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

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 2^{C} 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 * v·*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

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

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.

- Michael Dummett,
*Voting Procedures*, Clarendon Press, Oxford, 1984 - Michael Dummett,
*Principles of Electoral Reform*, Oxford University Press, 1997 - Michael Dummett, Submission to the Independent Commission on Electoral Reform, 3rd July 1998
- T. Nicolaus Tideman,
*Collective Decisions and Voting*(draft), 1993