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# Voting matters - Issue 15, June 2002

## On Dummett's 'Quota Borda System'

#### M Schulze

Markus Schulze is a physicist and mathematician from Germany

In two books[1][2], in his submission to the Jenkins Commission[3], and at a number of conferences, Michael Dummett has promoted a preferential voting method where one successively searches for solid coalitions of increasing numbers of candidates and where, when one has found such a solid coalition, one declares the candidates with the best Borda scores elected. Dummett calls his method 'Quota Preference Score' (QPS) or 'Quota Borda System' (QBS). He writes that his method 'has never been in use, but was voted the best at a conference on electoral reform held in Belfast with representatives of all parties' [3]. In his book Voting Procedures, he describes this method as follows (where v is the number of voters, S is the number of seats, C is the number of candidates, and the 'preference score' is the Borda score) [1, pp. 284-286]:

The assessment will proceed by stages, all but the last of which may be called 'qualifying stages': it will of course terminate as soon as all S seats have been filled. We may first describe the assessment process for the case when S is 2 or 3. At stage 1, the tellers will determine whether there are any candidates listed first by more than 1/(S+1) of the total number v of voters: if so, they immediately qualify for election. If seats remain to be filled, the preference scores of all candidates not qualifying at stage 1 will then be calculated. At stage 2, the ballot papers will be scrutinized to see if there is any pair of candidates, neither of whom qualified at stage 1, to whom more than v/(S+1) voters are solidly committed: if so, that member of the pair with the higher preference score now qualifies for election. If seats remain to be filled, the tellers will proceed to stage 3, at which they will consider sets of three candidates, none of whom has already qualified. If more than v/(S+1) voters are solidly committed to any such trio, that one with the highest preference score qualifies for election. In general, at the qualifying stage i, the tellers determine whether, for any set of i candidates none of whom has so far qualified, there are more than v/(S+1) voters solidly committed to those candidates; if so, the member of the set with the highest preference score qualifies for election at stage i. If there still remain seats to be filled after all the qualifying stages have been completed, they will be filled at the final stage by those candidates having the highest preference scores out of those who have not yet qualified. ( . . . )

When S = 4, however, it may be thought that a body of voters, amounting to more than two-fifths of the electorate and solidly committed to two or more candidates, is entitled to 2 of the 4 seats. To achieve this, the assessment process must be made a little more complex. Stage 1 will proceed as before, and, at stage 2, the same operation must be carried out as described above. Before proceeding to stage 3, however, the tellers must also consider every pair of candidates of whom one qualified at stage 1 and the other did not: if more than 2·v/(S+1) voters are solidly committed to such a pair, that one who did not qualify at stage 1 qualifies at stage 2. (Note that, if more than 2·v/(S+1) voters are solidly committed to two candidates, one of them must qualify at stage 1.) Likewise, at each qualifying stage i, the tellers must ask, of every set of i candidates of whom at most one has already qualified, whether more than 2·v/(S+1) voters are solidly committed to those candidates. If so, and none of them has previously qualified, the two with the highest preference scores will now qualify; if one of them qualified at an earlier stage, that one, of the rest, who has the highest preference score will qualify at stage i. ( . . . )

In general, at stage i, the tellers must ask, of each set of voters solidly committed to i candidates, what multiple of v/(S+1) members it contains, up to i·v/(S+1). If it contains more than v/(S+1) voters, at least one of the i candidates will qualify for election; if it contains more than 2·v/(S+1), at least two will qualify; if 3 £ i and it contains more than 3·v/(S+1), at least three will; and so on, up to the case in which it contains more than i·v/(S+1) voters, when all i candidates will qualify.

This description of QBS seems unnecessarily long. Usually, Dummett offers a significantly shorter description. For example, in his submission to the Jenkins Commission he writes[3]:

The scruntineers can first mark as elected any candidate ranked highest by a sufficiently large minority (one-sixth of the voters in a five-member constituency, etc.). Then, having calculated the Borda counts of all remaining candidates, they can discover whether any set of from two to five candidates receives solid support from a sufficiently large minority: if so, that candidate in the set with the highest Borda count is marked as to be elected. The remaining seats will be filled by the candidates most generally acceptable to the electorate as a whole, i.e. those with the highest Borda counts.

In my opinion, a problem of the shorter description is that readers could mistakenly believe that the order in which the solid coalitions are considered at each stage and the question at which stages the different candidates have qualified were unimportant. However, example 1 demonstrates that they are decisive.

Example 1 (v = 100; S = 2; C = 5):

```29 DBCEA.
17 ABDCE.
17 BADCE.
17 CADBE.
13 ACDBE.
7 CABDE.
```
The Borda scores are 243 for candidate A, 250 for candidate B, 227 for candidate C, 251 for candidate D, and 29 for candidate E. Table 1 lists all solid coalitions. At stage 1, no candidate qualifies for election. At stage 2, it is observed that more than v/(S+1) voters are solidly committed to the candidates A and B and that more than v/(S+1) voters are solidly committed to the candidates A and C. When one uses only the short description of QBS, then one could mistakenly believe that there are two different possibilities how to proceed resulting in two different sets of winners. First: When one starts with the set A and B, candidate B qualifies for election because he has a better Borda score than candidate A. Then one has to consider the set A and C; candidate A qualifies for election because he has a better Borda score than candidate C. As no seats remain to be filled, QBS terminates and the candidates A and B are the winners. Second: When one starts with the set A and C, candidate A qualifies for election because he has a better Borda score than candidate C. Then one has to consider the set A and B; however, as this set has already won one seat no additional candidate qualifies at stage 2. At stage 3, one observes that more than v/(S+1) voters are solidly committed to the candidates A, B and D; however, as this set has already won one seat no additional candidate qualifies at stage 3. At stage 4, one observes that more than 2·v/(S+1) voters are solidly committed to the candidates A, B, C and D; as candidate D has the best Borda score candidate D qualifies for election. As no seats remain to be filled, QBS terminates and the candidates A and D are the winners.

However, the long description in 'Voting Procedures' states clearly that when one has to decide how many additional seats a given solid coalition gets at a given stage then one has to consider as already qualified only those candidates who have already qualified at strictly earlier stages. In example 1, when one starts with the set A and C, candidate A qualifies for election because he has a better Borda score than candidate C. Then one has to consider the set A and B; as none of these candidates has already qualified at a strictly earlier stage, candidate B qualifies for election because he has a better Borda score than candidate A.

In short, to guarantee that the result doesn't depend on the order in which the solid coalitions are considered at a given stage, it is important that one looks only at those candidates who have qualified at strictly earlier stages. For example, suppose, at stage 10, one finds a set of 10 candidates such that more than 5·v/(S+1) voters, but not more than 6·v/(S+1) voters, are solidly committed to these 10 candidates. Suppose that already 4 of these 10 candidates have qualified at stages 1-9. Then that candidate of this set who has the best Borda score of all those candidates of this set who did not qualify at stages 1-9 qualifies at stage 10 even if this set has already won additional seats at stage 10.

At first sight, it isn't clear whether the QBS winners can be calculated in a polynomial runtime since there are 2C possible sets of candidates. However, a set of candidates has to be taken into consideration only when at least one voter is committed to this set. In so far as at each of the C stages there cannot be more than v sets of candidates such that at least one voter is committed to this set, one has to take not more than C sets of candidates into consideration to calculate the QBS winners. Therefore, a polynomial runtime is guaranteed.

When not each voter ranks all candidates, then Dummett's intention is met best when in each stage i those voters who don't strictly prefer all the candidates of some set of i candidates to every other candidate are allocated to no solid coalition.

Nicolaus Tideman writes about QBS [4]:

To avoid sequential eliminations, Michael Dummett suggested a procedure in which a search would be made for solid coalitions of a size that deserved representation, and when such a coalition was found, an option (or options) that the coalition supported would be selected. If the solid coalition supported more than one option, the option (or options) with the greatest 'preference score' (Borda count) would be selected. Preference scores would also be used to determine which options would fill any positions not filled by options supported by solid coalitions. I find Dummett's suggestion unsatisfying. Suppose there are voters who would be members of a solid coalition except that they included an 'extraneous' option, which is quickly eliminated, among their top choices. These voters' nearly solid support for the coalition counts for nothing, which seems to me inappropriate.

At first sight, it isn't clear whether Tideman's criticism is feasible. It is imaginable that whenever there are 'voters who would be members of a solid coalition except that they included an "extraneous" option' there is also an STV method (i.e. a method where surpluses of elected candidates are transferred according to certain criteria to the next available preference and where, when seats remain to be filled, candidates are eliminated according to certain criteria and their votes are transferred to the next available preference) where this 'nearly solid support for the coalition counts for nothing'. If this is the case, then it is not appropriate to criticize QBS for ignoring this 'nearly solid support'. However, example 2 demonstrates that there are really situations where the QBS winners differ from the STV winners independently of the STV method used.

Example 2 (v = 100; S = 3; C = 5):

```40 ACDBE.
39 BCDAE.
11 DABEC.
10 DBAEC.
```
The Borda scores are 252 for candidate A, 248 for candidate B, 237 for candidate C, 242 for candidate D, and 21 for candidate E. Table 2 lists all solid coalitions. At stage 1, the candidates A and B qualify for election because both candidates are preferred to every other candidate by more than v/(S+1) voters each. At stage 2, it is observed that more than v/(S+1) voters are solidly committed to the candidates A and C and that more than v/(S+1) voters are solidly committed to the candidates B and C; but as both sets of candidates have already won one seat each, no additional candidate qualifies for election at stage 2. At stage 3, it is observed that more than v/(S+1) voters are solidly committed to the candidates A, C, and D and that more than v/(S+1) voters are solidly committed to the candidates B, C, and D; but as both sets of candidates have already won one seat each, no additional candidate qualifies for election at stage 3. At stage 4, it is observed that more than 3·v/(S+1) voters are solidly committed to the candidates A, B, C, and D; as this set has already won 2 seats, candidate D, the candidate with the best Borda score of all those candidates who haven't yet qualified, qualifies for election. As no seats remain to be filled, QBS terminates and the candidates A, B, and D are the winners. However, STV methods necessarily choose the candidates A, B, and C because, independently of how surpluses are transferred, candidate C always reaches the quota. In my opinion, example 2 questions whether compliance with proportionality for solid coalitions is sufficient for being a proportional preferential voting method.

Dummett's justification for his method is his claim that, unlike traditional STV methods, QBS is less 'quasi-chaotic'. He writes [3]:

The defect of STV is that it is quasi-chaotic, in the sense that a small change in the preferences of just a few voters can have a great effect on the final outcome. This is because it may affect which candidate is eliminated at an early stage, and thus which votes are redistributed, this then affecting all subsequent stages of the assessment process.

However, in my opinion, example 3 demonstrates that also QBS is 'quasi-chaotic'. This is because a small change in the preferences can affect which candidate qualifies at an early stage, this then affecting all subsequent stages of the assessment process.

Example 3 (v = 100; S = 2; C = 5):

```26 BCAED.
24 DCEBA.
10 EADBC.
8 ABCED.
7 EABDC.
7 EDBCA.
6 CDEBA.
6 DEBCA.
3 DCEAB.
2 EBADC.
1 DCBEA.
```
The Borda scores are 142 for candidate A, 216 for candidate B, 215 for candidate C, 204 for candidate D, and 223 for candidate E. Table 3 lists all solid coalitions. At stage 1, candidate D qualifies for election because more than v/(S+1) voters strictly prefer candidate D to every other candidate. At stage 2, it is observed that more than v/(S+1) voters are solidly committed to the candidates C and D; but as this set of candidates has already won one seat, no additional candidate of this set qualifies for election at stage 2. At stage 3, it is observed that more than v/(S+1) voters are solidly committed to the candidates A, B, and C; as none of these candidates has already qualified, candidate B, the candidate with the best Borda score, qualifies for election. As no seats remain to be filled, QBS terminates and the candidates B and D are the winners.

When a single DEBCA ballot is changed to BDECA, the Borda scores are 142 for candidate A, 218 for candidate B, 215 for candidate C, 203 for candidate D, and 222 for candidate E. Table 4 lists all solid coalitions for this modified example. At stage 1, no candidate qualifies for election. At stage 2, it is observed that more than v/(S+1) voters are solidly committed to the candidates C and D; as candidate C has a better Borda score, candidate C qualifies for election. At stage 3, it is observed that more than v/(S+1) voters are solidly committed to the candidates A, B, and C; but as this set of candidates has already won one seat, no additional candidate of this set qualifies for election at stage 3. At stage 4, it is observed that more than v/(S+1) voters are solidly committed to the candidates A, B, C, and E and that more than v/(S+1) voters are solidly committed to the candidates B, C, D, and E; but as both sets have already won one seat each, no additional candidates qualify for election at stage 4. At stage 5, candidate E qualifies for election because he has the best Borda score of all candidates who have not already qualified. Thus, by ranking candidate B higher candidate B is changed from a winner to a loser. By changing a single ballot the QBS winners are changed from the candidates B and D to the candidates C and E.

### References

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