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                       IS A MAJORITY SUFFICIENT?

The purpose of a voting system is to combine the wishes, or preferences
of individuals or voters into a decision for society or a group of people.  In
assessing the desirability of different voting systems in single-winner
elections two basic questions need to be asked.  First, what criteria can
we use to judge which candidate is the most representative of the
electorate given individual voter's preferences?  And second, which voting
system is most likely to insure the election of that candidate?  In the case
of a two man election in which each voter casts one vote the answer
seems quite clear.  The candidate with the majority support is the most
representative and any voting system that chooses that candidate, such
as our system of first-past-the-post or plurality voting, is considered
legitimate.  But in some cases of multicandidate elections, elections
with three or more candidates, the appearance of a majority winner may
be misleading in terms of electing the most representative candidate.
Such elections can be classified as pseudomajority elections rather
than real majority elections.

One example of pseudomajority elections can be found in the top-two
runoff system adopted in California in 2010.  Under this system, if no
candidate wins a majority in the first round, a second election is held
between the top two candidates with, sans a tie, the candidate having
the most votes winning.  The problem with this system, which is also
labeled the double ballot majority system, is that it can result in the
election of a candidate that does not in reality have credible majority
support.  The 2012 Egyptian presidential election is an example of
this.  In that election you had the Muslim Brotherhood's candidate
Morsi in the runoff against Shafik who was the last Egyptian Prime
Minister appointed by Mubarak.  Despite the fact that Shafik was a
representative of the then very unpopular Egyptian political
establishment, he lost to Morsi by a "narrow margin" according to
Wikipedia.  That and data on the Wikipedia entry tell an interesting
story.  The data for the elections are:

     Morsi          24.78%          51.73%

     Shafik         23.66%          48.27%

     Sabahi        20.72%

     Fotouh        17.47%

     Moussa       11.13%

In the Wikipedia piece on the Egyptian election there were data on a
poll that Al Ahram did on some pairwise runoffs which from the above
and what is known about that election seems to be quite credible.
They are as follows:

     Moussa        77.6%
     Morsi           22.4% 

     Fotouh          74.7%
     Morsi            25.3%

Even if these polls might be seriously skewed, they are so lopsided
that they clearly suggest that in a head to head (pairwise) competition
against either of the two lowest top five candidates in the first round
Morsi would have lost handily.

Instant runoff voting (IRV) is an example of another voting system that
is vulnerable to pseudomajority elections.  Under instant runoff voting,
voters list their preferences for candidates one by one.  If no candidate
had a majority of first place votes, the candidate with the least first
place votes is dropped and that candidate's supporters' second place
votes are reallocated for a second count.  If that gives no majority to
any candidate the process is repeated until a majority in votes counted
is reached or the process runs out of candidates to drop for a recount.
Such a process can easily lead to the election of a  pseudomajority
candidate as happened in the 2009 IRV election for mayor in Burlington,
VT.  In that election, in the penultimate round of vote counting the
Republican candidate had 3,297 votes, the Democrat 2,554 votes and
the Progressive 2,982.  Under the IRV system the Democrat candidate
was dropped and the Progressive candidate won by 4,314 votes
to the Republican's 4,064 votes or a margin of 250 votes.  Had the
Republican candidate been dropped instead and his voters' second
choices been reallocated, the vote would have been 4,067 for the
Democrat and 3,477 for the Progressive for a margin in favor of the
Democrat of 590 votes, more than twice the original margin of 250

Under our first-past-the-post voting system in which a voter only casts
one vote, a plurality,  just more votes than any other candidate has
where none have a majority, is sufficient to elect a candidate.  The
argument here being that in such a situation a plurality is the closest
to getting a majority and therefore presumably the closest to choosing
the most representative candidate.  But as the data on the 2012 Egyptian
presidential election above show had it been conducted as a simple
plurality election, Morsi would still have won though clearly he was not
a very representative candidate.  The problem here was already recognized
in the 18th Century by two Frenchmen the Marquis de Condorcet and
Jean-Charles de Borda.

In such single winner elections with multiple candidates, Condorcet
argued that the most representative candidate is the one who could
beat all the other candidates in all the pairwise, one-to-one, contests
possible from the field of candidates.  The number of pairwise contests
possible in such multicandidate elections is given by the formula
p = n(n-1)/2 where p represents the required number of pairwise contests
and n represents the number of candidates in a multicandidate election.
Thus if we have three candidates we would require three pairwise
contests, for four candidates it would require six pairwise and for five,
ten pairwise contests. 

Prior to the advent of computers conducting a Condorcet election was
not very practical.  But with computers one could have voters list their
preferences for candidates one by one and calculate from that all the
possible pairwise elections from the field of candidates to determine
a Condorcet winner.  However there is just one little, but rather serious,
problem here, the Condorcet paradox.  The paradox is that while
preferences for individual voters are transitive; that is if a voter prefers
A to B to C (in standard notation A > B > C), he does not prefer
C to A; preferences may not be transitive for voters taken collectively
and a Condorcet leader cannot be determined.

And example of the Condorcet paradox can be seen from the simple
case of three voters, X, Y, and Z and three candidates A, B and C.
If the preferences of the voters in descending order are as follows:

                              X         Y          Z
                              A         B          C
                              B         C          A
                              C         A          B

We get for each pairwise contest possible from the field of
candidates the following for:

                    A vs B     a win for A by 2:1

                    B vs C     a win for B by 2:1

                    C vs A     a win for C by 2:1

Or stated in more standard notation we have A > B > C > A.  Various
complex fixes have be proposed for this which in reality are not so

Borda's answer to the question of which candidate is the most
representative in a single-winner multicandidate election is that
candidate who has the highest "Borda count."  The Borda count
is calculated by asking voters to list one by one all candidates in
their order of preference and assigning numerical scores according
to the listings.  In the classic Borda count, if one has ten candidates
the candidate that a voter lists first is given a score of 9, or the
number of candidates minus one, the second 8 points, the number
of candidates minus two and so on to the last candidate who is
assigned zero points.  The count is summed over the preference
listing of all the voters and the candidate with the highest Borda
count is considered the most representative candidate and the one
who should be chosen.  As Sir Michael Dummett points out in his
book "Principles of Electoral Reform" the count represents the
total number of votes each candidate would get in all the pairwise
contests possible from the field of candidates in which voters
vote consistent with their rankings of the candidates.

Simulations on the Internet by the mathematician Ka-Ping Yee
suggest that Condorcet and Borda voting systems tend towards
similar results, but with one notable exception, Borda results are
not always of a majoritarian outcome as are Condorcet results
where there is a clear Condorcet leader.  Sir Michael Dummett in
his book "Principles of Electoral Reform" presents an interesting
hypothetical example of this which raises the issue of whether a
majority is sufficient to identify the most representative candidate.
His example consists of 56,000 electors and four candidates A,
B, C, and D.  In the example A is a highly polarized or divisive
candidate who is supported by 29,000 of the electors but is
considered the least desirable by 25,000 of them.  The rankings
and their support levels used in Dummett's example are as

Voters       29,000       24,000       2,000       1,000
                    A               B             C              D
                    B               C             A              C
                    C               D             B              B
                    D               A             D              A

In this example A would clearly win in a regular plurality or IRV election
having an absolute majority of first place votes.  And he is clearly the
Condorcet leader as Dummett would label him.  But in this hypothetical
election B's Borda count of 133,000 is higher that A's of 91,000.  In
this case Dummett points out that:

     There is undoubtedly a case to be made for saying that B
     is a more representative candidate than A.  All of A's
     supporters reckon him [B] the second best candidate, but no
     elector thinks him the worst: he would surely represent
     opinion in the constituency better than A.

In short, a polarized or divisive candidate may have a majority, but may
not be the most representative candidate.

Heretofore we have tacitly been assuming that voters are voting sincerely
and not tactically or strategically.  In the case of our two candidate election
that surely would be the case.  But as the Gibbard-Satterthwaite theorem
establishes, tactical voting can happen in any multicandidate election.
As Michael Dummett put it in his book on electoral reform,

     Under all electoral systems, some voters may vote tactically,
     that is to say, in a way that does not conform to their true
     preferences.  It can be mathematically demonstrated that no
     system can avoid this.  That is to say, there can be no system
     under which, given his preferences, every voter will always have
     only one way of voting that will bring about an outcome as
     desirable as possible from his point of view, however the others
     choose to cast their votes.  Nevertheless, different systems
     vary greatly in the degree of incentive they give for tactical
     voting.  A system that debars a voter even from expressing
     all his preferences (or takes minimal account of most of
     them) gives the most potent of all incentives for it.  This is
     why tactical voting plays so large a role in elections under
     the 'First Past the Post' system.

Tactical voting generally involves either feedback on the preferences
of other voters (e.g., it would be a wasted vote to vote for a third
party candidate like a Ralph Nader) or some a priori assumption of
how a given voting system can be gamed to one's advantage in
voting.  In the case of ranking voting systems, like one has in an
IRV or Borda election, feedback on the preferences of other voters is
generally a daunting task.  In the case of a five candidate election one
is dealing with 120 different rankings (5! in mathematical notation)
with differing frequencies of occurrences.  Given the inherent complexity
of such a situation Samuel Merrill's assertion that IRV is relatively
resistant to manipulation does not come as a surprise at least on
the individual level.  But on a collective level that does not seem to
be the case as the wide spread use of "how to vote cards" in
Australia attests in its IRV elections.  Based on information from at
least one survey, half of the electorate in Australia may rely upon
such cards in voting.  Such tactical voting is surely pernicious and
undesirable.  In the voting process of determining a winner it is
certainly fair to weigh one vote against the others in the algorithm
determining a winner or the most representative candidate.  But
to engage in a voting pattern on the part of one group that affects
how the preferences of other groups are or are not expressed surely
cannot give credible outcomes in terms of the voting system
identifying the representative candidate.  The erratic way in which
Hare based systems like IRV behave with shifts in preferences
that Dummett's analysis suggests, surely makes such manipulation
of serious concern.

In terms of feedback on others' preferences, Borda elections face
similar problems but have a different response in terms of tactical
voting strategy.  As Dummett and others have pointed out the Borda
system offers a fair incentive to tactical voting.  Generally this takes
the form of voters ranking the candidate who is most perceived as
threatening their first choice low in their rankings even though
otherwise that candidate is closer to theirs in terms of preferences.
If a great many voters engage in such behavior, the outcomes are
hardly likely to bear any relation to their actual preferences, i.e.,
they are not representative.  Given that there is a lot of anecdotal
evidence of this problem, especially in Borda elections held in
academic departments, it is not surprising that a Michael Dummett
would conclude that "the Borda system is arguably too vulnerable
to tactical voting to be adopted" in single-winner elections.

Approval voting, in which in multicandidate elections voters are
allowed to give one vote each to the candidate or candidates they
support with the candidate having the most votes winning, may be
a much more benign voting system when it come to the issue of
tactical voting.  Given the similarities of outcomes in Ka-Ping Yee's
simulations of approval, Condorcet and Borda voting systems, it may
well be that an approval voting election may better reflect a sincere
Borda election than an actual Borda election.  As far I can see, an
approval voting election's tactical voting would not involve the
manipulation of the expression of other's preferences nor
misrepresentation of one's own preferences thereby contributing to
unintended outcomes.

In an approval election, if one's favorite candidate has little chance
and a voter does have a definite preference among the credible
contenders, it is not a misrepresentation to vote accordingly (e.g.,
to give both a Ralph Nader and an Al Gore an approval vote in a
Florida-style election).  And certainly under approval voting the
information needed to make intelligent choices given the constraints
of other voters' preferences is rather simple and straight forward.  For
instance, in a five candidate contest the relevant information that one
needs in considering the tactical aspects of voting is simply the
percentage support levels for each candidate, five pieces of information
as opposed to 120 pieces.  Given modern polling techniques, such
information is likely to be fairly available with reasonable accuracy.

Since approval voting does not always choose the Condorcet leader
but rather the candidate that is acceptable to the largest fraction
of the electorate, it well resist choosing a highly polarized and
divisive candidate as in the case of the Borda leader in Dummett's
example above.  On this score, approval voting may well be a very
good approach to electing the most representative candidate and for
opening up our elections to independent and third party candidates.