A method for measuring and decomposing electoral bias for the three-party case

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A method for measuring and decomposing

electoral bias for the three-party case
Galina Borisyuk†, Ron Johnston*, Colin Rallings† and Michael Thrasher†

LGC Elections Centre, University of Plymouth, UK

* Department of Geographical Sciences, University of Bristol, UK

Measuring bias: moving from two-party to three-party elections
Simple plurality voting systems invariably lead to elections that have disproportional outcomes. This disproportionality usually favours the larger of the two main parties which – as identified from Duverger’s (1954) classic work onwards – tend to dominate such systems. What is not as well attested is whether that disproportionality is unbiased. A system rewarding the largest party with a ‘winner’s bonus’, (a ten percentage points greater share of the seats than of the votes, for example), is disproportional. However, if main party A obtains that bonus but main party B, with the same vote share, gets a bonus of only five points, then the system is not only disproportional but also biased towards A.
Grofman and King (2007), describe the term partisan symmetry (see also King et al, 2005), a requirement that:
‘… the electoral system treat similarly-situated parties equally, so that each receives the same fraction of legislative seats for a particular vote percentage as the other party would have received if it had the same percentage’ .
Measuring this symmetry has led to experimentation and debate (King, 1990; Grofman et al, 1997; Gelman and King, 1994; Gelman et al, 2004)– some of which has sought only to identify the extent of bias, without also decomposing it to uncover its sources.1
A method for measuring and decomposing bias was developed by Ralph Brookes (1953, 1959, 1960). From our point of view this approach has the major benefits of using a readily-appreciated metric and being decomposable into the various bias sources that he identified.2 It has been widely applied to the analysis of election results in Great Britain in the last two decades (e.g. Johnston et al, 2001, 2002, 2006).3
The problem is that Brookes’ method was formulated for the analysis of a system where two parties predominate. Its application to British elections since 1974 is thus constrained by the growth of three-party politics (Rallings and Thrasher, 2007). Although a third party victory component was subsequently added by Mortimore (1992: see Johnston et al. 1999) the method nevertheless remains focused on the two-party situation.
Our goal in this, as it was in a previous paper (Borisyuk et al. 2008), is to devise an extension for the three-party case whilst retaining much of the approach embedded in the Brookes method. The earlier attempt developed two methods for measuring and decomposing bias in a three-party system. The initial analysis of the 2005 British general election appeared promising and more or less in line with the findings using the Brookes’ method. However, when we examined previous elections dating from the early 1980s onwards some worrying discrepancies appeared both between the two methods and also with the findings from earlier applications of Brookes’ method. In a statement that is all too rare in the social sciences we published our findings and admitted it was time to go back to the drawing board!
This paper represents the result of that re-consideration. It begins with an outline and a demonstration of the principles that lay behind Brookes’ original procedure. We believe that certain elements in that procedure, particularly the idea of retaining as much of the shape of a party’s vote distribution as possible whilst establishing a norm of comparison, should lie at the heart of any extension of the method to the three-party case. At the conclusion of this section we re-cast the original formulation into a form that permits such an extension. In the next section we describe the distribution of votes and seats at the 2005 British general election and other recent elections there to reinforce the point that a different method for measuring bias is needed. The following section outlines the new procedure and uses the 2005 election by way of illustration. Having measured overall bias we then proceed to identify its components using the new method and again use the 2005 election result to present those findings. Finally, we summarise for each of the last six general elections the overall bias as calculated by this new method and compare this with that produced by the Brookes method..
The measurement of bias
We begin by stating a definition for electoral bias. Let ‘x’ be the number of seats the leading party wins with given share, α, of the two-party vote (i.e. that portion of the total number of votes cast won by the two leading parties), and ‘y’ the number of seats the second party could win if it got the same share of votes. Then, bias towards the first party is defined as the difference between the number of seats gained by this party, ‘x’, and the mean of seats gained by both parties, i.e. the mean of ‘x’ and ‘y’.
Hence the bias to party A is defined as

biasA(a) = x - (x + y)/2 = (x - y)/2 (1)

which is simply the negative of bias towards its rival, B:

biasB(a) = y - (x + y)/2 = (y - x)/2 (2)
In short, should we find a positive seat bias towards party A then there is, by definition, an equal though negative bias towards party B.
We can best illustrate this method by considering the two-party share of vote at the most recent general election held in 2005. Figure 1 shows the vote share for the largest party (in this case Labour) on the horizontal axis while the vertical axis is the vote share of the second party, the Conservatives. All points lie in a straight line because the graph represents the two-party share which must sum to one. Each point on the diagonal gives the vote share of Labour and Conservative parties in a given constituency. Projecting these values onto the horizontal axis gives the shape of Labour’s vote distribution of the two-party vote. Likewise, the values projected onto the vertical axis produce the shape of distribution for the Conservatives. Since we consider the two-party vote share these distributions are the mirror image of each other. Also indicated on both the diagonal and also horizontal and vertical axes is the overall vote share for each party. In this example, Labour has 52% and the Conservatives have 48%,4

Figure 1: 2005 British General Election: the two-party vote share

Brookes’ method begins by asking what would happen to the distribution of seats if the party that came second at the actual election had received the vote share of the first-place party. Using the principle of reverse vote shares the method applies a uniform swing to each constituency such that party B wins 52% of the overall two-party vote share. Figure 2a-d illustrates this process. Figure 2a shows the vote distribution for party A at the actual election (as shown earlier on the horizontal axis in Figure 1). Figure 2b is the vote distribution for party B at the actual election (shown on the vertical axis in Figure 1). Following the application of uniform swing to the vote share of party B (Figure 2c) the vote distribution slides to the right such that its overall vote share is now 52% (the result of applying the reverse vote shares procedure). Figure 2c represents the situation following reverse vote shares. Finally, Figure 2d shows the superposition of the distribution of party A at the actual election and the distribution of party B at the reverse vote share election (literally a combination of Figure 2a and Figure 2c).





Figure 2: Brookes’ two-party method: Calculation of bias

Because the example is considering the two-party case it follows that all those constituencies that lie to the right of 50% are won by the respective party. This gives us the number of seats won by party A at the actual election (i.e. ‘x’ in previous notations) and now the number of seats (‘y’) that would have been won by party B had it received the same share of the overall vote received by party A. It is now time to bring in the earlier definition of bias. In Brookes’ formulation the bias towards party A is measured as the actual number of seats it received, x, (the AB election – where A is the victorious party and B came second) minus the average of x and y, where y is the seats won by party B (following the application of uniform swing to create the reverse vote shares situation (the BA election – where B wins the notional election and A comes second). In effect, Brookes is comparing the distribution of seats at the actual election with a norm for comparison that is the superposition of the AB and BA elections (see Figure 2d).5
Having established some of the principles of the thinking that lay behind Brookes’ original formulation we now wish to prepare the ground for its extension to the three party case. In order to do this, however, we depart from the original formal notation and introduce our own. For example, seatAAB identifies seats won by party A at the election where A was the leading party with share of the two-party vote (seatAAB() = ‘x’ in previous notation).

Then, bias towards the leading party is defined as the difference between the number of seats actually gained by this party, seatAAB() and what on average a party could win given its overall vote share of and given the shares’ distribution of a particular shape (as it was at the actual election for parties A or B):

biasA()= seatAAB() –(seatAAB() + seatBBA()) /2 = (seatAAB() - seatBBA()) /2 (3)

biasB(1- ) = seatBAB(1-)–(seatB AB(1- )+ seatABA(1- ))/2 =

= (seatBAB(1- ) - seatABA(1- )) / 2

If we also assume that minor parties do not win any seats and where N equals the total number of seats then:

biasB(1- ) = ( (N - seatAAB() ) – (NseatBBA() ) ) / 2 =

= (seatBBA() - seatAAB())/2, (4)

which is simply the negative of bias towards its rival, A.
Moreover, we can specify total electoral bias for the two-party case as:

Three-party Britain
It should become immediately apparent when we begin to examine the detail of recent general election results in Britain that some radical overhaul of the Brookes method is required. The 2005 election saw Labour win 355 seats and a legislative majority with just 36% share of vote. The second placed party, the Conservatives, captured 33% of votes but won only 198 seats while the Liberal Democrats came a strong third with 23% and 62 seats – almost 10% of the total. In addition the two nationalist parties in Scotland and Wales won six and three seats respectively. Figure 3 demonstrates the extent to which recent elections have ceased to be strictly two-party affairs. The scale shows Labour’s share of the combined Conservative/Labour vote at the 2005 election. The points show that while Conservative and Labour captured the majority of seats, in a large number of constituencies the performance of other parties made it a more than two-party electoral race.

Figure 3: Three players at 2005 British General Election

Table 1 further demonstrates the extent of three-party Britain in the modern era. It shows, for example, that in 1983 the gap between the second and third-placed parties was just two percentage points although the difference in seats was huge – Labour won 209 to the SDP/Liberal Alliance’s 23 seats. Although the overall vote share of the third party declined from this point until a small rise at the 2001 election and a further rise in 2005 it achieved much more success in winning seats. The 62 seats won by the Liberal Democrats in 2005 is the largest for a third party since the early 1920s. And, as noted earlier, the gap between first and second-place party at this election coupled with the relatively low vote share of the winning party shows the extent of the move away from two-party electoral competition in the British case.
Table 1: British General Elections 1983-2005, three players


Overall (national) share of vote, %

The largest party


































Reformulating Brookes’ measure for the three-party case
Having re-cast Brookes’ original algebra we now extend it to the three-party case. In the two-party case it is clear that the establishment of a norm of comparison with the actual distribution of seats is central to the measurement of bias. This is also true for our extension of the measure. Stated simply, the bias towards a party is measured as the difference between the actual number of seats gained by that party and a norm which is the expected unbiased number of seats that, on average, three parties could win under equal conditions. For the three-party situation this is stated as:




In our earlier paper (Borisyuk et al. 2008) when constructing the expected norm we considered the actual election and just two notional elections. The first notional election saw the actual second-placed party awarded the same vote share as the actual first-placed party. The second notional election instead saw the original third-placed party given the vote share captured by the first-placed party at the actual election. In this way the actual number of seats won is compared with a norm that is the mean of seats gained by the leading party under three scenarios – the actual election and two notional elections.

As stated earlier, although initial findings proved encouraging subsequent analyses suggested some problems with the revised measure. A re-evaluation led us to believe that the establishment of the norm of comparison was incomplete and that the construction of notional elections should consider the whole set of possible outcomes. Although previously we had considered replacing the leading party we now believe that it is necessary to consider the situation when the second and third placed parties swap their national vote shares also; these additional notional elections should also be included in the set of ‘equal conditions’ scenarios that are required for constructing the norm for comparison. In going back to the drawing board we have not so much started afresh but have instead retained some of the principles that lay behind the original Brookes formulation and now extend them in a different manner than before.
Before stating this method formally we can best describe the underlying process in a series of Figures that show the distribution of three party votes. Because we use three-party vote share (the votes for other parties are discarded for this purpose) we know that α + β+ γ =1 and can show this in three-dimensional space as points in a plane. Figure 4 is the first such demonstration and shows the distribution of the three parties’ vote within individual constituencies at the 2005 general election. Points towards the peak of the triangle are constituencies where the largest party (in this case Labour) performed well. Points located towards the right and left hand base respectively are constituencies where the second (Conservative) and third-placed (Liberal Democrats) parties performed best. This method of displaying the data captures well the overall distribution of votes between the three parties.

Figure 4: Distribution of the three-party vote shares
Figure 5, which shows the same distribution as Figure 4, demonstrates that electorates tend to be smaller in areas dominated by Labour and are larger in Conservative areas. There is a clear correlation between each party’s vote share and electorate size and an even stronger correlation between each party’s share and the combined number of votes for Labour/ Conservative/ Liberal Democrats (correlation of -0.6 for Labour, 0.6 for Conservative, and much smaller but statistically significant 0.2 for the Liberal Democrats). Because the overall share of vote is a weighted average – weighted by total number of votes, the mean of the distribution (42%, 34%, 24%) is different from the overall three-party vote shares (39%, 36%, 23%).

Figure 5: Distribution of the three-party vote shares: Electorate Size
Figure 6 is the same as Figure 4 except we now display the point for the overall vote share as well as dividing the triangle into sections that show where each of the three parties won their seats. It is these that we will want to compare against once we have constructed the norm for comparison. In constructing this norm we want to find something that will satisfy a list of conditions. These are, the overall vote shares for the parties at the actual election, the distribution of electorate size, a zero-correlation between electorate size and a party’s vote share and finally, a symmetrical distribution of three-party vote shares. Regarding this last condition we believe that it is important that the distribution retains some characteristics of the original distribution at the actual election. This is ensured in the Brookes’ method in that the superposition of AB and BA distributions echoes the original A and B distributions (see Figure 2).

Figure 6: Distribution of the three-party vote shares: Seats Won
In constructing the norm for comparison we take three parties, A, B and C with overall vote shares, alpha, beta and gamma. The principle is to consider all possible combinations assigning alpha, beta and gamma to parties A, B and C. There are, of course, six possible combinations viz., ABC (actual election), ACB, BAC, BCA, CAB, and CBA. The superposition of these six configurations will be used as the ‘norm of distribution’ and all actual figures (including the number of seats won by each of the parties) will be measured against those extracted/calculated from this norm. It is important to note that the top of the triangle will always show the largest party, the right-hand side shows the second-placed party while the third-placed party is shown on the left-hand side.
The first such superposition is shown in Figure 7. It shows, in addition to the distribution at the actual election (ABC) the distribution ACB whereby the position of the leading party is retained but where the positions of the second and third-placed parties are reversed. In the context of the 2005 general election this means in effect that the overall vote shares of the Conservatives (33%) and Liberal Democrats (23%) are reversed. In practical terms this means that in each constituency we add ten percentage points to the Liberal Democrat share and subtract the same amount from the Conservative share. The distribution of ‘blue’ points in the plane is the consequence of adjusting vote shares for the second and third-place parties in this manner. It is important to note the vertex of the triangle refers not to a particular party but rather the place that party occupies in the finishing order. It follows that the Conservatives that occupied the right side of the triangle in the actual election (ABC) now occupy the left side of the triangle (ACB). The effect of so doing may mean that some constituencies now have a new ‘winner’.

The largest

The third

The second

Figure 7: Distribution of the three-party vote shares: ABC + ACB Scenarios
Figure 8 captures the result from another step in the process of establishing the norm for comparison. Alongside the distribution at the actual election we now see the consequences of constructing the scenario BAC. This has entailed a new leading party, the relegation of the original first-placed party into second place but the third party is retained. Given the overall vote shares in 2005 this has meant a three point drop for Labour and a corresponding rise for the Conservatives.
Figure 9 shows the superposition of all six combinations (ABC, ACB, BAC, BCA, CAB and CBA).

The largest

The second

The third

Figure 8: Distribution of the three-party vote shares: ABC + BAC Scenarios

Figure 9: Distribution of the three-party vote shares: ABC+ACB+BAC+BCA+CAB+CBA Scenarios
The next stage of the process requires that we compare the actual number of seats won by each party with the expected unbiased number of seats following construction of the ‘norm’ distribution. So, for example, the top section of the triangle shows the complete range of seats that would have been won by the largest party (in effect, twice for A, including the actual election; twice for B and twice for C).
Overall this distribution of points differs from a Gaussian/normal distribution and has a very distinctive shape. This means that we cannot calculate expected values by reference to some normal distribution. There are two possible approaches to this problem. One approach is to use Monte Carlo simulation such that points/values should be randomly drawn with a sample size equal to the number of seats at the actual election. The number of points located within each of the three patterned areas may be used as an estimate for the unbiased number of seats won. Samples of the same size could be drawn randomly from the distribution repeatedly and points/values calculated for each re-sample. Taking an average of these sample results would give an approximation for the unbiased number of seats for each party (the first, second and third parties). This approach has a possible additional advantage that we could calculate not only expected values but also errors and confidence intervals, but this has yet to be explored fully.
A second approach considers all points in the scatter plot and identifies the number in each patterned area. The estimate for unbiased number of seat now becomes 1/6th of the number of dots within each corresponding patterned area. We use this fraction because these dots represent the superposition of six scenarios and altogether there are six times as many dots as there are seats at the actual election. Technically, we get the same outcome by considering separately six scenarios, calculating the values for each of them, and averaging the results. The second approach is the one used here, partly for ease of use and partly because it facilitates the formal description of the process provided below.
We can now elaborate further on the equations provided earlier [equations 6-8]. When calculating bias towards party A we begin by taking the actual number of seats won and subtracting from that the average of the norm of seats for party with vote share alpha. In similar fashion the bias towards party B is calculated in the same way but this time referencing vote share beta. Finally, bias affecting party C is calculated by using vote share gamma. The formal description is shown as:




The total electoral bias in the two-party case may be negative or positive dependent upon the direction of bias towards or against the leading party. For three-party competition, however, there is no simple dichotomy of bias and theoretically it may be in one of six possible directions. Three of these directions depict the situation when just one party has a positive bias while the remaining two parties have a negative bias. Three other directions are when two parties show a positive bias of seats while there is a single unlucky party that has the negative bias. For this reasons we show total electoral bias as the absolute value of bias:

We can now show (see Table 2) the measurement of electoral bias at the 2005 general election using the new procedure. The first column indicates each of six scenarios beginning with the actual election while the second column denotes each party’s share of the three-party vote within that scenario. The third column is the number of seats won by each party. Obviously, the first scenario equates to the actual election and shows the largest party with 355 seats, the second-placed party with 198 and the third party with 62 seats. The second scenario is that also shown in Figure 7 and calculates that the same leading party would now get 374 seats, parties B and C, which have now swapped positions, would receive 112 and 129 seats respectively.
At the bottom of Table 2 the calculation of bias for each of three parties are shown. Recall that bias towards party A is its actual number of seats minus the norm of distribution for parties with vote share alpha (the average of the sum of seats for parties with vote share alpha in six scenarios). Substituting vote shares beta and gamma gives the bias towards parties B and C respectively. For the 2005 general election the procedure measures a positive bias towards Labour of 83 seats with a negative bias of 30 seats to the Conservatives and a negative bias of 52 seats to the Liberal Democrats (these figures do not sum to zero because of seats allocated to ‘others’). Total bias for this election is calculated as 165 seats. This analysis agrees with that of the modified Brookes method, that the 2005 general election was strongly biased towards Labour (Johnston et al. 2006) but we can now show that the compensating negative bias affected the Liberal Democrats rather more than it did the Conservatives.
Table 2: Measuring three-party bias: 2005 General Election

Relative three-party share of vote

Seats won

ABC (actual election)

Party A



Party B



Party C






Party A



Party B



Party C






Party A



Party B



Party C






Party A



Party B



Party C






Party A



Party B



Party C






Party A



Party B



Party C







Bias towards a party =

actual number of seats –

(MEAN across SIX scenarios seats for party with the same share of vote)

Party A:


= 355 - (355 + 374 + 249 + 304 + 172 + 178) / 6

Party B:


= 198 - (198 + 129 + 308 + 130 + 349 + 255) / 6

Party C:


= 62 - ( 62 + 112 + 57 + 178 + 93 + 180) / 6

Total bias = SUM of absolute values of three biases

= ABS(83.0) + ABS(-30.2) + ABS(-51.7)

= 165

Decomposing bias for the three-party case
One of the great strengths of the Brookes method is that it not only measures total bias but it also decomposes that bias into one of four categories. The first of these has been labelled differently (gerrymander, vote distribution, efficiency) but we prefer to use the term ‘geography’ (denoted by G in the equations following). It shows as an asymmetry in the distribution of partisan voting strength across constituencies (Gudgin and Taylor 1979). In a first past the post voting system a party performs well (in terms of the geography of its vote) by winning small and losing big. In other words it should avoid accumulating surplus votes (those additional to the number required to win the constituency) and if it cannot win then attract as few as votes as possible since these are literally ‘wasted’. The second component within electoral bias stems from malapportionment, i.e. differences in electorate size across constituencies. This is denoted by the term ‘E’. The level of abstention (‘A’) is the third component and becomes relevant when one party wins its seats but where electoral turnout is low compared with its rivals whose victories are achieved in constituencies with on average higher turnouts. Finally, there is the minor party effect, or component ‘M’, and here it is restricted to those parties outside of the main three.
We begin this decomposition by rearranging the definition of bias (in this case towards party A with vote share alpha) in the following form:
In this way the bias towards party A is partitioned into one of five terms6. The first term is bias resulting from an interaction between party B and party C where vote share for party A remains constant (in effect the scenarios ABC and ACB). The following two terms of bias derive from a non-symmetry between party A and party B – in one case where the position of party C is unaltered and in the other where it also is allowed to change. The final two terms express a non-symmetry between party A and party C. These effects may all move in the same direction or one may partly cancel out another as they move in opposite directions.
The style of notation used here replicates that used for the two-party method. The subscript relates to the party under consideration while superscripts describe the finishing order for the three parties. Hence:

seatAABC - number of seats won by party A at actual election,

seatAACB - number of seats won by party A under ACB scenario,

seatBBAC, seatBBCA - number of seats won by party B under BAC and BCA scenarios respectively,

seatCCAB, seatCCBA - number of seats won by party C under CAB and CBA scenarios respectively;

PAABC, PAACB, PBBAC, PBBCA, PCCAB, PCCBA - total number of combined votes for three major parties where corresponding party won seats under particular scenarios;

RAABC, RAACB, RBBAC, RBBCA, RCCAB, RCCBA - average electorate;

DAABC, DAACB, DBBAC, DBBCA, DCCAB, DCCBA - average number of abstentions;

UAABC, UAACB, UBBAC, UBBCA, UCCAB, UCCBA - average number of minor party votes.
We can now specify the formulae for the four components of bias, in this particular case towards party A.


Decomposition of bias towards party B and party C yields formulae similar to those above. For example the formula for the decomposition of the component relating to the electorate size effect as relevant to party C would read as:
In other words, we compare the actual position of the third party C with that of the third (in terms of overall vote share) party under each of five notional election scenarios.
The result of decomposing bias at the 2005 general election is shown in Table 3 and Figure 10. As stated earlier, the overall positive bias towards Labour is 83 seats but now we have more details about the source of that bias. Almost half of it, 40.6 rounded to 41 seats, derives from Labour’s vote distribution or its geography. Labour’s narrow wins in a number of constituencies coupled with its poor performance in seats held by its rivals largely explains this. Lower turnout in Labour held constituencies is a feature of the 2005 election and the decomposition shows that this is worth 16 seats in the decomposition. Labour’s advantage in terms of electorate size – its urban-based seats are on average smaller than most others – is just 16 seats. Turning to the negative bias for each of the other parties shows that for the Conservatives the decomposition shows that this bias stemmed almost equally from its victories coming in larger constituencies (in terms of electorate) and the relatively higher turnout (lower abstentions) in its constituencies. Meanwhile, the Liberal Democrats are greatly disadvantaged by their geography – third parties that contest constituencies everywhere, winning only one in five votes, are almost bound to suffer from a poor vote distribution unless much of their vote-winning is highly targeted on relatively few seats. It is also worth noting that the bias components regarding the votes for minor parties are generally rather small. This is to be expected given that the procedure is specifically designed for the three-party case and ‘others’ captured just 12 of the remaining seats at the 2005 election.
Table 3: 2005 British General Election: components of bias

Figure 10: 2005 British General Election: Components of Bias

A more detailed examination of Table 3 shows, for example, the advantage/disadvantage accruing from vote distribution to and from the different parties. Labour’s 41 seat advantage is almost entirely derived from its advantage vis a vis the Liberal Democrats; there is only a small advantage of three seats from the Conservatives. For its part the Conservative party gains from the Liberal Democrats (+28 seats) but loses (-24 seats) relative to Labour. Finally, the Liberal Democrats’ negative bias of 46 seats comprises 33 seats relative to Labour and 14 seats to the Conservatives. The impact of electorate size shows that the Conservative negative bias of 12 seats is almost entirely derived from the nine seat disadvantage relative to Labour. Both of Labour’s rivals are the source for its positive bias in respect of abstentions – the Conservatives have a negative bias of 12 seats and the Liberal Democrats 9 seats.
Comparing Brookes and the three-party procedure
We began the search for a new method for the decomposition of bias largely because the results of recent British general elections showed that the party system is now quite different to the one envisaged by Brookes when he developed his procedure (for what was then an even more strongly two-party system – New Zealand’s in the 1950s – than was the British prior to 1970). Although others subsequently modified his procedure to take more account of the growing impact of a third party there is always a sense that this is a half-way house and that a radical redesign would be preferable. We should, therefore, compare the two methods as they apply to general elections from 1983 onwards.
Table 4: British General Elections 1983-2005: Comparing two- and three-party methods

Figure 11: Three Party Bias at British General Elections, 1983 – 2005
Table 4 and Figure 11 show this comparison for elections between 1983 and 2005. For the purpose of comparison the two-party method refers to Brookes as modified by Mortimore and is estimated using the reverse-vote rather than equal votes procedure (i.e. bias is computed by comparing the number of seats won by party A with  share of the two-party vote with the number that B would win at the notional election when it obtained  share, rather than the difference between the two parties’ in the number of seats that would be won in a notional election where both received an equal share of the two-party vote, which is the position adopted by Johnston et al, 2001). There are some obvious differences relating to specific elections. For example, for the 1983 election, whereas total two-party bias estimates a bias of 11 seats the three-party method calculates overall bias at 176 seats. The small positive bias towards the Conservative (six seats) now becomes a negative bias of nine seats but the real difference lies in the large pro-Labour bias, 89 seats and the big disadvantage (negative 78 seats) for the SDP/Liberal Alliance. This should not come as a real surprise, however, given the narrow Labour lead over the Alliance but the large disparity in the seat distribution.
For the three-party method the least biased election of this set is the 1997 contest that saw Labour win what commentators widely termed an electoral landslide. The Brookes’ method shows total bias as 62 seats and gives a pro-Labour bias of 31 seats. By contrast, the three-party method calculates total bias at less than half that figure, shows only a modest pro-Labour bias and rather small negative biases towards Conservative and the Liberal Democrats. In our view this is confirmation of the three-party method. There is no doubting that the election result was disproportional – Labour won more than two-thirds of the seats (418 of 641) with just 44% of the vote – but the decomposition suggests that it was not particularly biased. The Conservative party won just 31% of the votes while the Liberal Democrats demonstrated the success of their targeting of seats tactic – their overall vote share fell (from 18% of the three-party vote in 1992 to 17% in 1997) but the party more than doubled its share of seats (20 of 634 seats, i.e. 3.2%, in 1992 and 46 of 641, i.e. 7.2%, in 1997).
This paper set out to revise a method for the measurement and decomposition of electoral bias – an inequality in seats when parties receive equal vote shares. That method, developed by Brookes in the 1950s, was based on the assumption that two main parties would contest a first past the post election and that minor parties would be just that – minor and of little electoral significance. The changing dynamic of general elections in the UK from the early 1970s onwards, however, prompted a revision of the Brookes method to accommodate the increasing role of the third party. The problem remained that bias continued to be expressed in terms of the two main parties – a bias in this case towards or against Labour or Conservative.
The search for a method that is more suited to the three-party case has been rather long and has proceeded incrementally. The first iteration appeared promising at first but subsequent application to a range of general elections suggested problems that needed to be resolved. We are more confident that the second iteration, presented in this paper, will prove helpful in measuring and decomposing electoral bias for the three-party case.

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1 Grofman and King (2007, 32) do claim, however, that ‘The degree of deviation from symmetry of treatment is known as partisan bias, and is easily quantified, and made specific as to direction’.

2 An alternative approach, developed almost contemporaneously with Brookes’, identifying the same basic bias components, is Soper and Rydon (1958), who developed early ideas of Brookes (1953).

3 The only other attempts to measure and account for bias in Great Britain have been those by Curtice (2001; see also Curtice and Steed, 1986), which although it identified the various sources of bias did not quantify their relative importance, and Blau’s (2001) important critique of the Brookes’ method.

4 These values differ from the mean values of the distribution (54% and 46% respectively) because of the unequal size of constituencies.

5 For superposition AB and BA, vote shares now has zero correlation with size of constituency and has symmetrical shape of distribution (a norm distribution). Because of zero correlation with constituency size, overall vote share equals the mean of the distribution.

6 We might expect to see six rather than five terms here (because six scenarios are used in the calculation of the norm of distribution) but if we compare the actual result with itself then we will get zero.

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