Dr David Chapman, who has carried our research on the electoral system at the Universities of Lancaster and Virginia, is currently Director of the Democracy Design Forum, a consultancy on electoral systems.
The advantage claimed for PAV is one of equity, that as compared with other systems, it gives candidates and parties a stronger incentive to be equally responsive to the different sections of the electorate. Also, PAV appears to be a highly practicable method of election. It is not complicated to count, having about the same level of complication as the Alternative Vote, and it could easily be counted by hand, not needing to be counted by computer, however large is the number of candidates.
PAV can best be explained by means of its relation to Approval Voting. The procedure of Approval Voting is simply this: the electors vote (non-preferentially) for as many candidates as they like, for one or for more than one, and the candidate who gets most votes is elected. PAV simulates this procedure by use of preferential voting (that is, voting where the elector votes by marking the candidates in order of preference, 1 for a first preference, 2 for a second preference, and so on, for as many candidates as he wishes).
Now under Approval Voting, the voter will always vote for the candidate whom he most prefers. But under what circumstances will he vote further down his preference ordering, voting in addition for his next-preferred candidate, or for several of the next-preferred candidates? It seems likely that he will do so if he expects that a candidate whom he very much less prefers has some chance of being elected, and if he thinks that voting for the next-preferred candidate or candidates will reduce this chance. For example, a voter whose first preference is Labour, second is Liberal Democrat, and third is Conservative, will always vote for the Labour candidate, and might vote for the Liberal Democrat in addition, if he thinks that the Conservative has a significant chance of winning.
PAV approximately simulates this voting behaviour, by use of the preference orderings provided by the voters. Thus PAV always counts the voter as voting for his first-preferred candidate. PAV counts him as voting for his next-preferred candidate when the latter is preferred to the leading candidate, that one who so far in the counting has obtained most votes. In other words, this leading candidate is treated as one who has a significant chance of being elected, and therefore voters are assumed to vote for the candidates they prefer to him.
The first stage. In respect of each ballot paper, a point is given to the candidate marked as first preference on that paper. The points of each candidate are counted, and the leading candidate is found (that is, the candidate who has most points). If there is a tie between two or more candidates, one of them is selected by lot to be the leading candidate.
Any further stage. Those ballot papers are considered, in respect of which a point has not so far been given to the leading candidate of the previous stage. In respect of each such ballot paper, a point is given to the candidate next-preferred to the last candidate to receive a point, provided this next-preferred candidate is preferred to the leading candidate of the previous stage. The leading candidate (who will possibly be a new one) is then found, that is, the candidate who has obtained most points up to and including the current stage.
These further stages are repeated, each one giving more points to the candidates, until the final stage is reached, at which none of the electors' next preferred candidates is preferred to the leading candidate, so that no candidate is entitled to receive any further point. At this final stage, the candidate who has most points is elected.
It will be seen that the method of counting the votes for PAV, is somewhat similar to that for the Alternative Vote. Under both PAV and AV, the first stage is to count the first preferences on all ballot papers. In each later stage, the next preferences are counted on a limited number of the ballot papers, until the winning candidate is found.
A preferential system which bears some resemblance to PAV is that of Descending Acquiescing Coalitions (DAC). DAC is a new preferential election method for filling a single seat, which was recently proposed by Woodall[2,3], as an improvement on the Alternative Vote (which is discussed more fully below). DAC resembles PAV in that both can be regarded as a preferential simulation of Approval Voting. However, Woodall[2] admits DAC is `much more complicated than [the Alternative Vote]', and would be likely to require a computer to carry out the counting. Thus it is clear that PAV will be much simpler than DAC (see below).
(The notation used to describe the election is explained as follows. The first lines show the voters' preference listings of the candidates. Thus in the top line, 35 voters rank L first, C second, and R third. The subscripts against some of the candidates in a preference listing, show in what stage points are given to the candidate. Thus in the third line, 16 points are given to C in the first stage, and 16 points are given to R in the second stage. After the preference listings, each column shows the total points which have been obtained by each candidate by the specified stage. Thus by stage 2, L has obtained 35 points, C 65, and R 49. The greatest total of points, that of C, is shown in underlined, C being the leading candidate at stage 2.)
Election 1
35 L1C R 16 C1L R 16 C1R2L 33 R1C2LStage 1 Stage 2 Stage 3
L 35 35 35 C 32 65 65 R 33 49 49In stage 1, each candidate gets one point for each first preference. L is the leading candidate, getting most points. In stage 2, candidate C (who is the next preference of the 33 first-preference supporters of R, and who is preferred by them to L, the leading candidate of the previous stage) therefore gets 33 more points. Similarly, R gets 16 more points, by being preferred to L by 16 first-preference supporters of C. C, now having most points, becomes the new leading candidate. In stage 3, none of the next-preferred candidates is preferred to C, the leading candidate of the previous stage, and so no candidate gets any more points. Thus C, having most points in the final stage, is elected.
We can use the results of Election 1 to illustrate how PAV deals with incomplete preference listings, that is, ballot papers which do not express a preference for all the candidates. It makes no difference whether or not a last preference is expressed by the voter. For example, if the 33 voters voting RCL voted RC instead, this would not alter the result, since we would still know, for stage 2, that they preferred C to L, the leading candidate, so that C would still get 33 extra points. However, it does make a difference if a non-last preference is not expressed. For example, if the 33 voters voted just R, that is, first preference for R, with no preference given for any other candidate, then C would get no extra points in stage 2, since no preference for C over L would have been expressed.
But let us return to the original results of Election 1 as shown above. In this situation of single-peaked preferences, PAV has elected the centre candidate in the left-to-right dimension. This candidate elected by PAV is also the so-called Condorcet winner, that is, the candidate who beats each other candidate, always being preferred to the other candidate by a majority of voters. (C is preferred over L by 65 voters to 35 and over R by 67 to 33.) Note that PAV achieves this result (that is, of electing the centre candidate or Condorcet winner) despite the fact that C has fewest first preferences, which would prevent C from being elected under the Alternative Vote, that form of preferential system which is most commonly used for electing to one seat.
However, if PAV is actually in use for a series of elections, then it is unlikely that the electors' preferences between the candidates will remain single-peaked. For candidates L and R will surely come to realise that under PAV, their respective extremist positions are going to lose them election after election, and so they will adjust their appeals to give themselves a better chance of winning. Thus L will appeal to the supporters of R, to persuade more of them to change their preference listing to RLC instead of RCL, and R will appeal to supporters of L to get them to change to LRC. The pattern of the electors' preferences will then no longer be single-peaked, but will tend towards what might be called a symmetrical pattern, where there is about the same number of voters with each possible preference listing (that is, in this case, one-sixth LCR, one-sixth LRC, and so on). Thus a typical election might be something like Election 2.
Election 2
18 L1C4R 17 L1R3C 17 C1L4R 15 C1R2L 17 R1C2L 16 R1L3CStage 1 Stage 2 Stage 3 Stage 4 Stage 5
L 35 35 51 68 68 C 32 49 49 67 67 R 33 48 65 65 65Thus by broadening their appeal, L and R have got more points, and L has succeeded in getting elected. L now gets second preferences, not only from first-preference supporters of C as before, but also from the first-preference supporters of R, and similarly R now gets second preferences from the first-preference supporters of L. This illustrates how PAV gives a candidate or party the incentive to appeal to, and to be responsive to, all sections of electors.
Election 2 can be used to illustrate the general strategy by which a candidate will seek to win under PAV. A candidate wins by getting a point from the most voters. A candidate C gets a point from any one voter V either if C gets V's first preference, or otherwise if C is preferred by V to that one of the leading candidates who is least preferred by V. Thus in Election 2, L gets a point not only from the 18 LCRs and 17 LRCs, but also from the 17 CLRs and the 16 RLCs. This has implications for a candidate's general strategy. He will be primarily concerned to persuade voters to prefer him over their least preferred leading candidate. Once they do this, he will not seek to persuade them to give him a still higher preference (that is, a first preference in Election 2), since this will tend to be difficult to achieve, and in any case it will not bring him any more points. Thus when there are three leading candidates, as in Election 2, each one will direct his appeal primarily at those electors who have tended to give him last preference, and in general, each candidate will be seeking to get second preferences rather than first preferences.
However, PAV is unlike the Alternative Vote in that the candidate with fewest first preferences can be elected, as was the case in the single-peaked example of Election 1 above. Indeed, PAV can enable a candidate to get elected who has very few or even no first preferences. A non-single-peaked example of this, which might well occur occasionally in practice, is Election 3. Here a candidate C, who has few first preferences, gets more points than either A or B (each of whom have close to half the first preferences) by persuading many of As and Bs first-preference supporters to give C their second preferences.
Election 3
32 A1C3B 16 A1B4C 15 B1A4C 32 B1C2A 2 C1B2A 3 C1A3BStage 1 Stage 2 Stage 3 Stage 4 Stage 5
A 48 48 51 66 66 B 47 49 49 65 65 C 5 37 69 69 69This lack of the need for first preferences under PAV, can be expected to reduce the entry barrier against new candidates. For it is likely to be easier to gain second preferences than first preferences, thus making it easier under PAV for a new candidate to compete successfully with already established candidates, than it would be under the Alternative Vote, or in particular under Plurality. Thus under PAV, at least when it has been in use for some time, it is likely that few candidates will obtain a majority of first preferences, and that the most usual situation in each constituency will be for there to be three strong candidates (or perhaps sometimes more than three) in not very unequal competition. In other words, it is likely that under PAV, there will be a tendency towards a symmetrical situation like that shown in Election 2.
In all the examples given above, Elections 1, 2 and 3, there were only three candidates competing. How then will PAV operate, if there is a larger number of candidates? The same procedure will be followed, that of sorting and counting the next preferences stage by stage, until that stage is reached, where no next-preferred candidate is preferred to the leading candidate, and thus no candidate is entitled to receive any further points. Because there are more candidates, there will of course be more next preferences to sort and to count. But the extra counting need not be in proportion to the number of extra candidates. The reason for this is that on any one ballot paper, only the top preferences need to be counted, down to the preference for the candidate who is one preference step above that one of the `leading candidates' whom the voter least prefers. It is likely that the extra candidates will be given a very low preference (or no preference) by most of the voters, and that because of this their preferences for them will not need to be counted.
Election 4 is given below, as an example of a four-candidate election. Election 4 is assumed to be a re-run of Election 2, in which one party, the party which previously ran L as its candidate, now runs two candidates L and M, one a woman and one a man, in order to give the electors a wider choice. Electors are assumed to put L and M in the same position in their preference listings as they put L in Election 2.
Election 4
10 L1 M2 C R 8 M1 L2 C R 9 L1 M2 R C 8 M1 L2 R C 9 C1 L2 M R 8 C1 M2 L R 8 C1 R3 L4 M 7 C1 R3 M5 L 9 R1 C3 L4 M 8 R1 C3 M5 L 9 R1 L4 M C 7 R1 M3 L CStage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6
L 19 44 44 70 70 70 M 16 43 50 50 65 65 C 32 32 49 49 49 49 R 33 33 48 48 48 48
But how far does PAV tend to elect the Condorcet winner (CW)? The CW was elected in Election 1, where preferences were single-peaked, and also in the more likely preference situation of Election 2 (L, the PAV winner, being preferred over C by 51 voters to 49 and over R by 52 to 48). However, in Election 3, where C, the PAV winner, got most of his votes from second preferences, the CW was not elected, the CW being candidate A (who was preferred over B by 51 voters to 49, and over C by 63 to 37). It thus appears that in practice, in the preference situations most likely to occur, PAV has a very high probability of electing the CW, but that it might not elect the CW in some unusual situations, where the PAV winner obtains an especially high proportion of his points from lower preferences.
A voter is said to acquiesce to a set of candidates if there is no candidate outside the set whom he prefers to any candidate in the set. (In other words, in respect of any pair of candidates, one in the set and one outside the set, he always either prefers the candidate in the set, or expresses no preference between them.) The set of all those voters who acquiesce to the candidates A and B is referred to as the coalition acquiescing to A and to B, or as {A, B}. For example, if there are only three candidates A, B and C, then {A, B} will be all those voters voting as follows: ABC, AB, BAC, BA, A or B.
That candidate is elected who obtains the acquiescence of a greater number of voters than any other candidate. This is determined as follows. A candidate A is said to beat a candidate B if the greatest coalition acquiescing to A and not acquiescing to B, is greater than the greatest coalition acquiescing to B and not acquiescing to A. That candidate is elected who beats each other candidate. This can be illustrated by the following two examples, taken from Woodall[2].
Election 5 (Election 3 of Woodall)
11 AB 7 B 12 CThis produces acquiescing coalitions as follows, in descending order of size.
{A, B, C} 30
{B, C} 19
{A, B} 18
{A, C} 12
{C} 12
{A} 11
{B} 7
B beats A, because {B, C} > {A, C}. B beats C, because {A, B} > {A, C}. Thus B is elected.
Election 6 (Election 4 of Woodall)
5 ADCB 5 BCAD 8 CADB 4 DABC 8 DBCAThis produces a set of the greatest acquiescing coalitions as follows.
{A, B, C, D} 30
{A, B, C} 13
{D} 12
{A, D} 9
{A, C} 8
{B, C, D} 8
{B, D} 8
{C} 8
A beats B, because{A, D} > {B, C, D}. A beats C, because {A, D} > {B,
C, D}. A beats D, because{A, B, C} > {D}. Thus A is elected.
Let us now compare DAC with PAV. Under DAC, A beats B if more voters are in the greatest coalition acquiescing to A and not acquiescing to B, than are in the greatest coalition acquiescing to B and not acquiescing to A. Under PAV, A beats B if there are more voters who give A a first preference, or otherwise prefer A to a `leading candidate', than those who give B a first preference, or prefer B to a leading candidate.
DAC is like PAV, and unlike the Alternative Vote, in that it does not require a candidate to get first-preference votes in order to get elected, and so it can elect the candidate with fewest first preferences (as it does in Election 5). The two systems DAC and PAV are similar to each other, and to Approval Voting, in that each of them can give value to one or more of the highest non-first preferences of an elector, and in that if it does, the value of a non-first preference is the same as that of a first. DAC can thus be regarded as a preferential simulation of Approval Voting, as can PAV.
10 A1 B2 9 B1 2 C1 B 9 C1 8 D1 A2Stage 1 Stage 2 Stage 3
A 10 18 18 B 9 19 19 C 11 11 11 D 8 8 8Thus B is elected.
Now suppose that in Election 8, the two voters who voted CB in Election 7, change to voting BC instead. The stages of the count will then be as shown below, and A will be elected. Thus by moving up the preference listing of these two voters, B will have lost the election.
Election 8
10 A1 B 9 B1 2 B1 C3 9 C1 8 D1 A2Stage 1 Stage 2 Stage 3 Stage 4
A 10 18 18 18 B 11 11 11 11 C 9 9 11 11 D 8 8 8 8
Alternatively, suppose that in Election 9, the profile is as in Election 7, except that three new voters enter the election, and vote first preference for B, so that the second line in the election profile is 12 B instead of 9 B.
Election 9
10 A1 B 12 B1 2 C1 B3 9 C1 8 D1 A2Stage 1 Stage 2 Stage 3 Stage 4
A 10 18 18 18 B 12 12 14 14 C 11 11 11 11 D 8 8 8 8
Thus A is elected. Again, B has lost the election, this time by getting more voters to vote for him.
It should be pointed out that the Alternative Vote is also non-monotonic, whether more or less so than PAV I am unable to determine. DAC, on the other hand, was designed to satisfy as many monotonicity properties as possible, and in fact satisfies eight out of ten of them.
How far, then, would this lack of monotonicity in PAV be a problem not just in theory, but in actual practice in real elections? The main objective of PAV is to give each candidate the incentive to be responsive to each section of electors. Thus the important question is, how far will lack of monotonicity interfere with this incentive? Will a candidate (such as B in Elections 7 to 9 above) ever have the incentive to displease the electors, so that they give him a lower preference, or so that fewer of them vote for him?
This seems unlikely, for two reasons. First, a non-monotonic profile of votes such as those of Elections 7 to 9 seems itself unlikely when candidates are competing strongly, not only for first preferences, but for second and third preferences as well. Then the profile tends towards a more symmetrical pattern such as that shown in Election 2 above, which would be monotonic. Second, in order for the candidate to be provided with this negative incentive, he must be able to predict that the overall profile of votes at the next election will be such as to produce this non-monotonicity, and furthermore that his own votes will be in that presumably narrow range where he will benefit from losing votes. In the absence of this prescience, the candidate will have the incentive to respond positively to the electors, in the expectation that nearly always it will be beneficial for him to get more votes rather than fewer of them. Thus it seems unlikely that this lack of monotonicity will affect the candidates' incentives, or will be of practical importance.
The other systems similar to PAV are liable to strategy in a similar way. Thus under normal (non-preferential) Approval Voting, a similar strategy is very likely to be used-ABCs voting only for A and BACs voting only for B, when the election is expected to be a two-horse race between A and B. Under DAC, a preferential system with some similarity to PAV, this same strategy of the truncated preference listing is likely to be used (according to Woodall[4]).
How far, then, is it a problem, that there is this opportunity for strategy under PAV? The strategy will be used in constituencies where two of the candidates are clearly stronger than the others, and it is expected that the winner will be one or other of them. But in constituencies where there are three or more strong candidates, and it is unclear which of them is going to get most points, the electors will tend not to vote strategically, but to express fully their preferences between these candidates.
However, there are reasons to expect that any constituency will tend to move from the former situation towards the latter, that is, from one with two strong candidates to one with three or more. Firstly, as it was shown above, PAV does not require a candidate to get many first preferences in order to win, and so it presents relatively little entry barrier to an effective new candidate. Secondly, when there are two strong candidates, let us say A and B, and a weaker candidate C, the strategic voting which this situation encourages actually benefits C. For some ABCs will vote only first preference for A, which will reduce Bs votes, and some BACs will vote only first preference for B, thus reducing As votes. This reduces the number of first or second preferences which C needs to get, to approach about the same number of votes as A or B, making it easier for C to become a third strong candidate. It will then be uncertain which of the three candidates is going to get most votes, and strategic voting will become unlikely.
Thus in conclusion, it seems that the tendency in any constituency is towards a situation where there are three (or perhaps more than three) strong candidates, each with some chance of winning. To the extent that this situation occurs, the truncation strategy will tend not to be used, and voters will express fully their preferences for the candidates.
To answer this question, let us consider the examples of Elections 1 and 2 above. In Election 1, candidates L and R fail to respond to all sections of electors, L not responding to the right-wing electors, and so getting a last preference from them, and R not responding to the left-wing electors. Consequently, they lose points, and neither of them has any prospect of getting elected.
However, in Election 2, each of them has broadened his appeal to include the whole electorate, L responding to right-wing electors, and R to left-wingers. L now gets second preferences, not only from centre electors as before, but also from right-wing electors, and similarly R gets second preferences from left-wingers. Thus by broadening their appeal, L and R get more points, and L succeeds in getting elected. This illustrates how PAV gives each candidate the incentive to respond to each section of electors.
Note that in Election 2, the situation between all three candidates is symmetrical in the sense that any two candidates compete with each other for the second preferences of the third candidate's first-preference supporters. Thus L and R compete for the second preferences of centre electors (just as they did in Election 1). But now L competes with C for right-wingers' second preferences, and similarly R competes with C for left-wingers' second preferences. Any one candidate thus needs to be responsive to the first-preference supporters of any other candidate, in order to compete with the third candidate for their second preferences. For example, L needs to be responsive to centre electors to compete with R, and to right-wing electors to compete with C. Thus PAV gives each candidate the incentive to be responsive to each section of the electorate.
Another way of understanding the incentives provided by PAV is as follows. In the likely situation where there are three candidates competing, and each becomes a leading candidate at some stage in the counting, a candidate receives one point for each first preference and one point for each second preference. Thus (assuming all voters express their second preferences), a candidate needs to get either a first-preference or a second-preference vote from at least two-thirds of the voters in order to get elected. He is not likely to achieve this, in competition with two other candidates also trying to do the same thing, unless he appeals to each section of electors. Thus the candidate has the incentive to respond to each section of the electorate.
Furthermore, a first preference is worth no more than a second preference--both are worth only one point. Thus there will be no need for a candidate to appeal to a given section of electors any more strongly than is necessary to get second preferences from it, and no reason to give the section any specially favourable treatment, in order to obtain from it a higher proportion of first preferences. This is clearly a factor making for the candidates' more equal responsiveness to each section.
It is interesting to compare the situation under PAV as described above, with that under the Alternative Vote. Here, in order to get elected, a candidate needs to obtain the support not of two-thirds of the voters, but of only one-half. Thus he is likely to appeal less widely. Further, each candidate must strive for first preferences, since the candidate with fewest first preferences will be excluded. This seems likely to create an incentive for a candidate to favour some sections of electors over others, in order to get first preferences from them.
To illustrate this, let us consider an example with three candidates, A, B and C, where it is expected that C will be excluded, and that it will be a close finish between A and B. Each of A and B will have his core supporters, to whom he is strongly responsive, in order to obtain first preferences from them. Also, each of A and B will be strongly responsive to those voters giving first preference to C, in order to compete with the other candidate for these voters' second preferences. But A will tend to be unresponsive to the core supporters of B, because of the difficulty of persuading them to switch from first preference for B to first preference for A. Similarly, B will tend to be unresponsive to the core supporters of A. Thus the Alternative Vote, by forcing candidates to strive for first preferences, makes for their unequal responsiveness to the different sections of electors. In comparison, PAV, which makes no requirement for first preferences, will give candidates the incentive to respond more equally to each different section of electors.
What then would be the effect on the parties' shares of seats? The present Plurality system, which essentially gives a seat to the candidate with most first preferences, discriminates strongly against the Liberal Democrats, who have third most first preferences. However, under PAV, they would be likely to get many more seats than now, since there seems no reason why they should not get about as many second preferences as either of the other two major parties. Thus it seems likely that the three major parties would be more equal in their seats than they are now, and that no one party would get a majority; so that a coalition government would need to be formed, by some two of them.
The point of most interest, and the main advantage claimed for the new system, is that it would give parties the incentive to change their policies to be more inclusive, more equitably responsive to the different sections of the electorate. For example, the Conservative Party currently tends to be unresponsive to strong Labour supporters, since under the present Plurality system few of them could be persuaded to switch to voting for the Conservatives. But under PAV, the Conservative Party would become more responsive to them, in order to compete with the Liberal Democrats for their second preferences. Similarly, the Conservatives would become more responsive to strong Liberal Democrat supporters, in order to compete for their second preferences with Labour. Thus the three major parties would tend to converge in policy, towards a policy more equally responsive to each section of electors; and as a result of this convergence, a coalition government formed by any two of them would be likely to be stable, and acceptable to all sections of the electorate.
For similar reasons, it might be desirable to use PAV for purposes such as the following: the election of a president, or of a chairman, by the members of a legislature; the election of the party leader by the party membership, or by the party's MPs.
PAV could also be used for a multi-option referendum, to enable the electorate to choose one option out of three or more. This can be justified as follows.
In the usual type of referendum, electors choose between two options, these options being some proposed action, let us say A, and the status quo S. Proposers will be concerned to find an A which will get a majority over S, and in doing so they may come up with an A which is very harmful to the minority, while perhaps only marginally beneficial to many people in the majority. Thus the two-option referendum might lead to very unequal treatment of different sections of the electorate, and to division and conflict.
However, if a PAV-using multi-option referendum is introduced, a compromise option C is likely to be proposed, one which is better than A for S preferrers and some A preferrers, and better than S for other A preferrers. Thus there will be three options on the ballot paper, A, S and C, for the electors to place in order of preference. Since C will have many second preferences, it is likely that C will be adopted. This illustrates how a PAV-using multi-option referendum tends to improve the outcome, reducing the risk that any section of the electors will be severely harmed.