E J Nanson's method

[Tom Round]

Some weeks ago, it was asked on this list how Edward Nanson's
method works. By coincidence, only a fortnight ago an article
appeared discussing Nanson's contribution to the literature:
Iain McLean (1996) "E J Nanson, Social Choice and Electoral
Reform." 31(3) AUSTRALIAN JOURNAL OF POLITICAL SCIENCE 369-
385. On page 372, McLean gives a brief summary of what Nanson
proposed:

     "He [Nanson] offers a method that combines the
     merits of Borda and Condorcet. That is, initially,
     to have the voters rank the candidates, and conduct
     a Borda count. After the initial Borda count, all
     candidates who fail to achieve more than the average
     score of all candidates are eliminated, and the
     Borda count is re-run among the survivors. This
     continues as often as necessary until a single
     winner is selected. Nanson proves that, if a
     Condorcet winner exists, this method, unlike the
     ordinary Borda count, is guaranteed to choose her
     (it, him)."

Nanson's method, according to McLean, was adopted by the
following bodies:

a)   Trinity College (the Anglican residential college of Uni
     of Melbourne)

b)   elections to the Assembly and Canonry [the House of
     Clergy, I think?] of the Melbourne Anglican Diocese

c)   the University of Melbourne, 1926-82, for election of its
     University Councillors and academic committees.

In 1983, however, it was replaced by STV for Council
elections, though retained for some internal elections by the
Council. The University Registrar gave the reason that
Nanson's version was seen as "advantaging inoffensive but not
outstanding candidates against those with strong support"
(page 375). Some subscribers on this list would agree with
this sentiment so far as applied to multi-seat elections,
while also advocating that a single-seat winner indeed SHOULD
be the least disliked, "everyone's second choice".

Unfortunately the article is unclear whether MU used Nanson's
method for single-seat, multi-seat or both kinds of elections.
The rules provided did allow for multi-seat elections:

Rule 24:  "If a further vacancy is to be filled, the column
     and row assigned to each elected candidate shall be
     excluded, and the process of election carried out in
     precisely the same manner as before."

As can be seen, there is no provision for reduction in value
of votes each time they elect a candidate. Thus this would
have produced a "block vote" result rather than proportional
representation. If this was in fact practiced, then one can
indeed see the Registrar's point. Filling a single seat with
one Centre candidate may appease both Left and Right, but
filling seven seats in a row with Centre candidates -- rather
than, eg, a result of Left 3, Right 3, Centre 1 -- is likely
to aggravate the larger blocs (and justly so, given the denial
of proportionate representation to them).

[I note parenthetically, and with much annoyance, that the
trend in Australia now is to view University senates/ councils
as analogous to "cabinets" rather than to "legislatures" --
even though they exercise sweeping (delegated) legislative
powers. Eg, this trend is seen in the replacement of PR-STV by
block vote systems at Uni of Queensland and some other
campuses, and by reductions in council size, usually from
around 30 to around 15, proposed by several recent official
enquiries. Thus statutes imposing $20-per-day fines for
overdue library books can now be enacted by 8 out of 15
members of a university's governing body of 15, without any
guarantee that minority viewpoints will be heard and
considered!)

d)   since 1968, election of the Council of the University of
     Adelaide

e)   the municipal elections of Marquette, Michigan, in the
     1920s.

At page 374, McLean quotes verbatim the excerpts from the
Melbourne Uni statutes. The crucial provisions are that,
first, a "Dodgson matrix" is set up, along the following
lines. (This is a simplified version of the example McLean
gives at page 382).

              Votes against:
             A           B          C
Votes
for:
A          --            13             20

B          16            --             14
     
C                9            15         --

Thus, A versus B pairwise means 13 votes for A, 16 for B (ie,
the same as 13 votes against B and 16 against A, which easily
translates into "largest pairloss" terms).

Then, once the matrix is worked out, Nanson's "elimination
rule" kicks in:

Rule 23:  "The numbers in each column shall be summed. The
     column with the lowest sum, and the corresponding row,
     shall be excluded, and the remaining numbers in each
     column shall again be summed. The column with the lowest
     sum at this stage, and the corresponding row, shall be
     excluded, and this process shall be repeated until only
     two columns are left. Of the candidates to whom these
     [last two] columns refer, that one who has a majority of
     preferences over the other shall be declared elected."
     (page 374).

Some questions I am curious about -- maybe others can help:

1.   When the Borda score is re-calculated after a candidate
     is eliminated, are the weights of the preferences re-
     adjusted? For example, if a ballot's first three
     preferences are eliminated, does its fourth-preference
     then receive a number of points equal to that of the
     original first-preference candidate? Maybe this just
     happens automatically when a column and row are excluded
     and the remaining vote totals are re-calculated. I think
     that Borda's original method -- [candidates minus 1]
     points for each first preference, etc -- produces the
     same scores as would adding up each candidate's total of
     "votes for" in each un-eliminated column.

2.   Attention Deane Crabb in particular: Does this sound
     anything like the method the Democrats use for their
     internal party ballots?

3.   Mr McLean's statement that Nanson's method "must elect a
     Condorcet winner" -- more accurately, a Beats_All
     candidate -- I am tempted to take with a grain of salt.
     As Steve Eppley pointed out, some methods (eg Borda's, Copeland's)
     will indeed elect a Beats_All candidate IF everyone
     votes sincerely, yet at the same time contain built-in
     incentives for voters to vote insincerely, making the
     Condorcet-efficiency of such systems much weaker in practice than on
     paper.

     I'm curious to know what other subscribers on this list,
     particularly those whose powers of mathematical analysis
     are less shaky than mine, would make of the Nanson
     system, and what incentives it offers for manipulation.

Tom Round

-------------------
Overflow-Cc: 100245.2440 at compuserve.com ('Geoff Powell'),
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   dunnmj at ozemail.com.au ('Martin Dunn'),
   GGoode at VTRLMEL1.TRL.OZ.AU (Goode, Geoff),
   j.pyke at qut.edu.au (John Pyke, QUT Law School),
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   lee at cs.mu.OZ.AU ('Lee Naish'),
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