Lady Warsi certainly didn't enamour herself to the gay rights group Stonewall with her literature about how nice Section 28 was long after Micheal Howerd had abandoned this policy.
And she also managed to come up with this campaign leaflet about  how if...


Dewsbury is by all accounts an interesting and lively place where debate is, to borrow a word from Mr Cameron, "robust".  After Shahid Malik won the seat in 2005 and not long after the 7/7 terror attacks who's protagonists put Dewsbury on the national map for all the wrong reasons this lively debate escalated into a memorable libel trial when Mr Malik sued Tory councillor Jonathan Scott and Dewsbury and District Press editor Danny Lockwood over a letter printed in a 2006 edition which insinuated Mr Malik was complicit in using asian thugs to intimidate voters. 


Mr Scott claimed that during the local elections Malik's "cohorts" used "physical intimidation and verbal abuse" at a polling station in Saville Town.  Mr Scott incidentally was Ms Warsi's election campaign manager.  We know this because of a complaint made by Pink News, Stonewall and a group of Labour Cllrs to David Cameron in 2007 over the continued circulation of the "homophobic leaflets" used in the 2005 election campaign.  The case seems to have revolved around the principle that while it was in the public interest for the paper to publish the accusations they should have contacted Mr Malik ahead of the publication of the letter to allow him a chance to put his side of the story.  Mr Lockwood insisted that this had been done and wrote a letter to Mr Malik claiming that attempts had been made to contact him before publication which were later retracted in court.

"We quite clearly don't have phone records. I trust Martin Shaw implicitly.  If Martin says to me he tried to ring I'm pretty certain he tried to ring.  Yes I was angry, yes I was lashing out. We do not have phone records. It was a big porky pie."  Defending his decision to publish the letter Mr Locwood went onto add that:  "Dewsbury had severe problems with elections over the last nine years.  We've had stories of postal fraud and intimidation. That's not specific to Labour but it is specific to the Asian community"

Indeed in 2010 Lady Warsi opined that at the 2010 General Election the Conservatives lost at least three seats "based on electoral fraud”.  She refused to identify the seats concerned but said the problems were “predominantly within the Asian community” and that Labour had been the beneficiary.  “I have to look back and say we didn’t do well in those communities, but was there something over and above that we could have done? Well, actually not, if there is going to be voter fraud".  Pressed further on what she meant she clarified  "I am saying there are seats at the last election in which those constituencies are concerned, quite rightly, that electoral fraud took place.What I can say to you is that individual constituencies are raising these issues with the police and with the Electoral Commission.   You'll be surprised to know that 81 complaints were made to the police after the last general election and I think over 25 of them have now become formal complaints.  Yes, this process is in place; yes, the appropriate authorities have been informed; yes, it is part of the electoral reform process and discussions I am having with Nick Clegg. "

Of course the Prime Minister prior to the General Election raised the idea of severly cutting the funding of the Electoral Commission.  Stating nostalgically that


It is fair to say postal vote fraud is not an exclusively asian phenomenon - we have previously explored the phenomon of postal voter fraud by Conservatives in Guilford.  And indeed there have also been allegations of voter fraud against the Conservative party in Dewsbury and Halifax which is "a hot spot for voting fraud".

The Malik libel case eventually went to a fully jury trial, but unfortunately the Jury was not able to reach a majority verdict.  The Hon. Mr Justice Eady indicated that he would accept a majority verdict of at least 10-2 but the jury were unable to reach this level of unanimity.  They were not even able to agree that the publications were defamatory, let alone whether they were substantially true and requested for permission to “google” the parties on the basis that they required more information to decide the cases.   Unsurprisingly, the Hon. Mr Justice Eady said :


A retrial was planned but both parties pulled out when they realised that costs had now reached in excess of £300,000 and avoiding a retrial would mean avoiding anyone having to cough up any of this money.  Dewsbury's local newspaper continues its unique brand of interesting journalism.  Mr Malik lost his seat in 2010 when pro-AV and anti-sealze campaigner and ex independent MP Martin Bell financed Khizar Iqbal to stand against him on an "anti-sleaze ticket"....


...Mr Malik blamed the vote split effect for his losing the seat saying Mr Iqbal had been "brought forward not to win but to make sure that I lost".  Ironically Mr Bell had actually demonstrated one of the positive aspects of FPTP.  Deeply unpopular MPs and candidates can lose in defiance of national party trends if they're ...erm... unelectable.  Mr Iqbal has now rejoined the Conservative party and is back on the local Council.

What really happened in Dewsbury in 2005 I guess we'll never know fully so let us instead have a look at how FPTP elections could play out using AV.

Using data from Croydon Central for 2010 I've done a basic statistical analysis of the various possibilities.  I could have used 2005 results for this but it makes more sense when considering if a losing candidate could actually win under AV to take FPTP results where there is more of a discrepancy between the winner and loser.  This election is more interesting too as for complicated reasons the Conservative Party had schismed in Croydon by 2010 and Andrew Pelling was running against his own party.  There was some argument over whether he had resigned or been sacked by Conservative Central Office.  The payoff for ordinary Conservative Party members now having a vote in the leadership election is that they have lost control of their Constitency associations which do not utilise OMOV to select candidates.  Central Office can and does impose candidates on local party Associations.  The old federal Conservative Party structure was replaced by a more centralised one to try and resolve some of the bitter disputes above and "modernise the party".  The Conservative Home website is full of amusing articles on how to circumvent these rules as the party comes up with new ideas such as open primaries to excluded the poor who cannot afford to canvass an entire constituency rather than a few hundred members.  The membership at the moment is 160,000.  The membership of the Labour Party is about 250,000-300,000 but has been as low as 180,000 at the tail end of the Brown years.  So why is any of this sillier than say PR with closed party lists?  Some have postulated that Mr Pelling actually stood order to "split the vote under FPTP and let Labour in".  It has not gone unnoticed Mr Pelling has since joined the Labour Party...

It is of course highly unlikely that any voter would transfer their vote through every permutation but if they did the results would look something like this



This is the case if all the minority candidates redistribute their votes in order of general candidate popularity.  However, it is more likely that the reallocation would be slightly random.  In which case the results would look more like this


In either case it is the last round that is decisive. 

So is there a senario in which the BNP or another such minority group could "hold the ballence of power"?

Well, statistically the worst case senario to be found would be if Candidate 9 votes gave all their 2nd preferences to candidate 8 and these were then ALL transfered to candidate 7 and then Candidate 6 so that all the minor party candidates all expressed identical second preferences.  This would be an extreme statistical, social and political freak event and is probably impossible as if they are all of the same mind why are they all giving their first preferences to different parties/candidates in the first instance.  But as it is a mathermatical possibility here is what such a senario would look like.


If that looks rather complicated here are the solutions again.  Rounds 1 to 6 are pointless exercise of reallocating votes of one no hoper to another.  The decisive rounds in an AV election are the final two.  Before these only the candidate who is first in the first round can win.

At the end of the third to final round the candidate in 4th place's votes can now push the candidate in 1st place over the 50% finishing post if they all transfer their votes to the leading candidate.  However, they cannot push the candidate in 2nd place over the finishing line.  As the minor parties collectively seldom poll more than 10% as a whole it means the 1st place candidate needs to break the 40% barrier in order to win.  Here's an animation in which the Labour and Conservative percentages go up and down while the Others and the Liberal Democrat percentages stay fixed...


We could complicate the animation by making the Liberal Democrats percentage change, but if it changed much they wouldn't be the 3rd party any more and then Labour or the Conservatives would be in the middle of the diagram.  The important point to note is that at the 3rd and final round the leading candidate must poll over 40% for the Others 10% to be able to push them over the finish line - this can only happen if a very large percentage of the others transfer their votes to the leading candidate.  Even if ALL the "Others" (the BNP only usually poll 2-3% of the others 7-10%) collectively transfer their votes to the candidate in 2nd place at the penultimate round they cannot push the candidate in 2nd place over the finish line in any statistical permutation.  Therefore the ballance of power in an AV election resides with the Liberal Democrats.

So in response to William Hague's claim that under AV the party that comes 3rd in the 1st round can win. 
Yes, but only if the party in 2nd place gets less than 33% of the vote.
The party that comes 2nd can win on second preference votes of ALL the other parties.
And the party that comes 1st in the first round can win with the 2nd preference votes of the 4th, 5th, 6th to nth votes or with the 2nd preference votes of the 4th, 5th, 6th to nth votes.  Here's another diagram showing all the permutations possible where parties A and B are neck and neck and we vary the percentages of C and D
In any NON 1st round winning senario, for A,B,C,D to be 1st, 2nd, 3rd & 4th place the following conditions MUST be true:
A + B + C +D = 100%
A <= 50% (if A is greater than 50% then A wins on the first round and there's no point in calculating anything)
A > B
B > C
C > D


Results

A (1st in the 1st round) + D (votes redistributed in the penultimate round) are just capable of causing the 1st round winner to win without going to a final round

B (2nd in the 1st round) + D (votes redistributed in the penultimate round) are NOT capable of causing the 1st round loser to win without going to a final round

If the Substantial portion of D's votes are transfered to C in the penultimate round B (the second place candidate in the 1st round) can be eliminated by C if they poll < 33%

A gridding program like Surfer can give us a better ideal of the interdependency of A+D, B+D and C+D as B approaches A.  It looks like this.  The blue area of the grid represents > 50%.  Viewing from the side we can see how D votes can just push the leading candidate A over the finishing line...

...and viewing from the top we can see how the gird never extends quite up to or beyond the 50% mark of the C + D and B + D axes because it is impossible for candidate B or C to cross the finish line with the help of candidate D.

If voting goes to the penulitmate round where all the minor party votes have been redistributed and we only have three candidates left A, B & C where A + B + C = 100%
A <= 50%
A > B
B > C
then, of course either A or B can win with the help of C.  The possible vote share permutations here are much easier to visualise and looks like this:



However, the No Campaign still managed to come out with some stand out nonsense graphs.  Here's probably their best example which shows BNP candidate E (not even D) pushing candidate B over the finishing line - mathematical impossibility...


After I pointed this out to them they gave up on that particular con trick and tried creating graphs that conceal the starting point.


However, being very good at Maths we have managed to take the Conservatives Race poster and back calculate the starting point...


...which reveals that the "athletes" have run less than 9 Metres and this picture too is impossible. 

Although they have a point that "the person in 3rd place can win" - not by the margin shown.  Actually (B and C) and (C and D) must both be within 15% of each other and (B and D) within 22% of each other to generate a valid senario and B must poll < 33%.  Since this is a Conservative No to AV poster it must have been approved by the Prime Minister.  Erm ... not inspiring.  To be fair to the NO campaign though - this is not "as simple as 1,2,3".  Interestingly although there is quite a large statistical window it's only happened once in Australian Electoral history.  And not under the AV system that was suggested for the UK but under a full preferential system by which as Lord Ashcroft points out Henry Arthur Hewson was elected to the Australian House of Representatives in 1972.

After bringing in AV the Australian politicians got the hump that not enough people used their preferences and made voting compulsory.  Since an AV election is a simulation of a series of runoff ballots this effectively made using all your preferences compulsory too and in the end this resulted in "donkey voting":

The statement that the person in 4th place can win ...


How inspiring.  These tactics however, are very effective in keeping the 3rd placed party very low in any marginal constituency:

Anyway all of this statistical analysis is silly anyway as all that was really needed to win the referendum was this picture:


Chris Huhne, of course got very cross at Cabinet about the use of this picture causing George Osborne to retort that "This is cabinet not a sub Jeremy Paxman interview".  Mr Huhne was, of course, Mr Clegg's rival for the leadership of the Liberal Democrats back in 2007.   A contest that was so close it was decided by a margin of only 1%.


Devotees of the late Right Honourable Francis Urquhart MP can do nothing but admire the speed with which tabloid newspapers wheeled out his ex-wife.  Huhne won his seat in 2005 with only 568 votes ...




...revealed to the press that actually he was divorcing his wife for his mistress Carina Trimingham  (a technique often employed by MPs wanting to dump "loyal" wives at the optimum point of the electoral cycle known as "doing a Pelling").  Ms Trimingham is also campaigns director at the Electoral Reform Society making the referendum loss by such a large margin a personal as well as political failure for Mr Huhne.


Unfortunately by the time of the referendum this affair was rather old news and all Mr Huhne's children now seem to be grown up making the "leaves wife to bring up family" angle a no goer ... but fortunately his ex-wife was able to revive a claim that he had asked people to take speeding points from him to prevent him losing his licence.  Mr Huhne lost his driving licence eventually anyway due to speeding a lot while trying to cover the enormous distances ....


...between events in his Eurpean Union Constituency when he was an MEP.  Of course the large size of the constituency is down to the de-localisation of MEPs that the proportional repesentation system used by the European Union electoral system creates.  The irony that this driving ban reflects a practical problem of PR seems to have been lost on everyone.


When the driving points story and the fact Ms Trimingham is bisexual failed to ignite anyone's imagination the Sunday Times ran an story about Mr Huhne somehow incriminating himself in a "secret tape" that was "obtained legally".  We were not informed who was on the other end of the phone by Mr Huhne (according to the Telegraph it was his ex-wife) but he apparently said “I honestly don’t think, I really don’t think, that it is sensible to have these sorts of conversations over the phone,” as “the last thing” the witness wants is a “half-baked story” appearing in The Sunday Times.  An official complaint has been made to Essex police by Simon Danczuk, a Labour MP about the diriving points matter and they are now very busy reviewing 8 year old speed camera footage which I'm sure is a very sensible use of police time and resources.  Inside the paper regular columnist Jeremy Clarkson was very busy opining that there is a need for Super injuctions and Privacy laws to prevent people like himself becoming the victims of malicious tittle tattle and unsubstantianted claims.  Then again maybe Mr Huhne is just a bad man.  The accusations and counter accusations are complicated so we asked a detective what he thought.


The £250,000,000 "cost of AV" proposed by the No campaign was beyond satire.  But it has to be said the Nick Clegg chose his words carefully when he replied that the votes would be counted "in the normal way" without counting machines.  This is not a denial that it would have taken more time or money. 


Of course cost is a ridiculous argument.  Particularly from the likes of David Blunkett who's abandoned ID card scheme cost £260,000,000 and was expected to have cost £800,000,000 if it had gone to completion.  Blunkett even admitted this himself when cornered.  Gamely responding that during an election campaign people just make figures up.  While there are laws about telling lies about individual candidates personal character... statistical lies and outright mathematical lies are pretty much regarded as simply all part of the free debate... so tell as many outright provable lies as you can.  It doesn't matter if it gets votes in the box.


Of course the other accusation against AV is that it's not full blown PR.  There's a reason for this.  In order to approximate the number of seats to the number of votes in the most mathematically rational way so that number of seats = proportion of the votes you have to use multi-member constituencies.  The larger you make these the nearest you can get an equal correlation of the votes to seats.  The oldest and most common system is STV. 


Under STV a candidate must poll above a set "winning" threashold.  If not all seats are filled on the first count the winner's "extra" votes are reallocated to a second preference.  If still no one passes the threashold then the candidate in last place is eliminated and their votes are reallocated.   As soon as there is a second winner if there is another seat to fill this winner's "extra" votes are reallocated to a second preference.   The process repeats till all seats are filled.  There  are different ways of course of setting the threashold but usually the "Droop quota" is used which is:


It's really an attempt to refine AV for multi-member constituencies by allowing extra votes from a constituency where the winning candidate got > 50% flow into one where they got < 50% by taking the two constituencies and melding them into one big one and using a formula to calculate proportional vote transfer...  if you follow that.  The difference between STV over closed list PR is that under STV the voter can chose for their votes to be re-allocated along non-party lines if they have a strong personal dislike for an individual Candidate.  In order to make this easier to visualise here's a theoretical example that I have made up showing what might happen if you merged Croydon Central, North and South into one STV constituency...


If you only have a one member constituency then the winner's extra votes reallocation proceedure is meaningless and thus a 1 seat STV election is simply an AV election. 



However there wont be a referendum on this as the maths really is hard and politicians dont think anyone would vote for multi memeber constituencies ... although the UK actually had them until as late as 1950 which used the Bloc vote system whereby electors could cast a vote for up to as many candidates as there were seats to be filled. The elector could not vote more than once for any candidate, but was free not to use all the possible votes.  This was abandoned as it was even more non-proportional than FPTP ...which is quite hard. 

Anyway the problem with STV is that proportionality effectiveness is a function of constituency size and number of seats.   We could go into some quite complex mathematics to prove this but here's the back of a fag packet explanation instead.  3rd party and minor party votes are spread about geographically over a wide area in patches and this is why small parties do very badly under FPTP and only a little bit better under AV.  The STV mathematics is an attempt to reintroduce proportionality but because of the wide geographical spread of 3rd and small party voters there is no mathematical way of doing this without delocalising MPs to some extent.  On the plus side both AV and STV negate the inherant gerrymandering problems of FPTP.  Under FPTP as MPs are highly localised where you actually draw your constituency boundaries can have an extremely disproportionate effect on proportionate representation This can result in the party with most votes not actually having the most seats in parliament as happened 3 times in the 20th century.  The last was 1974.

The alternative to AV or STV is PR using open or closed party lists and a largest remainder method of calculating how many MPs of each party there should be.  The European elections utilise the D'Hondt method of PR from closed party lists with its easy to comprehend formula...

The total votes cast for each party in the electoral district is divided, first by 1, then by 2, then 3, then 4, then 5, right up to the total number of seats to be allocated for the district/constituency. If the district contains 8 seats, the highest 8 numbers are chosen from all the numbers resulting from the divisions. The parties under which each of these 8 highest numbers were produced get the seat.

Party A:	*100,000	*50,000	*33,333	*25,000	20,000	16,666	14,286	12,500	→	**** = 4 seats won
Party B:	*80,000	*40,000	*26,666	20,000	16,000	13,333	11,428	10,000	→	*** = 3 seats won
Party C:	*30,000	15,000	10,000	7,500	6,000	5,000	4,286	3,750	→	* = 1 seat won
Party D:	20,000	10,000	6,666	5,000	4,000	3,333	2,857	2,500	→	No * = no seats won

....using a closed party list system that creates very large ballot paper...

Unless you're in Northen Ireland where they use STV because they are special.  And the resuts look a bit like this


Or one can use the the fudge between AV and PR known as AV+ whereby the bulk of MPs are elected using AV from single member constituencies with the remainder needed to achieve statistical proportionality chosen from party lists to "top up" the legislature.  Again this leaves some MPs without physical constituencies. 


AV+ was the invention of the 1998 Jenkins Commission into all the different forms of Proportional Representation set up by Tony Blair.  Mr Blair wanted to woo Liberal Democrat voters with promises of reform, but fortunately as there are multiple forms of Proportional Representation he could spend a long time getting Jenkins to investigate them all and thereby avoid doing anything... 


Aware that all forms of PR for the Westminster Parliament are not in their immediate self interest the Labour Leaderships have long played cat and mouse with the Liberal Democrats over Electoral Reform.  Claiming to be committed to it in some form but never enough to do anything about it - just enough to pick up Lib Dem voters.  Fortunately AV+ with its "two types of MP" invented by Jenkins has certain logistical problems to be resolved and no history of implementation so it can never be implemented.  For example what if a candidate is rejected by a constituency but ends up in Parliament anyway to achieve proportionality...?  Such questions have never been fully clarified.  Arguably STV is more logical than AV+ (a form of Additional Member system) which is a best of both world solution and so likely to satisfy nobody... but it would require quite large constituency regions to achive proper simulated proportionality.

That said the Blair government did experiment with Additional Member systems in the Scottish Parliament and Welsh and London Assemblies so this wasn't a compete dead end.  Of course the reason for using these systems was to stop any of these regional bodies getting single party government that could lead to a referendum on Independence.  A policy that has spectacularly failed now the SNP have a majority in the Scottish Parliament.  Which just goes to show if you think about winning and not the system ... then you're not the 3rd party.  Aditional Member systems do have the advantage of the greatest statistical proportionality.  Obviously because you can vary the number of aditional members to achieve this.  Here's a picture I stole off wikipeda of relative proportionality for voting systems...


Based on Patrick Dunleavy (London School of Economics and Political Science), Helen Margetts (Department of Politics and Sociology, Birkbeck College, London), Report to the Joseph Rowntree Foundation, 15 April 1999.  As we can see AV actually is more proportional than FPTP but that didn't stop the No to AV campaign claiming it both was and was not.

Of course in regards to the AV referendum the Party Political question is who would have benefited and most analysis would seem to suggest that in general Labour and the Conservatives would lose seats to Liberal Democrats under AV while the very small parties are frozen out.  The Conservatives would lose more seats which is probably why they are almost totally against it, while the Labour party would lose a few which explains why they are split on it.  If you're wondering how the party split on it here's a histogram of Yes/No/Dont know Labour MPs vs Majority showing that actually...



...it's a pretty random split.  At least it is independent of majority.  The input data is taken from Labour List's list.  ...Of course, AV is a statisticians wet dream as there are so many potential permutations as variables. 

A simple attempt at the most basic simulation can be achieved by downloading the 2010 election results off the Electoral Commission website and doing some cell operations to try and figure out what happens when you divide all the minor party votes up equally between the top three polling parties in each constituency and then divide up the 3rd parties votes between the 1st and 2nd place parties for various percentages.  Anyway after running all the various permutations I came up with this election result:



As we can see the Labour party could push the Conservatives out of power if they got more than 60% of 3rd party votes.  However, they could also find that they had 80% of 3rd party votes but did worse...?  Bear in mind that the 1st, 2nd and 3rd placed parties are not always the main three parties and to some extent this explains the extremely interesting graph shape.   As we can see the Liberal Democrats have a lot to gain...

If you divide up the all the 3rd, 4th, 5thm 6th to nth party votes and generate a similar percentage plot (i.e. dont share out the minor party votes equally) you see this:


One might imagine that given it's been their biggest policy plank to which everthing has been sacrificed to implement the Liberal Democrats should be as happy as a manic depressive in the manic stage by the prospect of AV but in reality they've been so committed to full blown STV PR systems that would give them even more seats that given the chance to actually change something they're not happy either.  There's even a No to AV because it's not PR campaign spearheaded by the one and only Mr Principles-over-Power himself Dr David Owen...



I could explore further such things as Duverger's law of 2-party domination which attempts to explain how the number of viable political parties is a function of the type of voting system you use but really, at this point I got a bit bored and adapted my program to analyse the mathematical solution to Deal or no Deal.  Deal or no Deal like First Past the Post involves guessing what's inside a box before it is opened so as to make a seemingly important choice. 

First one must consider the number of ways of arranging 22 boxes.  This is fairly simple it is 22 factorial.  The number of ways of arranging 2 things is 2, of 3 things 6, of 4 things 24 ...don't understand why?  Neither do I it's just a mathematical fact that the number of ways of arranging two things = 1 X 2 = 2, of three things = 1 X 2 X 3 = 6, of four things 1 X 2 X 3 X 4 = 24.  This function is called a "factorial".  With it we can calculate the "decrease in possible box arrangements with each round".  From 1,124,000,727,777,610,000,000 at the start of the game to 0 at the end.


As you can see the rate at which the available choices drop is very rapid.  Even on a logarithmic scale it looks like this:


The chance of correctly guessing which box is going to be opened next is 1 / the factorial number for each round above which is:


To find out how the boxes behave and the passages from all boxes to 1p or £250,000 is a simple matter of writing an iterative program to select 22 random numbers and use these to randomly chose a box opening order.  After 100 iterations I started to see a pattern

As you can see if you calculate a standard deviation for the data scatter in linear space then by box 19 the Mean - the standard deviation < 0.  It is of course impossible to win negative - £s so what this means in practice is that by box 19 there's a very high probability of dropping down to less than £100 and never recovering.  In log space the deviation looks like this.



If you run 60,000 iterations and sort them in order of maximum money remaining in the grame from box 22 down to box 1 so that 0 stands for the unluckiest iteration and 60,000 winning the £250,000 you see something like this:





Of course most people stand in a TV studio in Birmingham for up to four weeks in the hope of winning the £250,000.   The chances of having either 1p or £250,000 if you go right to the final box are approximate 4.55%.  And of course as the financial scale is logarithmic it's the top five boxes that control the average within the game.  The rate at which these boxes drop out of the game over the course of the 22 rounds and 60,000 random iterations is remarkably uniform...