Problems |
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0. What are we trying to achieve?
Introduction
The UK’s electoral system, First Past The Post (FPTP), has worked to produce majority governments, but does not treat voters or parties fairly or equally. This thesis shall propose a new system of Delegated Voting (DV) for the UK that could do so, whilst reinforcing our traditions of local representation and majority-government.
The methods
This thesis will use qualitative and quantitative analysis, including computer-simulations of voting, created using Microsoft Excel1, and statistical analysis2 of UK election results. It will compare, where possible, how different systems might treat UK election results, although the difficulties of transferring election results between electoral formulae limit this process. Other literature will be used mostly as a guide, to enable direct focus on the issues.
The problems
Much of the literature examines various systems and identifies problems with them3, or establishes parameters and compares systems’ performance on them. Examining existing systems like this may separate the innovation from the analysis, so this discussion will try to focus more on issues than on existing systems. It will examine flaws in existing systems, but try to draw lessons from the discussions of how a system ought to respond to a particular problem.
Adapting from a basic set of problems identified by Reeve and Ware, 1992; the body of this thesis considers each of these issues:
Communicating a message of a desired type from voters to
representatives
Determining factors of result
Preferences to be expressed; Condorcet-winners; monotonicity
Proportionality and party systems
Producing majorities
Each issue is examined in a UK context, considering its existing electoral system, which is found to be problematic. There is extensive reference to the Single Transferable Vote (STV) system, which is shown to have different problems than those of FPTP. The nature of the problems in FPTP and STV should demonstrate how a system might best implement democracy in the light of these five issues, and they conclude roughly this:
A system needs to allow voters to show the proportions in which they
support minority parties, and also to allow them to choose a
majority government if there is sufficient support.
The distribution of votes under FPTP and STV greatly affects the
overall result, and the options facing individual voters. A better
system would make distribution matter less, and would try to present
a more similar decision to each voter, by allowing for votes to be
counted into a national result, rather than only at a
constituency-level. It would give extra Parliamentary votes to the
winning party through a consistent formula, as FPTP has in an
inconsistent manner, because of distributions.
Categorical preferences are preferable to ordinal preferences,
because they can be fairly counted in a way that gives the necessary
determinacy, and they encourage consensus and a strong constituency
link.
Proportionality of representation comes with most benefits where it
is direct, and is most important for smaller parties. For larger
parties, proportionality of both representation and tenure is
important.
For Britain to have majority governments, the system needs to give
the winning party a greater proportion of Parliamentary votes than
it has electoral votes, and it is important that this bonus be
sensitive to the degree of victory.
The tools
Most electoral systems tend to function in a similar manner. Firstly, the polity is divided into a number of distinct districts, from each of which a certain number of legislators4 will be elected to a one-member, one-vote, legislative assembly. Then, an election is held in each district, whereby voters complete a ballot by either selecting one or more candidates or parties, in a categorical ballot, or ranking one or more candidates or parties in an ordinal ballot. Finally, these votes are counted, and according to a formula, these data are converted into a choice of which representatives will be selected for the district, and these representatives are accordingly elected, and sit in the legislative assembly until the next election.
Electoral systems can be described using three parameters:5
District Magnitude
Ballot Structure
Electoral Formula
Varying these factors can give many different electoral systems. In light of what will become clear when discussing determining factors, proportionality and majorities, it will be best to talk about the effects of these factors together, and it would be misleading to treat these aspects of electoral systems as discrete. For example, the same ballot structure used with the Alternative Vote (AV) and the Single Transferable Vote (STV6) gives very different results due to different district magnitude, and so the specification of all three parameters is necessary to consider their effects, as they interact with each other.
The underlying values
This investigation is empirical, but also evaluative according to democratic principles, particularly liberty and equality. It will be guided by the concerns that voters are not unnecessarily constrained in their choices; that every voter’s preferences matter equally; and that a national political community is fostered.
The system
This discussion proposes a new electoral system, and shows how existing systems suffer problems that it should not. This system, which focuses on carrying every vote through to Parliament, is called Delegated Voting, (DV).
Under DV, instead of a one-member one-vote assembly; each MP, elected by the plurality of the vote in their constituency (as in FPTP); would carry the votes they received (and a share of their party’s votes for unsuccessful candidates) as their votes in Parliament, delegated to them by the voters. Apart from a top-up to the winning party, this would make every vote at an election count for a vote in Parliament, safeguarding equality, and ensuring transparency.
An MP whose constituency voters give them 50,000 votes would cast these 50,000 votes in Parliament, in addition to their share of their party’s unsuccessful votes. Those voters who vote for a candidate who loses in their constituency, would have their votes added to their party’s other losing votes nationally, which would then be distributed equally amongst the party’s elected MPs, so that each would have a share of these votes as well. An MP might have a Parliamentary vote of 70,000, comprising 50,000 from their constituency and 20,000 from their party’s losing votes (e.g. if their party won 100 constituencies, and received 2,000,000 votes besides).
To give majorities to parties that have convincingly “won” elections (see Section 5), the party with most votes would receive a top-up to their national vote. These extra votes would be equal to three-times their vote margin over the nearest party7, and would be distributed equally to their MPs together with their votes for unsuccessful candidates.
Parties with a considerable national vote that won no constituencies would have their candidates with most votes elected as national MPs. The number of national seats they would win would be determined by how many times the average Parliamentary votes of a sitting MP was their total vote, i.e. the number of MPs their total could supply with an average Parliamentary vote. These MPs would have their constituency votes, and an equal share in their party’s unelected candidates’ votes, similarly to constituency MPs.
1. Communicating a message of a desired type
The issue
The debate between majoritarianism and proportionality may turn on different ideas of what an election is for8. If elections are to choose majority governments then the system must deliver majorities, but if they are to decide how groups of voters should be represented in parliament, the system may need to deliver proportionality.
An election consists of more than a victory for a single party; it comprises millions of individual decisions, which may be strategically influenced by the voting system. For its results to be fair, the electoral system needs to be aligned so that voters face similar choices, and their votes can be treated alike. Voter need to understand the effects of the system, and to be able to vote accordingly, so that the decisions fed into the system reflect their eventual use, and there is reflective equilibrium.
The effect of voting a particular way must be clear, unambiguous, and not affected by arbitrary constraining factors. This is achieved in DV by having each vote cast delegated to the Parliamentary vote of an MP of that party, and using an explicit formula to produce majorities. Section 2 shows how STV and FPTP do not produce similar clarity and certainty. This section considers what an election ought to do, considering that complicating the objectives comes at the expense of clarity, and of the responsiveness of a system to each of them.
A contest between two alternative governments
The adversarial set-up of the Commons and its front benches, between the Government and the main opposition party, encourages a view of them as alternative governments. On average 82% of votes have been cast for these two parties9, ensuring majority-government in all but one Parliament, so some votes in British elections are cast to choose between alternative majority-governments.
Many parties, e.g. the SNP, don’t post candidates in all seats, and voters don’t expect the Liberals to parlay 20% into triumph at the next election, but still vote for them. While votes for the top two parties are part of a contest for government, those for other parties are not, and should not be treated as such. This is why DV distinguishes this contest from other voting, in an explicit formula.
Distributing power between parties and arranging the circumstances of coalitions
Most elections are forecast before they happen, so voters for the winning party expect it to win with or without them, and voters for the second-placed party are likely aware that they are voting not to form a government, but to strengthen the opposition. Voters seek to affect the balance of power, whether or not they choose the victor. This is especially important for voters whose parties do not win the election.
Where majorities are small or non-existent, the balance of power between the parties can become a very live issue. If voters control parties’ numbers of Parliamentary votes, (as in DV, and other very proportional systems), they determine the relative importance of parties and the circumstances of possible coalitions, however voters have limited power to affect coalition formation. There may be many ways in which parties could combine, and this only matters if a majority is not expected.
Selection of advocates or of delegates for groups of voters
MPs often operate as advocates for their voters, and may also act as delegates. Representative democracy expresses people’s rights to be involved in government through their representatives. MPs are rightly thought of as their voters’ delegates, and so in DV their right to vote in the Commons is modelled on the delegation of power from voters to them, unlike in systems where MPs win a seat.
Conclusion
British elections should and do comprise two main elements at the national level: picking a government from two alternatives, and determining the popularity of that government and of other parties. On the local level, constituencies pick their representative, and every voter chooses which political party should get their vote: the embodiment of their democratic power. The electoral system should clarify the contest for government, allowing all voters to participate in it if they wish, and should allow them to give their vote effectively to any party they choose. DV delivers clarity at all levels, with fair contests for government and proportional results on the national level, and fair contests on the local level.
2. Determining factors of result
Different factors affect electoral results in different systems, favouring and disfavouring particular patterns of votes. The maxim, “Votes win elections” is rather inadequate, as the expression of votes tends to depend on their interaction with other votes and with the system itself. This even without the complexity of considering that votes will be cast with consideration of the system in which they are cast, so that the system affects what votes are cast as well as what outcomes votes produce.
This section explores what factors affect results, particularly under STV and FPTP, showing how these distort votes, and prevent them from mattering equally. To the extent that votes are distorted in these systems, they are less democratic than DV, in which votes are collated nationally, and distribution does not matter, except in terms of the winning party’s majority.
In very few electoral systems is the result purely determined by who wins most votes. Votes tend to interact with each other, so that the location, party, number, and timing of votes, together forming the distribution, become crucial, and for each vote its outcome is a function of these aspects and of the distribution. This interaction is inherently complex, and the greater its influence, the harder it becomes to predict the exact effect of given distributions of votes without running through the entire process. So the effects of votes are uncertain and contingent, which is detrimental to the clear and free choice of the voter. There are arguments for some factors other than numbers of votes to exert an influence, but these do not remove democratic requirements, such as voter equality.
Distributions (location, party, number and time) in FPTP
In FPTP, the result is determined by the number of constituencies in which parties have a sufficient concentration of votes achieve a plurality. As only first-place matters, only votes that make the winner beat their nearest rival make a real difference. Parties get nothing for votes where they lose, so they are ineffective.
The effect of the concentration of votes in FPTP is to reward those parties with localised support, so if a party dominates a town then it is almost guaranteed to win a seat. It penalises parties with low geographical concentrations of votes, e.g. 30% in each constituency can be rewarded with no seats. If those 30% of votes were in 50% of constituencies (giving 60% in each) then this would guarantee 50% of seats. Average second-placed vote-shares in the 2001 UK election were 29%, and average winning-shares were 52%, so 30% could do even more. Evenly distributed between the right constituencies, a 30% national vote share could have beaten 76% of second-placed candidates, or 56% of winners. The 2001 Liberal vote of around 19%, spread equally between the right constituencies, could have beaten around 57% of second-placed candidates, or around 38% of winners. No party’s votes are spread this efficiently, but their efficiency varies greatly, as Table 2.1 shows.
Table 2.1: GB 2001 general election | Conservative | Labour | Liberal | Nationalists (PC + SNP) | Others |
% National vote | 32.7% | 42.0% | 18.8% | 2.6% | 3.9% |
% Seats | 25.9% | 64.4% | 8.1% | 1.4% | 0.2% |
% Seats placed 2nd | 55.5% | 18.7% | 17.2% | 8.4% | 0.2% |
Votes per seat | 50,323 | 26,006 | 92,583 | 73,439 | 985,549 |
This effect changes over time, so that at different times different parties benefit and lose out from it, as Table 2.2 shows.
Table 2.2: GB general elections ’92 and ’05 votes | Conservative | Labour | Liberal |
% National vote ‘92 | 42.8% | 35.2% | 18.4% |
% Seats ‘92 | 53.5% | 42.6% | 2.8% |
Votes per seat ‘92 | 40,957 | 42,346 | 334,900 |
% National vote ‘05 | 33.2% | 36.1% | 22.6% |
% Seats ‘05 | 32.2% | 57.7% | 10.1% |
Votes per seat ‘05 | 44,355 | 26,908 | 96,540 |
The effect of a vote for a particular party changes between elections. To prove that this is due to votes being well placed in terms of constituencies, one appropriate metric would be the ratio of constituencies in which a party is first-placed, to those in which it is second-placed. Another would be the proportions of votes contributing and not contributing to winning constituencies, (winning and losing votes). These metrics are calculated in Table 2.3, with a higher score in each indicating votes concentrated where they contribute to winning seats, and showing whether election results were more than or less than proportionate to vote totals.
Table 2.3: GB general elections ’92 and ’05 placing | Conservative | Labour | Liberal |
Ratio 1st-places/2nd-places ‘92 | 136.7% | 148.4% | 10.6% |
Ratio winning-votes / 2nd-place votes ‘92 | 250.1% | 208.8% | 13.4% |
Ratio winning-votes / losing votes ‘92 | 69.8% | 59.3% | 6.0% |
Ratio seats (%)/votes (%) ‘92 | 123.2% | 120.9% | 17.4% |
Ratio 1st-places/2nd-places ‘05 | 73.6% | 253.6% | 32.8% |
Ratio winning-votes/2nd-place votes ‘05 | 85.9% | 268.9% | 38.5% |
Ratio winning-votes/losing votes ‘05 | 163.9% | 247.8% | 71.6% |
Ratio seats (%)/votes (%) ‘05 | 94.4% | 156.3% | 43.6% |
Table 2.3 indicates that in the 1992 election, Conservative and Labour voters were both well distributed, gaining similar advantage from this. They did much better than the Liberals, who came second ten-times as often as winning a constituency, and had many more losing than winning votes. By 2005, the Conservative and Labour performances had diverged, whilst the Liberals were catching up. Labour voters benefited massively in 2005 from where they lived, with about 5 winning votes for every 2 losing votes, to the Conservatives’ 3:2. The location of their votes enabled Labour to win a share of seats 25.5% points above that of the Conservatives, with only 2.9% more votes. Voters are rewarded with more seats per vote, when they vote for these parties in the right numbers in the right locations.
Chart 2.1 (see overleaf) shows the national distribution of votes in 2001 between the parties. Labour’s vote is concentrated in blocks of around 20,000 per constituency, with relatively few votes in those constituencies where they lose. They outstrip the Liberals in some Conservative constituencies, but this represents only 14% of their votes, and in Liberal constituencies they have only 3% of their votes. In contrast, the Conservatives have 57% of their votes in constituencies won by others, and the Liberals have 78% of their votes in such seats.
Is distortion needed?
FPTP’s tendency to produce majorities enables British general elections to be choices of majority government10, by awarding Parliamentary majorities to parties gaining a large share and a plurality of the vote, without requiring that they win an absolute majority.
Table 2.4: GB swings since 1992 | Conservative | Labour | Liberal |
Votes 1992 (%GB) | 14,048,399 (42.8%) | 11,560,484 (35.2%) | 6,028,205 (18.4%) |
Swing 1992-97 | -11.4% | +9.2% | -1.2% |
Votes 1997 (%GB) | 9,591,082 (31.4%) | 13,541,380 (44.4%) | 5,243,440 (17.2%) |
Swing 1997-2001 | +1.3% | -2.4% | +1.6% |
Votes 2001 (%GB) | 8,353,666 (32.7%) | 10,740,347 (42.0%) | 4,814,339 (18.8%) |
Swing: 2001-05 | +0.5% | -5.9% | +3.8% |
Votes 2005 (%GB) | 8,782,197 (33.2%) | 9,552,436 (36.1%) | 5,985,454 (22.6%) |
NB: A fall in absolute votes can represent an increase in vote share, as turnout falls. | |||
Table 2.4 shows that the absolute vote shares and swings in party support in recent British elections have been too small for it to seem realistic that government would alternate through different parties winning 50%+ of the vote, and none have done so since 194511, (Section 5 reinforces this conclusion). To produce majorities with such results, the system must produce majorities in seats without majorities in votes.
Is the system aligned to produce results according to certain distributional factors?
British General Elections are national in character, and there is pressure on parties to gather support around the country, as exemplified by the criticism of the Conservative Party for winning none of around 72 Scottish seats in the 1997 election, and only one subsequently, but the formula is not working to ensure even representation in Scotland and Wales.
The Conservatives’ one seat in Scotland came from over 360,000 votes, whilst the Liberals won 10 from just under 380,000 votes, and Labour 55 from just over 1,000,000 votes. The situation in Scotland is, in short, even more distorted than in Britain overall, and in a way that changes over time, to the changing benefit and detriment of various parties, as Chart 2.2 demonstrates12.
There is no identifiable correlation between parties winning in Scotland or Wales and winning the election, as Labour have won the most seats in every case but two since 1945 (the Conservatives equalled Labour’s seats tally in Scotland in 1951, and beat it only in 1955, winning more votes both times, as in 1959, when they won fewer seats).
The effects of the location of votes have changed over time, and seem consistent only in that they have favoured the first party above the second party and produced majority government about 90% of the time, and have always disadvantaged the third party.
Defences of the FPTP system13 tend to defend it on grounds of a national choice of government, between two alternatives, so that this bias is appropriate, but its great variation across time and place14 does not seem to conform to any particular rationale. The holding back of the third party can be partly justified as enabling there to be two clear alternatives for government. It means that voters who wish to choose between two governments can see that their vote might make more difference if cast for one of two parties with the potential to win power, but this does not justify holding the Liberals back from challenging for second-place.
Chart 2.1 clearly showed a vast divergence in the situation in different constituencies, which means that even were the national situation aligned to give voters a particular choice (e.g. of majority government), then voters in different constituencies face different choices. This is incompatible with voters being unconstrained in their choices, and their preferences being treated equally. The ability to choose majority-governments may foster a national political community, but this divergence undermines it.
Factors relating to the distribution of votes matter greatly under FPTP, and allow it to produce majority governments. This function comes at the expense of treating individual voters very differently and unequally, again without much seeming justification except for in the production of the macro effects.
Benefiting winners
16/17 (94%) of British post-war governments have been formed by single-party majorities. The distribution of votes under FPTP has always favoured winning and second-placed parties over third-placed parties , and favoured winning over second-placed parties in 14/17 post-war elections. FPTP uses the distribution of votes to benefit parties likely to form a government, but not entirely consistently. To see how consistently, the benefit to the first and second-placed parties can be examined in each election. Comparing this to their share of votes, it can be seen how the system is working to reward winning parties with majorities in the Commons, by rewarding local pluralities. The top two parties’ benefits can be compared to the vote share of the third party, to see whether the system responds to their margin over that party, e.g. if it reduces the benefit to the second-placed party when that party’s margin over the third party shrinks, to reflect the similarity of their position.
Chart 2.3 shows that as the vote share of the third-party increased, the bias of the system in favour of the first two parties actually increased. As these parties lost vote share to the third-party, the system compensated, by increasing their benefit from the system. The benefits to the top two parties look uncorrelated with their vote shares. In fact, the winning party’s benefit from the system had a -0.57 negative-correlation with its share of votes, whilst the second-placed party’s benefit from the system was insignificantly negatively-correlated with its share (–0.09).
In 1997, the relation for the second-placed party suddenly shifts. This was a key election, characterised by a backlash against the Conservative Party, which had become very unpopular. There was some “tactical voting”15, where voters backed whichever of Labour and the Liberals was likely to beat the Conservatives in their constituency. The Conservatives shifted from a benefiting of 7.2% in 1992, to –5.7% in 1997, demonstrating how distribution determines results.
It has been shown that the effect of a vote varies dependent on where, when, and for whom it is cast, and this effect varies depending on how others vote, and hence varies between elections. Voters are not given consistent, free, or equal choices, and the effect of voting a certain way is unclear to them. It is particularly detrimental to equality and national political community for voters in different constituencies to face radically different choices. Systems that aggregate votes nationally and treat them consistently, such as DV or the German Additional Member System16, ought to solve these problems.
Where anything other than people voting choices contributes to a result, this must be rigorously justified. While the provision of majority government has been a macro effect of distributional factors under FPTP, the large micro-level effects are significant and unjustified.
Localisation of support
Rewarding pluralities in local constituencies means that FPTP ensures seats for those whose support is sufficiently localised, (in greater or lesser proportion than they are supported). Parties running on popular localised platforms, such as Plaid Cymru, the SNP, and the northern Irish parties, will win local seats. It seems appropriate that strongly nationalist areas can elect nationalist representatives, although this may effectively disenfranchise minorities in such constituencies who have no chance to elect a sympathetic representative.
For parties with non-localised support, e.g. those that campaign on broadly national issues, there is little chance of election without very large shares of the vote, and this may hold back people from voting for them. The Green Party in the UK, campaigning on an environmental platform, and gaining more votes in the 2005 General Election than Plaid Cymru or any northern Irish party, has never managed to elect an MP, whilst the former collectively won 21 seats in 2005. This makes widely dispersed voters count for less, and denies freedom of choice, as well as the chance to organise nationally rather than locally. A democratic system must be sensitive to voters who are dispersed, even whilst it connects local representatives to their constituents by making them reliant on local votes. Hence DV allows those parties with widely dispersed, yet significant, support to have national MPs. These are distinct from constituency MPs, so that the latter may retain the constituency link and specific local legitimacy. This contrasts sharply to the German AMS, where half of the members of the Bundestag are elected from regional party lists17, rather than for constituencies, which are hence around 3-times the size of UK18 constituencies.
Distributions in STV
District magnitudes greater than one may make the distribution of votes less important in STV, since votes need fewer other similar votes to be expressed, but the complex interaction of votes means that distribution still matters. The numbers of votes needed to stay in at different rounds of a contest will vary depending on the distribution of votes in each constituency, and so different distributions of preferences will do well. Tables 2.5 and 2.6 demonstrate how votes in STV can have different effects depending on the constituency where they are cast. In these examples, there are 3 candidates to be elected, and 199,996 voters, so the minimum votes to guarantee election is 50,00019.
Table 2.5 | Voters | Rounds | |||||
Type-V | Type-W | Type-X | Type-Y | Type-Z | |||
Party-A | 2 | 2 | 2 | 2 | 2 | 1 | No candidate has quota, A eliminated with no first-preferences |
Party-B | 4 | 1 | 3 | 5 | 6 | 2 | No quota, F eliminated with fewest first-preferences. |
Party-C | 6 | 6 | 4 | 3 | 1 | 3 | Type-V votes go to E (F and A eliminated), E elected. |
Party-D | 5 | 4 | 6 | 1 | 3 | 4 | 23,000 Type-V and Type-X votes redistributed from E to B. |
Party-E | 3 | 5 | 1 | 4 | 5 | 5 | B elected, 9,000 votes redistributed from B, including 5,492 to D. |
Party-F | 1 | 3 | 5 | 6 | 4 | 6 | D elected. |
No. of voters | 26,000 | 36,000 | 47,000 | 47,000 | 43,996 | (A was a Condorcet-winner.) Winners: E, D, B | |
Tables 2.5 and 2.6 are the same, except that Type-Z voters change their preference ordering between Party-A and Party-C. No-one’s opinion of Party-B changes, but Party-B loses the seat in Table 2.6. Reversing the analysis, from Table 2.6 to 2.5 B gains no support but gains a seat. Because they were in a different constituency, Party-B’s first-preference votes interacted with different votes, making them ineffective for Party-B.
So distribution matters, although due to STV’s complexity, the effects are more subtle, and maybe less consistent. Many changes would make little difference, e.g. if Party-A remains a universal second choice, then it will never win. Additional Party-E or Party-D voters could make no difference to their parties’ results.
Party-C and Party-F, neither of whom wins a seat, demonstrate the effect of party fragmentation. If a combined party had the higher of their two places in each preference ordering, then it would be first to win a seat in table 2.5.
Removing the option of Party-D from Table 2.5 would lead Party-C, Party-B and Party-E to be elected, whilst removing Party-A would have no effect, although over 75% of voters rank Party-A above Party-D. In fact, were preferences for other parties to change, it would be possible for each party to be guaranteed to win or to lose with no change in its support. STV clearly makes expression of each voter’s preferences contingent on those of others, so others’ votes can make their votes ineffective.
Table 2.6 | Voters | Rounds | |||||
Type-V | Type-W | Type-X | Type-Y | Type-Z | |||
Party-A | 2 | 2 | 2 | 2 | 1 | 1 | No candidate has quota, C eliminated with no first-preferences |
Party-B | 4 | 1 | 3 | 5 | 6 | 2 | No quota, F eliminated with fewest first-preferences; V votes go to A. |
Party-C | 6 | 6 | 4 | 3 | 2 | 3 | A elected. |
Party-D | 5 | 4 | 6 | 1 | 3 | 4 | 19,996 votes redistributed from A to E (7,428) and D (12,568). |
Party-E | 3 | 5 | 1 | 4 | 5 | 5 | D elected. |
Party-F | 1 | 3 | 5 | 6 | 4 | 6 | E elected. |
26,000 | 36,000 | 47,000 | 47,000 | 43,996 | (A was a Condorcet-winner.) Winners: A, E, D | ||
Conclusion
FPTP has worked to produce majorities, but this has come at great expense. Voters’ opportunities, and the effects of their votes, have been contingent on where, when, and for whom they cast their votes. STV similarly makes the expression of votes contingent on others’ votes in that constituency. Whilst local circumstances should affect local decisions, and so the conditions in a constituency should determine who wins it, only national circumstances should determine the national effects of elections, which has not been the case. A democratic electoral system for the UK must allow national results of elections to be determined by national totals of votes, whilst restricting the influence of local factors to local results.
DV responds to these problems, by making the national result depend purely on national totals of votes, and the local result depend purely on a candidate’s local support versus their rivals.
3. Preferences to be expressed; Condorcet-winners; monotonicity
Elections to legislatures involve decisions on several levels. In individual constituencies, the choice is of certain representatives, but nationally the choice concerns the overall composition of the legislature. There is then a tension between what makes sense at constituency level and what makes sense at national level. This is sometimes resolved by having a single national constituency, removing the local constituency link, which is very strong in the UK20. In deciding what preferences should form part of a ballot, there must be reference to both national and constituency-level results, i.e. attention to whether a constituency gets the right MP, and to whether the country gets the right Parliament, which is not necessarily the sum of “the right MPs” (in terms of individual constituencies).
This section explores what type of a choice might have legitimacy at a local level, although it is noted that choices at the national level, in terms of the composition of a Parliament, involve other criteria, which are explored in the next two sections. At the local level, it shows that although a Condorcet-winner would be appropriate, the choice may be indeterminate where there is no Condorcet-winner. It demonstrates that STV and AV, whilst determinate, can give inappropriate results, e.g. rejecting Condorcet-winners, and may not treat voters equally. The concept of a Condorcet coalition is then introduced, as a response to the problem of indeterminacy, showing that the merger of two candidates may produce a Condorcet-winner.
The prevalence of Condorcet-winners and coalitions is then statistically tested, showing that whilst the former are often absent, the latter occur almost always when this is the case. These results are used to suggest that the greatest legitimacy may come from encouraging the pre-electoral merger of candidacies and platforms, by using categorical preferences to simplify the contest for plurality. Hence, the criterion of the plurality of categorical preferences, as used in FPTP, is recommended for DV.
Condorcet-winners
If a majority prefers one option or candidate in a binary decision (a pair-wise comparison), then it is democratic to choose them over the other option. A Condorcet-winning candidate is one who commands a majority against all-comers in pair-wise comparison with them, i.e. who is preferred by majority to all other candidates. In terms of individual decisions, it is generally agreed that where there is a Condorcet-winner: this is the correct choice (Dummett, 1997). Where MPs have an important link to their constituency, it is important that a good result pertains at this level.
When voters rank alternatives, they describe which they prefer to which, i.e. any higher-ranked candidate is preferred to any lower-ranked candidate, in pair-wise comparison with them. If voters register categorical preferences, (choosing one candidate) then they only reveal that they prefer that candidate to each other candidate, not whom they prefer amongst the others.
Table 3.1 | Voter X | Voter Y | Voter Z | Pair-wise comparison | Win | |
Candidate-A | 1 | 3 | 3 | A‘v’C | Y & Z outvote X | C |
Candidate-B | 2 | 2 | 1 | A‘v’B | Y & Z outvote X | B |
Candidate-C | 3 | 1 | 2 | B‘v’C | X & Z outvote Y | B |
If they rank all candidates, this shows whom they prefer in every pair-wise comparison (e.g. Voter X in Table 3.1 prefers A to B, A to C, and B to C). Each candidate can be compared to every other candidate, to see which a majority prefers. In Table 3.1, Candidate-B is a clear Condorcet-winner, commanding a majority against every available alternative. Electing Candidate-B would seem the most democratic outcome. Candidate-A is a Condorcet-loser, losing against every available alternative.
In contests with more than two candidates, there may not be a Condorcet-winner, as in Table 3.2. The electoral system requires a determinate result in each constituency, and it is unclear what this should be where there are no Condorcet-winners, especially given Arrow’s Impossibility Theorem21.
Table 3.2 | Voter X | Voter Y | Voter Z | Pair-wise comparison | Win | |
Candidate-A | 1 | 2 | 3 | B‘v’A | X & Y outvote Z | A |
Candidate-B | 2 | 3 | 1 | A‘v’C | Y & Z outvote X | C |
Candidate-C | 3 | 1 | 2 | C‘v’B | X & Z outvote Y | B |
The group’s preferences here are non-transitive: they prefer A to B, and B to C, but not A to C, so there is a majority against every candidate. There is no one fair choice.
All candidates are compared to all other candidates, and voters are counted for their preferred candidate in each pair-wise comparison. Which alternative is picked depends on everyone’s votes, but everyone is counted in every comparison. Where the comparisons to be made are indeterminate, and not all known to the voter at the time of voting, this is not the case.
AV and STV
In STV, not all pair-wise comparisons are made, and it is only when all but two candidates are excluded in the single-member version, AV, that all voters’ preferences in a comparison are counted. These systems use rounds of voting, where candidates are awarded the votes on which they are first choice amongst remaining alternatives. Anyone reaching the quota is elected, and otherwise the candidate with fewest is excluded, and the comparison made again.
Rankings determine how votes are cast in rounds, i.e. for the highest-ranked remaining candidate on the vote; although when a candidate reaches the quota22 and is elected, their surplus votes over the quota are redistributed (when a candidate is excluded, all their votes are redistributed). Their highest-ranked candidate might not be the candidate whom a voter would choose in a round where that candidate will be neither excluded nor elected; where transferring their vote to a second-preference candidate might prevent them being excluded.
At each point, to whom a vote is attached depends on the ranking, but also on who has been excluded according to the aggregate preferences. This is not the same as in the comparisons above, whereby each pair-wise comparison would be made, and so voters could give their choice in each one through ranking. It makes voters’ effect dependent on others’ votes and therefore makes them neither free nor equal.
In Table 3.3, the AV winner would be Candidate-A, as in the first-round, Candidate-B would be eliminated, with least first-preferences (3,200), and Candidate-B’s first-preference votes passed to Candidate-A, who would then attain a majority over Candidate-C of 7,200 to 4,000. In the first-round, Type-X voters have their preferences for Candidate-B compared to the preferences of Type-Ys for Candidate-C, and Type-Zs for Candidate-A, so voters are having different relative preferences compared. Voters of Type-Z have no say in the contest between Candidate-B staying in and Candidate-C staying in. This method of comparison violates the equality of voters, by using different groups to make different decisions in which all have a stake.
Table 3.3 | Voters | AV rounds | ||||
Type-X | Type-Y | Type-Z | ||||
Candidate-A | 2 | 3 | 1 | 1st | B excluded with least votes (3,200) | |
Candidate-B | 1 | 2 | 2 | 2nd | A given Type-X votes. A elected with 7,200 votes to C’s 3,400. | |
Candidate-C | 3 | 1 | 3 | |||
No. of voters | 3200 | 3400 | 4000 | Condorcet-winner AV winner Condorcet-loser | B A C | |
This process doesn’t necessarily select a Condorcet-winner where there is one. In Table 3.3, Candidate-B is a clear Condorcet-winner, yet AV would elect Candidate-A, to whom a majority prefer Candidate-B. B loses the AV election because of the candidacy of C, whose voters would win the election for B, if Condorcet-loser C didn’t beat B out of the first round.
Strategically, Type-Y voters should hide their preferences, and give their first-preference to Candidate-B, as Candidate-C cannot win by their efforts, but voting sincerely results in their least favoured candidate, A, winning. If they are strategic, then Candidate-B wins, with a majority of first-preferences, which for them is preferable to Candidate-A winning.
Strategic voting introduces inequality as voters who better anticipate others’ votes can use theirs more powerfully, and it is contrary to political community to encourage people to give insincere preferences. Candidates are unequal here, as are voters. The presence of Candidate-C hurts Candidate-B’s chances, for example, but not vice versa. Voters of Type-X suffer from Type-Ys’ honesty, or Type-Zs suffer from Type-Ys’ hiding of preferences. The outcome is now dependent on voters’ knowledge of each other’s preferences, and whether they vote strategically. The relative value of votes varies depending on the other votes cast.
Table 3.4 is the same as Table 3.3, but with some Type-X and Type-Y voters converted to Type-Zs, whose first-preference is Candidate-A. First-eliminated here would be Candidate-C, passing C’s votes to Candidate-B, who would then have a majority over A of 5500 to 5100.
Table 3.4 | Voters | Pair-wise comparison | Win | |||
Type-X | Type-Y | Type-Z | ||||
Candidate-A | 2 | 3 | 1 | A‘v’B | X & Y outvote Z | B |
Candidate-B | 1 | 2 | 2 | A‘v’C | X & Z outvote Y | A |
Candidate-C | 3 | 1 | 3 | B‘v’C | X & Z outvote Y | B |
No. of voters | 2800 | 2700 | 5100 | Condorcet-winner AV winner | B B | |
So Candidate-A wins over more first-preferences; from both opponents; and then loses where previously A won. This violates the condition of monotonicity; that more higher-placed preferences will mean a candidate does better (or no worse), and vice versa with fewer. This is important not only dynamically: but because it begs the question: which result should prevail? In the case where Candidate-A is elected, 6,600 against 4,000 voters prefer B to A, but in the case where Candidate-B is elected, 5,500 against 5,100 prefer B to A. Even if one of these results is fair, the other isn’t, and voters could not be said to have liberty of decision or voting equality in such a case.
There is no obvious decision here other than the Condorcet-winner, who may lose under AV, and who has the lowest number of first preferences in Table 3.3, and the second-lowest in Table 3.4. In other circumstances there may be no Condorcet-winner.
AV, district magnitude, proportionality, and the Single Transferable Vote
If one MP is selected per constituency, and constituencies are similar, then the same party will be returned in every constituency23, which is unfair, given that other parties may be preferred by much of the electorate. This is a consequence of votes only contributing to the selection of one MP, who will have a single vote in Parliament. Aggregating more preferences together gives a more even result.
Candidate-A and Candidate-B would be the two STV winners in the constituencies represented in Tables 3.4 and 3.5, which does seem more appropriate than just Candidate-A or just Candidate-B, but despite having been chosen by very different numbers of voters, they would have equal status. Would they have equal legitimacy? Under AV, Candidate-A would have won in Table 3.3, with the majority of the population preferring Candidate-B, whilst B would have won in Table 3.4, (with less support than the B in Table 3.3), and Candidate-A would have lost (with more support than the A in Table 3.3).
Systems like AV24 and FPTP can give very disproportionate results because, for example, a party can get nothing for 30% of first-preferences in many constituencies where others beat it, or win many constituencies if its votes are better distributed. In STV, the concentration of votes needed to win is lower, and so results tend to be more proportionate and possibly fairer.
Talking about proportionality when there are ordinal preferences involved doesn’t really make sense however, as a number of incomparable preferences for candidates are expressed. Comparing only first-preferences would make a party receiving second-preference on every ballot come out as having 0% of the vote, whilst adding together all preferences would make a party listed ninth on every ballot receive the same proportion as one listed first on every ballot. Attaching a weighting to different rankings is inappropriate since different rankings mean different things to different people, and one can’t say that a certain number of x-ranked preferences are equal to a different number of y-ranked preferences.
Condorcet and categorical preferences
Table 3.5 shows Table 3.4 under FPTP, with voters’ categorical vote given to their first-preference candidate, and ignoring preferences that weren’t first-preferences.
Table 3.5 | Voters | Pair-wise comparison | Win | |||
Type-X | Type-Y | Type-Z | ||||
Candidate-A | 2 | 3 | 1 | A‘v’B | 5100 outvote 2800 | A |
Candidate-B | 1 | 2 | 2 | A‘v’C | 5100 outvote 2700 | A |
Candidate-C | 3 | 1 | 3 | B‘v’C | 2800 outvote 2700 | B |
No. of voters | 2800 | 2700 | 5100 | STV winner FPTP winner | B A | |
With categorical preferences, as shown in Table 3.5, the second and third preferences are indeterminate. Voters’ preferences are being used only in comparisons that they know will be made, as when Condorcet-winners were selected, and they may vote accordingly, not necessarily selecting their first-preference candidate, if they wish to help another to win the seat, e.g. Type-Y voters could switch their support to Candidate-B. In Table 3.5, Candidate-A might be termed the Condorcet-winner, as in pair-wise comparison it beats C, with Type-X voters registering no preference, and beats B, with Type-Y voters registering no preference. Alternatively, Candidate-B might be termed the Condorcet-winner, because, if asked, majorities of voters would prefer B to A or C.
In Table 3.6, there is no Condorcet-winner when all voters’ preferences are registered, but FPTP consistently gives a determinate outcome, which seems fair. There might be here a majority for Candidate-A (who doesn’t win the vote of all who prefer them to B) against Candidate-B, if all pair-wise comparisons were made, but with these ordinal preferences there is no fairer result than that for B, who more voters supported categorically25 than any other candidate.
Table 3.6 | Voters | Pair-wise comparison | Winner | |||
Type-X | Type-Y | Type-Z | A‘v’B | 4600 outvote 2900 | B | |
Candidate-A | 1 | 2 | 3 | A‘v’C | 2900 outvote 1900 | A |
Candidate-B | 2 | 3 | 1 | B‘v’C | 4600 outvote 1900 | B |
Candidate-C | 3 | 1 | 2 | Condorcet-winner STV winner FPTP winner | None A B | |
No. of voters | 2900 | 1900 | 4600 | |||
STV gives a result, but it does not treat voters equally, and it can give the result against a Condorcet-winner when there is one, as in Table 3.3, or in the case of Table 3.6 to a candidate who is neither a Condorcet-winner, nor the candidate with most first preferences. The Condorcet criterion itself gives a result only in the specialist conditions in which a Condorcet-winner exists. If there were always a Condorcet-winner then this might work, but it has been easy to demonstrate circumstances where none existed, so it would not produce the determinacy necessary at an election.
Condorcet pairs and coalitions
In each of these cases where there was no Condorcet-winner, a coalition could have beaten the alternative(s). Of course with a set of three options, this is trivial, as it reduces to a single pair-wise comparison, e.g. A & B against C, which either A & B must win or C, giving a determinate result in favour of one or two candidates. In these examples, more than one pairing could have beaten the alternative, e.g. any pair in Table 3.6. A pairing that beats any alternative can be labelled a Condorcet coalition, but this has two possible senses. The coalition can compete against each alternative, or the coalition members can compete independently against each alternative.
Table 3.7 illustrates the difference between these two conceptions. The pair AC, can beat B, as one of A and C is preferred to B by W, X, and Y voters, so AC wins 5,450 votes against B’s 2,200. Such a pair, which wins every pair-wise comparison when the votes of those who prefer either member to the alternative in the comparison are attributed to it; may be called a Condorcet pair. Alternatively, there is the Condorcet coalition A&C, which beats B because although C loses to B, A beats B, 4,850 to 4,800. The Condorcet coalition meets a stricter criterion, that independently of each other, at least one of its constituents beats each alternative.
Table 3.7 | Voters | Pair-wise comparison | Winner | ||||
Type-W | Type-X | Type-Y | Type-Z | A‘v’B | W,X outvote Y,Z | A | |
A | 2 | 1 | 3 | 3 | B‘v’C | X,Z outvote W,Y | B |
B | 3 | 2 | 2 | 1 | A‘v’C | W,Y,Z outvote X | C |
C | 1 | 3 | 1 | 2 | AC‘v’B | W,X,Y outvote Z | AB |
No. of voters | 1750 | 2100 | 1600 | 2200 | A&C‘v’B | C loses to B, but A beats B | A&C |
If there were a way to merge A and B, then there might be a Condorcet-winning option, which would be a legitimate winner. Political parties do merge, candidates withdraw and endorse their rivals, and parties take on each other’s popular policies, e.g. the three main parties’ manifestos in 1997 promise a referendum as a pre-condition of joining EMU, which was not policy prior to the Referendum Party’s campaign26. If there are Condorcet coalitions when there aren’t Condorcet-winners, then encouraging the merging of these coalitions could produce more winners.
In STV, similar candidates can benefit from their collective vote, as their votes may be passed on to each other. This means that there is less need to put up a single united candidate, and encourages many candidates to stand separately. Systems using non-transferable categorical preferences encourage fewer candidates to stand, and sympathetic interests to unite behind a single candidate.
If the members of potential Condorcet coalitions try to build those coalitions behind single candidates, then Condorcet-winners may emerge, filling the gaps where there weren’t already winners. Hence, if Condorcet coalitions are prevalent, then voting systems that encourage united fronts may respond better to the Condorcet criteria than those where more similar candidates will stand.
How do votes fit the Condorcet criteria?
The prevalence of Condorcet-winners, coalitions, and pairs, is a matter of statistics. Given the indeterminacy of parties’ ability to merge and to retain votes: it is more cautious to focus on the stricter coalition criterion, rather than the pair criterion.
There are a finite but very large number of distinct rankings of options: exactly (o!)27 rankings of o options. When more than one person is voting, there is a distribution of rankings, which is defined by the frequency of different rankings, e.g. 2 voters might rank candidates A,B,D,C. Regardless of the frequency of other rankings, if one fewer person has the ranking A,B,D,C, then a ranking is distinct from its predecessor. If 2 voters rank 2 choices, then there are 2 rankings (A,B and B,A), and 3 distributions of rankings ((A,B B,A) (A,B A,B) (B,A B,A)). If 5 voters rank 5 choices, they have over 225 million possible distinct distributions of preferences28, with 50 voters; there are over 2,500×1040 distributions. Random tests can be used to predict how winners will emerge. 5 million sets of 15 computer-generated voters’ random preferences between 10 options produce the statistics in Table 3.6.
Table 3.6: 15 voters' random ordinal preferences among 10 candidates | Number | % | Frequency (3 s.f.) |
Distributions tested | 5,861,130 | 100% | |
Distributions with Condorcet-winners | 3,164,779 | 54% | 1 in 1.85 |
Distributions with Condorcet coalitions | 2,696,305 | 46% | 1 in 2.17 |
Distributions with no Condorcet coalitions | 46 | 0% | 1 in 127,000 |
It seems that just over half the time there will be a Condorcet-winner, and that when there isn’t, there will usually be one or more Condorcet coalitions, although in 1 in 100,000 sets there will be none.
Condorcet-winners with n candidates
Table 3.7 shows how an additional candidate, D, changes a situation in which there is initially a Condorcet-winner (initial rankings are in parenthesis). The addition of candidate D prevents there being a Condorcet-winner, although likewise, an additional candidate E, picked first by two types of voters, would be a Condorcet-winner.
Table 3.7 | Voters | Pair-wise comparison | Winner | |||
Type-X | Type-Y | Type-Z | ||||
Candidate-A | 3 (2) | 4 (3) | 1 (1) | D‘v’A | X & Y outvote Z | D |
Candidate-B | 2 (1) | 3 (2) | 2 (2) | D‘v’C | Y & Z outvote X | C |
Candidate-C | 4 (3) | 1 (1) | 3 (3) | D‘v’B | X & Y outvote Z | B |
Candidate-D | 1 | 2 | 4 | Condorcet-winner FPTP winner | B A | |
No. of voters | 2900 | 1900 | 4600 | |||
As candidates are added, the frequency of Condorcet-winners decreases. Compared to 54% with 10 candidates in Table 3.6, in a test of 10,000 distributions with 15 voters and 3 candidates29, around 90% had Condorcet-winners, and the remainder had Condorcet coalitions. With 4 candidates, around 84% of distributions30 had Condorcet-winners, and the rest coalitions. This reinforces the arguments made above, that where systems encourage fewer candidates to stand, more Condorcet-winners may emerge.
Condorcet-winners with n voters
The more voters there are, the more possible distributions of preferences, so this may affect the frequencies of different results. In 1000 random distributions of 7 voters’ preferences between 3 candidates; 93% had Condorcet-winners, with the remainder having Condorcet pairs; whilst with 10,000 random distributions for 255 voters, it was 91%. With 4 candidates and 7 voters, 85% had Condorcet-winners in a test of 100,000 distributions, and 82% with 65,535 voters in 35,797 distributions. These tests are small relative to the massive numbers of possible distributions, and are vulnerable to poor random number generation. They suggest weakly that more voters mean fewer Condorcet-winners, but that the effect will be small, so that the results reported should scale up well to real constituency size.
Conclusion
It is clear that if there is a majority against a candidate and for another in their constituency, then the legitimacy of the former as a winner is very questionable. Yet there is a majority against every candidate fairly frequently, more often the more candidates there are (although constituencies may have only a few serious candidates, perhaps making the results for lower numbers most relevant). In these circumstances, there are two options with ordinal preferences: to elect a candidate to whom another is preferred; or to select Condorcet-winners where they exist, and to declare no result where they don’t, which is unacceptable for the UK electoral system.
Where there is no Condorcet-winner, there will almost always be a Condorcet coalition31, but there are sometimes several and sometimes none, and it is unacceptable to have a system that produces one winner sometimes, and other times an indeterminate multiplicity of winners. Whilst these results are discouraging for the pursuit of a system that uses ordinal preferences, (e.g. STV) they can provide support for a system using the plurality of categorical preferences as in FPTP and DV, which encourages unity behind single candidates.
If we operate a system of FPTP, there are likely to be fewer candidates in each constituency than under STV, as they will be going after a single seat and, without the transfer of votes, additional candidates will expect to do more harm with their candidacy to other candidates to whom they are sympathetic. In a smaller constituency there will also be a better chance of candidates and parties getting to know voters’ preferences. If there are possible Condorcet-winners or coalitions amongst the wider sets of candidates who would stand under STV, then these should be best placed under FPTP to undertake successful pre-electoral mergers (which may involve formal coalitions, or candidates manoeuvring to take on each other’s positions), knowing that they must concentrate support to win. Even without these strategic effects, in tests, around 68-69% of Condorcet-winners won the plurality of first-preferences32.
When we see a Condorcet coalition in a distribution of preferences, it is impossible to predict whether having both win would be a workable solution, as they may or may not be compatible, but by forcing concentration prior to the election, those who are compatible should concentrate to produce options that can command plurality support. Where a single Condorcet-winner exists, they should be similarly able to manoeuvre to ensure their election. Taking categorical preferences thus seems the best way to cope with the indeterminacy inherent in groups’ ordinal preferences.
The discussion of STV and AV demonstrated how they involved arbitrarily including or excluding voters with each decision, so that they violated the conditions of equality. FPTP also violates equality, not in determining who wins constituencies, but in their being awarded equal positions on unequal mandates. The winner of the most unanimous constituency in 2005 gained 75.5% of the vote there, whilst the most divided constituency was won with only 31.4%, and both MPs had one vote in the national legislature. Were decisions more complex, as in DV, to reflect the complexities of the result, then each voter might be equal.
With ordinal preferences, the Condorcet criterion does not always give a result, and STV may give a result against a Condorcet-winner, and lacks simplicity, fairness, and monotonicity33. STV also requires larger electoral districts, weakening the constituency link. Using categorical preferences to make the constituency-level choice of representative, as in the FPTP system, is a better way to reflect ordinal preferences. When only categorical preferences are registered the plurality winner is a Condorcet-winner in one sense34, and FPTP is simple, monotonic, and fair at the constituency level.
Though the plurality of categorical preferences always delivers a single winner, results are quantitatively different, because winners have widely varying margins of victory and totals of votes. Ignoring the margin, the total of votes, and the votes for other candidates as in FPTP, violates the democratic conditions already set out: although selecting the plurality winner as constituency representative, whilst taking into account every vote, as in DV, does not.
4. Proportionality
Basis
Proportionality is a recurring theme in the electoral systems debate, where the ideal of equality is extended to the proposition that equal groups of voters should have equal representation in the legislature. Directly proportional results can enhance transparency, making voters able directly to influence the composition of the legislature, and making their vote worth the same proportion for whomever it is cast. Without direct proportionality, there is generally uncertainty regarding the effect of a vote, with large numbers of votes sometimes leaving proportions of seats unchanged. Proportional results can enhance political dynamism, enabling the break-through to government of new parties, as happened for the Greens in Germany’s relatively proportional system, but not for the Liberal-SDP Alliance in the UK’s relatively disproportional system.
DV works differently than systems that try to secure proportional seats. As MPs’ voting weights in the Commons are different, it is a party’s collective votes that matter, not its number of seats. Since these votes are just the votes they received, they are exactly proportional35. This means that if there are significant problems with proportionality in seats, but proportionality remains an important criterion, then DV will be a good solution.
Direct proportionality and conditional proportionality
Proportionality is an explicit part of a directly proportional system, which distributes seats according to a formula using the proportions of votes, but is also an aspect of every electoral system. The FPTP system, for example, has given very disproportionate results in the UK, with prime examples being the Liberal-SDP Alliance’s 2.0% of seats from 18.3% of votes in October 1974 and Labour’s 55.0% of seats from 35.2% of votes in 2005. The proportionality delivered by systems that do not function by direct proportionality is conditional and varying, due to different determining factors. Nevertheless, there are trends in the proportionality of representation in all systems, although these trends may be stronger (Germany) or weaker (UK). Lijphart examined the various measures of proportionality, and found Gallagher’s least-squares index the best36. Lijphart ranks various systems in various countries, placing the UK (1945-92) at 32/37, with average disproportionality of 10.76 on this index, rising to 11.84 by 2005, compared to Israel (1951-69), with 0.86.
Problems
The simplest system with direct proportionality is the party list system, whereby people vote for a party, not individual candidates, and parties are then awarded seats in proportion to their national vote, which they fill from a prior list of candidates. It treats equally members who are elected at the top or the bottom of a party list. This system can transfer too much power from the voters to the party, unless modified to give voters choice from the party list, so that those first to fill the party’s seats are chosen by voters, not the party. A national list abandons the idea of constituency representation, and makes it very hard for voters to know all the people for whom they are voting, although it delivers high proportionality, and simplicity. If the country is divided into multi-member regions for this, then the former two problems, and the latter strength, are all lessened.
If members are elected to a one-member one-vote assembly, then perfect proportionality is impossible, because votes will not divide in the same proportions as seats e.g. if a party wins 20% of the vote in a 13-member constituency, then it can’t be awarded exactly 20% of seats, as 20% of 642 is 2.6. For smaller parties the disproportionality can be worse, e.g. 1% of the seats would be 0.6537. As the number of seats chosen from a particular pool of votes (district magnitude) decreases, proportionality deteriorates, as seats can be split less sensitively.
High-levels of proportionality are often linked to better results for smaller parties, although not universally. Some local parties in the north of Ireland have received more than proportionate seats under FPTP, although the SNP and Plaid Cymru both tend to receive fewer. Duverger’s law38 states that non-proportional systems favour 2-party systems, whilst proportional systems favour multi-partism. Different systems in different countries exhibit different trends in disproportionality, which varies between elections, so they cannot be clearly divided into proportional and disproportional systems, but there is correlation between levels of proportionality and the number of effective parties in a party system.
Calculating the effective number of parties according to Laakso and Taagepera’s index (1979), (1 divided by the sum of the squared percentage of seats for each party, e.g. 1 / (50%2+50%2) = 2), and using the least-squares index of disproportionality, there is a –0.53 negative correlation in Lijphart’s data39 between the two. A negative correlation indicates that as disproportionality decreases, or proportionality increases, effective parties increase. There is a well-known “chicken and egg” problem, in that a higher effective number of parties40 might lead to pressure for more proportionality, but it seems that more proportionality leads to more effective parties. The best evidence for this is the multiple entries in Lijphart’s data for 8 countries that have changed electoral systems. In every case where a country moved to a more proportionate electoral system, effective parties increased, and every time the system became less proportionate, effective parties decreased41. So party systems seem to have been changed by electoral systems.
More proportionality leads to more effective parties, which makes coalitions more complicated and more likely. The modern Israeli system has been criticised for leading to coalitions in which small parties are key partners, giving them too much power. In contrast, the Turkish system excludes entirely parties receiving less than 10% of the vote. It would seem that the main dangers of increasing proportionality are small parties becoming too numerous and powerful, and harming the prospects of majority government.
Proportionality may reduce the tendency for votes to coalesce in favour of big party blocs, detracting from effective contests for majority government between the two biggest parties. An ideal proportionality profile will enable equal blocs of voters to win equal legislative votes for their minority parties; but to enable a national contest to select a government, votes may need to be more effective when cast for parties that can win. DV’s systematic benefit to the winning party; based on its margin versus the second party; would deliver an incentive for opposing votes to go to the opposition (reducing this margin), without distorting the contest for second-place (and indeed all contests among parties that don’t win). A formula exaggerating the win might also prevent stagnation, a danger when seat shares correspond very closely to vote totals that change only at the margins.
Proportionality and ordinal preferences
STV is often offered as a very proportional electoral system42, however ordinal preferences leave proportionality an open question. Popular proportionality measures compare parties’ first-preference totals43; which as Hill points out, is inappropriate for STV; where second-preferences may be cast against the first-preference party, for example. As any weighting attached to e.g. second-preferences, would be arbitrary44, the only fair weighting seems to be counting each vote as one according to the first-preference. Only first-preference measures make any sense, but they make nonsense of ordinal preferences.
Proportionality of tenure
An alternative to the idea of proportionate representation would be to look at proportionality of tenure, that is time in government. This deals with the complaint that proportional results disproportionality favour small parties, who can be permanently in coalition government, whilst much larger blocs that can’t form a government without them are excluded. This may have been the case in Germany, where the third-placed Free Democratic Party was in government for 38 years 1949-1994, with either of the much larger CDU-CSU (32 years) and SDP (16 years). Taylor (1984) argued that proportionality was better thought of in terms of time in government, rather than seats in the legislature, and found that FPTP performed rather well on this metric, and better than a highly proportional system might have, if it led to constant coalitions. Table 4.1 computes proportionate tenure in the UK from 1945 to the time of the 2005 election, multiplying vote shares by the time until the next election45. It also shows the tenure achieved under FPTP, and what tenure might have been under DV and German AMS46 with the same votes. Years where no party would have had a majority, totalling 6.6 years, are shared (half each) between the Liberals and the winning party.
Table 4.1 | Proportionate tenure | DV tenure | FPTP tenure | AMS tenure |
Lab | 24.4 years | 25.1 years | 25.2 years | 14.1 years |
Con | 25.1 years | 31.4 years | 34.7 years | 15.8 years |
Lib | 7.8 years | 3.3 years | 0 years | 29.9 years |
FPTP in the UK has evidently been more proportional than Germany’s system, considering that the FDP received only 6-13% of votes, and yet was in power 84% of the time 1949-1994, which is very disproportionate, even if they remained a junior coalition partner. Counterfactuals of coalitions that would have otherwise occurred are arbitrary, so the results for DV and AMS are questionable, but on this evidence DV gives more proportionate tenure than either FPTP or directly proportional seats through AMS. Having the Liberals in coalition47 for 6.6 years seems preferable to either excluding them from government, having them in coalition 29.9 years, or even giving them 7.8 years of majority government48.
Conclusion
Proportionality in Parliamentary representation is an important measure of fairness at an election, and direct proportionality is fairer than conditional proportionality. It has important effects on the party system, with more effective parties resulting from more proportionality. This can enable the breakthrough of new parties, and keep the party system dynamic where otherwise there might be stagnancy, but it also can detract from majority government. Proportionality of tenure is another important metric, and can be in tension with proportionality in representation, with the former perhaps more important in the long run, and the latter in the short run. The electoral system should seek to secure both, as far as possible.
5. Producing majorities
All but one election since 1945 has resulted in a government being formed by a party with a Parliamentary majority. The election is said to have been “won” by this party, which is recognised as legitimate. The majority style of government is fundamental to how UK politics works. This presents a problem to an analysis that does not challenge this system, but which cleaves to a principle of equality between voters. If voters are equal, then this may demand that an equal number of electoral votes gain an equal number of votes in Parliament.
If government is by parties who have won electoral majorities, and alternates between different parties, then these different parties need each to win over 50% of the vote at consecutive elections. This section shows that this degree of dynamism has not been manifest, then goes on to look at how parties have won elections and majorities. Evidence is considered on the effect of the size of the majority, showing that this does have a large effect on a government’s power. It then concludes that a function other than exactly proportional representation is needed to allow the winning party a majority, and that the size of the majority is important and should be consistently sensitive to the scale of the victory. Thus, the majority-producing function in DV is found to be necessary to produce majority-government, with somewhat proportionate power according to the size of the majority, as well as the proportionate tenure demonstrated in the last section.
Chart 5.1 shows that swings to winning parties have failed to produce majorities in all elections since 1945, but also shows that if some swings, e.g. 1997, had occurred in other elections, then they could have produced majorities. Does this mean that there is sufficient dynamism but that it has been at the wrong time? The swing needed is measured by 50% minus the winning party’s previous election result (whether it won then or not). The swing tracks the swing needed to a large extent (they exhibit 0.54 correlation, or 0.64 if 2005 is omitted) indicating that the further away a party is from 50%, the larger the swing towards it, and vice versa.When far away from 50%, parties may experience large swings, but when they are just short, the swings are smaller. Parties can make up some of the gap, but the nearer they are, the harder it is to get any nearer, and they never make it. This shows how they win the votes of a certain number of swing voters, in addition to their core votes, but cannot persuade the other parties’ core voters.
Swing voters switch their vote between elections, and are widely argued to decide elections, whilst core voters never switch party. If each party has core support C<50% and there are also a number of swing voters S<50%, which are shared between the parties, according to their proportions P, then a party requires that C+PS>50% to win a votes majority. If C+S<50% then no swing will produce a majority, and if C+S≈50% then extremely lopsided swings will be needed. When there are only a finite number of swing voters, a party that gets many votes is receiving most of the swing votes, capping its own additional success.
To estimate core and swing voters, a combine survey data and election results can be used. It is likely that swing voters will be shared between the parties in proportions 0%<P<100%, and so any party’s core support at an election will be equal to (total votes) V – SP = C, where SP>0. Core support will change over time, if only through births and deaths, but as a proxy we can take a party’s minimum vote since 1945, probably an over-estimate. Swing voters can be measured by the highest margin between a party’s lowest and highest votes. In addition to this, recent survey data49 from 2003 recorded percentages of core and swing voters.
Table 5.1 | Core = min result (max) | Swing = Max - Min | Largest swing plus core | 90% swing + core | 80% swing + core |
Lab | 28% (49%) | 21% | 51% | 48% | 46% |
Con | 31% (50%) | 19% | 54% | 51% | 49% |
Lib | 3% (25%) | 23% | 25% | 23% | 21% |
Survey | Decided | Swing voters | 100% swing + core | 90% swing + core | 80% swing + core |
Lab | 30% | 23% | 53% | 51% | 48% |
Con | 28% | 52% | 49% | 47% | |
Lib | 19% | 42% | 40% | 37% |
The table compares both techniques, with the result-range measure of swing voters coming from the 23% gap in Liberal support between 1951 and 198350. The survey also identifies 23% as swing voters, making majorities possible only with completely lopsided swings. If 23% are swing voters, and no party’s core voters are more than 31%, then majorities would require that they persuaded 84.3% of swing voters, compared to 15.7% for all other parties, which seems very improbable.
Majority votes have not occurred, and they look very unlikely. In this case, elected majorities would need to occur without electoral majorities. The results since in Table 5.2 show how the electorate has voted for the winners of UK elections.
Two parties, the Conservatives and Labour, have fallen just short of attaining the swings they needed to gain 50% of the votes, but each election has resulted in victory for one of them. This is a two-party contest for a majority, with the Liberals achieving a best showing of 10% of seats in this period, whilst the Conservatives and Labour alternated in government. The winner’s margin of votes over the loser is very highly correlated with their Parliamentary majority both overall and versus the loser. Their margin of votes versus the rival is also highly correlated with their excess of share of seats over share of votes, (this excess enables the majority). This correlation is 0.69, or 0.86 before 2001.
Table 5.2 | 1945 | 1950 | 1951 | 1955 | 1959 | 1964 | 1966 | 1970 | 1974 | |
Winner | Lab | Lab | Con | Con | Con | Lab | Lab | Con | Lab | |
Winner vote share | 48% | 46% | 48% | 50% | 49% | 44% | 48% | 46% | 37% | |
Top 2 parties vote share | 88% | 90% | 97% | 96% | 93% | 88% | 90% | 89% | 75% | |
Govt. seat share | 61% | 50% | 51% | 55% | 58% | 50% | 58% | 52% | 47% | |
Overall vote majority | -4% | -8% | -4% | -1% | -1% | -12% | -4% | -7% | -26% | |
Parliamentary majority | 23% | 1% | 3% | 10% | 16% | 1% | 15% | 5% | -5% | |
Vote margin over rival | 8% | 3% | -1% | 3% | 6% | 1% | 6% | 3% | -1% | |
Parliamentary majority over rival | 28% | 3% | 4% | 11% | 17% | 2% | 17% | 7% | 1% | |
1974 | 1979 | 1983 | 1987 | 1992 | 1997 | 2001 | 2005 | Correlations | ||
Winner | Lab | Con | Con | Con | Con | Lab | Lab | Lab | ||
Winner vote share | 39% | 44% | 42% | 42% | 42% | 43% | 41% | 35% | ||
Top 2 parties vote share | 75% | 81% | 70% | 73% | 76% | 74% | 72% | 68% | 0.16 | |
Govt. seat share | 50% | 53% | 61% | 58% | 52% | 64% | 63% | 55% | ||
Overall vote majority | -22% | -12% | -15% | -16% | -16% | -14% | -19% | -30% | 0.16 | |
Parliamentary majority | 0% | 7% | 22% | 16% | 3% | 27% | 25% | 10% | ||
0.82 | ||||||||||
Vote margin over rival | 3% | 7% | 15% | 11% | 8% | 13% | 9% | 3% | ||
0.80 | ||||||||||
Parliamentary majority over rival | 7% | 11% | 29% | 23% | 10% | 39% | 37% | 25% | ||
In a competition between two parties for government, there is usually 1 winner, but there is a big difference between a margin of victory of 0.7%, the smallest since 1945 (February 1974, when the Conservatives won 37.9% to 37.1%), and 14.8%, the largest (1983, when the Conservatives’ won 42.4% to 27.6%). The margin of victory for the winning party is not relative to 50% of the electorate, but rather to whether it can sufficiently dominate its rival, in a contest where there are two clear options for government. If a party has clearly seen off its rival, and the two between them command the support of the lion’s share of the electorate, then can be seen as a majoritarian winner. In this context, the results 1945, 1959, 1966, and 1979-2001 look like fairly certain victories, as the winner commanded over 40% of the overall vote, and large margins over their rivals.
An appropriate function for a face-off election would incorporate these elements, producing majorities based on both absolute and relative performance, but on a consistent basis, giving greater clarity to the electorate. Elections do not consist only in the contest for government, but the contest to form a government can be between just two parties, when they achieve sufficient dominance relative to others. DV presents a clear method of producing a majority government, based on the two rivals’ contest, and on their importance relative to the minority parties. This method is consistent, unlike FPTP, and lets voters decide whether they wish to vote on the choice of majority between the two who might attain it at a given election, or to vote for a minority party. This choice is available to all members of the electorate.
Does size matter?
Italy’s electoral system in the lead up to Fascism51 guaranteed 60% of seats to the party achieving a national plurality of votes52, which simply and effectively guarantees majorities, yet this result removes the sensitivity of the size of majorities to election results. If majorities are not homogenous, and there is a link between the size of majorities and governments’ ability to push through their programmes, then electoral formulae that produce majorities sensitive to the degree of victory will be fairer. To test this governments’ majorities can be compared to their records of defeats in the Commons; where they imposed a whip on a vote, ordering their MPs to vote with them; yet lost the vote.
A government with a majority is able to control the Commons, because when its MPs vote with it, it commands a majority regardless of other votes, but it cannot always whip them successfully. The difficulty in winning votes comes from the difficulty of successfully whipping enough members to win a vote, which becomes harder with fewer MPs. With a majority of 1, 100% must be successfully whipped to ensure victory, whilst with a majority of 81, one can afford 40 rebels, needing 89% loyalty. Table 5.3 compares the minimum loyalty needed with the number of defeats in each Parliament 1945-present.
Table 5.3 | 1945 | 1950 | 1951 | 1955 | 1959 | 1964 | 1966 | 1970 | 1974 | 1974 | |
Gov seats | 61% | 50% | 51% | 55% | 58% | 50% | 58% | 52% | 47%53 | 50% | |
Min. loyalty | 81.4% | 99.2% | 97.5% | 91.2% | 86.4% | 99.4% | 86.8% | 95.4% | 105.5% | 99.6% | |
Gov defeats per year | 0.0 | 1.5 | 0.5 | 0.0 | 0.0 | 1.9 | 0.5 | 1.3 | 5.4 | 3.0 | |
Log defeats per year+154 | 0.0 | 0.7 | 0.3 | 0.0 | 0.0 | 0.8 | 0.3 | 0.8 | 1.3 | 1.6 | |
1979 | 1983 | 1987 | 1992 | 1997 | 2001 | 2005 | Correlation with seats | Correlation with min. loyalty | |||
Gov seats | 53% | 61% | 58% | 52% | 64% | 63% | 55% | ||||
Min. loyalty | 93.6% | 81.8% | 86.4% | 96.9% | 78.6% | 80.0% | 90.9% | ||||
Gov defeats per year | 0.5 | 0.7 | 0.5 | 1.3 | 0.0 | 0.0 | 2.1 | -0.71 | 0.75 | ||
Log defeats per year +1 | 1.0 | 1.4 | 1.0 | 3.0 | 0.0 | 0.0 | 2.0 | -0.78 | 0.81 | ||
There are very strong negative-correlations between the numbers of government seats and government defeats, demonstrating that the more seats the government holds, the fewer defeats it will suffer. There is an exponential relationship, as demonstrated by the greater negative-correlations with the log of defeats + 1. Again, this makes sense, as losing 18 times as opposed to 15 means little different in terms of a government’s power, whilst losing 3 times as opposed to none is much more significant. Defeats are even better explained by the loyalty needed to win votes, which is 0.81 correlated with the exponential measure of defeats.
So the size of the majority (more specifically the loyalty needed to win votes) clearly matters, and is exponentially related to the government’s ability to win votes. Election victories can be emphatic (1983) or marginal (February 1974), so if majorities are sensitive to the scale of victory, then this enables votes to determine the degree of power the government may be accorded in the Commons. Under DV, a government chosen by a large majority over its rivals will be less vulnerable to them in Parliament than one that only marginally beat them. The importance of the size of the majority means that its determination should be a clear and key part of the electoral system, not something that is conditionally related to votes.
Conclusions
In order to produce elected majority governments, winners of elections need to win a greater share of Parliamentary votes than they do of electoral votes, as alternating parties are very unlikely to win 50%+ of votes, due to insufficient dynamism in the electorate. The scale of majority does matter a lot for the power of a government and their ability to pass legislation, so it should be decided by the voters. It should reflect the magnitude of the winner’s victory over its rival for government, and its general popularity. DV produces majorities in line with this, according to the winner’s margin of votes over the second-placed party, and its overall vote, producing some proportionality in Parliamentary representation and voting power, whilst it has already been shown that these majorities allow proportionality in tenure.
6. A system of choice and fairness
Connecting people to their representatives
So, the effect of a vote must be clear, as Section 2 showed was not the case in FPTP and STV, because of indeterminacy and different determining factors. Votes must always be effective, regardless of when and where cast, which means that a voter must be able to use their vote to give Parliamentary power to their preferred party, whenever practicable55, as in DV.
To retain the constituency element in the British political system, district magnitude must be minimised, and constituency MPs must have a legitimate claim to have won their constituency. As ordinal preferences are indeterminate, and as categorical preferences may be more likely to select viable Condorcet coalitions: the plurality of categorical preferences in single-member districts, as used in FPTP and DV, is the best consistent criterion for this legitimacy. Categorical preferences also allow votes to be fairly compared nationally.
As constituencies can be won by very different amounts of votes, elected members should not have equal votes in Parliament. Making their Parliamentary vote directly dependent on their constituency vote strengthens the constituency link in DV56. Unlike in FPTP, where only plurality matters, every vote makes an equal difference to their Parliamentary vote regardless of their previous total.
Votes should be equal everywhere, regardless of other votes in their constituency, and whether or not they provide the margin of victory to a constituency candidate, so all votes should make a direct difference in Parliament, as in DV but not FPTP. In DV, whether a voter’s candidate wins or not, their vote feeds through with the same power, to be exercised in Parliament by their preferred party. A parliamentary vote based directly on votes cast also removes indeterminacy and makes proportionality transparent in the electoral formula.
Whilst regionally-concentrated parties such as the SNP will win constituencies; parties with broad national and even international agendas may have votes so spread out that even in large numbers they would not amount to any constituency pluralities, as we see from their lack of seats under FPTP. These parties are not treated equally unless, as in DV, some of their representatives are elected nationally. Having the number of these seats equal to their multiple of the average number of votes received by MPs who won constituencies, means that these national MPs would have similar individual votes to the average MP. Thus if the average constituency member had 30,000 constituency votes, plus 20,000 from their party, the Greens would receive a seat for every 50,000 votes57, filled in order of the votes their candidates received, with candidates again receiving their own vote plus an equal share of the votes of that party’s candidates not elected (amounting to votes close to 50,000).
Majority-government
The size of majority cannot be determined by the margin of votes over 50%, as parties rarely surpass 50%, however it should be a function of the votes parties receive, and how much in this total the winner dominates its rivals, as in DV. The majority is based on the winner’s votes, and the margin of victory over the nearest-placed rival, hence they have added to their total of votes (to be distributed to its MPs), a multiple of the number of votes by which they surpassed their nearest rival. The appropriate multiple depends on what majority seems right given a certain result, but a constant multiple is necessary to ensure transparency. Multiples are shown in Chart 6.1.
The contest for power is between the winning and second-placed parties, whose vote shares are shown by the red lines, never passing 50%. The FPTP results are then plotted above, in yellow, along with various functions, which add a multiple of the margin of victory to the winning party’s result, showing the DV result with that multiplier. Examining the victories in terms of votes, and the majorities (and minorities) DV would give for them with different multipliers, an appropriate multiplier would seem to be 3 or 4. A multiplier of 3 has been selected in this thesis, but this is not a definite choice. For the 1992 election, in which the Conservatives had a 42% vote share, compared to Labour’s 34.4%; on the margin×3 line, they have around 7,600,000 votes added to their total, amounting to around 23,000 per MP, bringing them to around 21,700,000 votes in Parliament, a share of 55.4% making them need 95% loyalty to ensure Commons victories.
With the government’s voting majority in DV based on the margin over the second-placed party, those who opposed the government in the pair-wise contest can reduce the margin and hence the government’s extra votes based on it, by voting for the opposition. Similarly, marginal votes for the winning party would have an increased effect, through their effect on the margin. Hence, voters would be incentivised to vote in the contest for government, which would combat the tendency to fragmentation of the party system, whilst voters would be free to vote for whom they chose, with votes for all parties except the winner providing equal benefit to that party’s Parliamentary vote.
Conclusion
In contrast to the other systems examined, DV treats voters equally, and provides results that are transparent, determinate, fair, and proportionate. It allows for majority government, but puts the provision of this in the voters’ hands, and allows them directly to influence the size of the majority. It discriminates in favour of the winning party, but against no other party. MPs are connected much more directly to the votes than in other systems, as they exercise the delegated votes of the electorate in Parliament. DV seems to be the most democratic electoral system, responding best to all of the different issues that have been considered under that criterion.
Further work
This thesis has concentrated on showing how existing systems are flawed in relation to various aspects of democracy, and how DV might respond better to them. DV has not been closely examined in isolation, and its effects, e.g. on voting patterns, and the party system, would be good grounds for future work.
1 All simulations are entirely original. Tests were conducted in Microsoft Excel 2000, generating preference rankings using its random number function, and Visual Basic.
2 Unless otherwise credited, all statistics are own analysis, using figures from Pippa Norris, Keele University, and Alba publishing.
3 E.g. Dummett (1984).
4 The number of districts may be 1, and the district magnitude equal to the size of the legislature, as in the Netherlands and Israel.
5 Farrell (1997), P.14-16
6 These systems are described in Farrell (1984) P.266-284
7 This multiplier is based on attempting to identify what constitutes a “substantial victory” in a General Election, and so is to some extent arbitrary. Different multipliers are illustrated in Section 6.
8 Taylor, (1984).
9 Own analysis, Keele, (2006)
10 Taylor, (1984).
11 Analysis of figures from Keele University (2005)
12 Based on data from Alba Publishing (2006).
13 E.g. Taylor (1984).
14 See Tables 2.1 and 2.2, and Chart 2.2.
15 Butler (1997), P.246.
16 This is fully explained in Kaase (1984).
17 German Culture (2006)
18 German constituencies average around 180,000 voters (German Culture (2006)), against around 70,000 in the UK.
19 See footnote 19.
20 For example, MPs undertake much advocacy work on behalf of constituencies, and are referred to in the Commons chamber according by constituency.
21 Arrow (1950)
22 For v voters selecting n members, a candidate needs (v/(n+1))+1 votes to be elected, which no more than n candidates could have, e.g. 50%+1 votes when selecting 1 member.
23 This is one reason why criteria for good local results differ from those for good national results.
24 AV is STV with district magnitude of 1.
25 Criteria for a candidate having received “more” ordinal preferences are indeterminate, as there is no fair way of adding up different ranked preferences, as discussed in Section 4.
26 The party proposed a wider referendum on European federalism. No other parties went this far, but Conservative policy changed 6 months after its launch, Butler (1997) P.6, and opinion polls suggested this won back votes from the Referendum Party, Comley (1997), P.9.
27 o!=o factorial, e.g. if o=5, o!=5×4×3×2×1.
28 There are o! ways to rank o options. For v voters, there are (n+v-1)×(n+v-2)×(n+v-3)×…(n+v-v)/v! distributions of n rankings, giving (o! +v-1)×(o! +v-2)×(o! +v-3)×…(o! +v-v)/v! distinct distributions of v voters’ rankings of o options. This can be calculated by ((o!+v-1)!/(o!-1)!/v!).
29 This represents over 60% of the possible distributions.
30 This was a test of 678,308 distributions, representing 0.004% of possible distributions.
31 A Condorcet coalition was defined as 2 options that independently beat all other options.
32 4 separate tests of 10,000 random distributions were conducted with 31 voters and 5 candidates.
33 Which comparisons are made, and with which votes, is contingent on the distribution of votes, so voters cannot know how their votes will be compared. Candidates with different totals of votes are elected as equal legislators, and candidates can gain votes and lose their seats, making winners’ legitimacy questionable.
34 Each pair-wise comparison does not involve every voter, because every voter has only registered a preference for one candidate, not in each comparison.
35 The winning party’s vote in the Commons would be slightly disproportional, due to the top-up of votes to that party, but other parties’ votes relative to each other would be perfectly proportional.
36 Farrell (1997), P.145-147.
37 The nearest proportion into which seats could split is proportionately further away from how votes are split.
38 Duverger (1972), P.23-32.
39 Own analysis, data reprinted in Farrell (1997)
40 According to Laakso and Taagepera’s index.
41 Many factors affect party systems, but these can be controlled for by looking at different electoral systems in the same country. To calculate correlation, one needs three points in each data series. The three countries that have had three different electoral systems during the period, Italy, Norway, and Sweden, exhibit correlations of 0.997, 0.789, and 0.863, respectively.
42 E.g. Farrell (1997) P.110
43 Hill (1997).
44 It cannot be said that a certain number of second-preferences is equal to a certain number of first-preferences.
45 This measures how long votes were in those proportions. If a party received 50% of votes at every election, its proportionate tenure would be 50% of the period, about 27.5 years.
46 Parties are taken to have won seats exactly proportional to their votes, delivering permanent coalition.
47 For parties that would never go into coalition, e.g. anti-system parties, the proportionality of tenure argument is not generally applicable.
48 They have been third-placed at every election since 1945.
49 Populus opinion poll December 5th-7th 2003, (http://www.populuslimited.com/pdf/2003_12_05_times.pdf), about 2 years from elections, which should include all swing voters, whose voting choices are made at elections.
50 This is problematic, since the 1983 result was for the SDP-Liberal Alliance, whereas the 1951 result was for the Liberal Party alone
51 Carstairs, (1980) P.156.
52 The system now guarantees 54% of seats in the lower house to the coalition winning a plurality. In the 2006 election, Romano Prodi’s coalition won this, but the coalition has “just enough seats” (BBC) for a majority in the upper house, which has different rules. The system may have led to Mr Prodi forming a coalition so broad that the BBC labels it “disparate”.
53 The February 1974 minority government could not attain a majority without support from other parties’ MPs.
54 Some parliaments have no government defeats, so 1 is added before taking a log value (there is no log of 0).
55 Some parties may be too small to justify Parliamentary representation.
56 The constituency link in DV could be weakened by making MPs’ votes in Parliament equal shares of their party’s overall vote, sharing out winning MPs’ votes, rather than having them retain them.
57 A quota might discourage fragmentation in such parties, similarly to the quota in the German AMS, which requires winning three constituencies, or 5% of national votes, (German Culture).
© Christopher Kavanagh, 2006
