[To see the submission in its original format, see the linked
document below]
ELECTING REPRESENTATIVE BODIES
In an election for President, Governor, Mayor or Chairman, a
fair electoral system should deliver "the most popular" candidate
in some sense. If there is a Condorcet Candidate---one
who is preferred by a majority to each rival candidate---he should
be elected. To the extent that there is a political
spectrum, a "centrist" candidate, perhaps the first choice of few
but the second choice of many, will be elected.
Of course the system itself should be designed to give the
voters maximal choice among candidates while protecting the
candidates against "vote-splitting"---that is, ensuring that
placing additional candidates on the ballot will not harm the
winning chances of more popular candidates whose appeal is to the
same groups of voters.
The technique for achieving fair elections to single offices is
simply Repeat Borda Eliminations. Start with ballots
marked to reflect each voter's preferences or rankings of
candidates (ties or equal rankings being permissible and treated as
half-preferences). Total the preferences
("matchpoints," to bridge players) "scored" by each candidate
across all ballots. Eliminate the candidate with the
lowest total as "irrelevant" and do a new tally with one fewer
candidate until only one candidate remains (the winner).
For multiple offices, such as City Councils, Boards of
Supervisors, Congressional delegations, and committees, a fair
electoral system should deliver a representative body.
If there are five seats to be filled (for example), the five
candidates elected should not be five "centrists" but should
represent the full spectrum of political factions … in
rough proportion to the numbers of voters supporting each
faction.
There are many in the United States who oppose representative
councils. In the New York City of my childhood, an
election system that delivered (although far from perfectly) a
representative (25-member) City Council was in use. The
Communist Party was supported by 4% or 5% of the city's voters, and
Benjamin J. Davis of Harlem, a Communist, was re-elected to the
City Council regularly. At the height of the
anti-Communist hysteria, posters urging voters to pass an
initiative repealing this basically democratic election system bore
the slogan, "Eliminate the last bastion of Communism in
America!" The sponsors of the initiative preyed on the
fears of a majority of voters, and succeeded in eliminating any
representation of political minorities in the City
Council. Any fear that Councilman Davis might bring
Communism to New York was ill-founded. Any specifically
Communist legislation (for example, a proposal for a municipal
takeover of the Italian and Jewish bakeries of Brooklyn and the
Bronx) would be voted down 24-to-1.
The election system of the New York City of my childhood was
called, redundantly, "Proportional Representation," but it was only
one system that went under that name. I believe it was
a version of the Hare System that used random drawings of "surplus"
votes; another version of the Hare System is called
"Single Transferable Vote," but its advocates generally do not
specify its details. All versions of the Hare System
constitute improvements upon the present American system of
single-member "districts" with either plurality voting, or
primaries and run-offs, but they are flawed; in a way that will
become apparent when I describe the improvements that produced MORE
("Multiple Office Representative Elections"), Hare can give voters
for very unpopular candidates a greater voice into choosing among
popular candidates than the voters for those popular candidates
have themselves, and permits "gaming" of the system with strategic
and insincere voting.
However, the Hare System embodies a valid concept, the
quota. To illustrate, consider a simplified
example. 999 voters seek representation on a council,
and 20 candidates (whose surnames we shall suppose start with each
of the first 20 letters of the alphabet) compete for 9
seats. It is clear that if among the 999 voters there
are 9 factions, each with 100 members, that each faction is
entitled to one representative. The other 99 voters
will go unrepresented unless they support one of the candidates of
the 9 factions, but that is the best for which we can hope under
the circumstances ... and far better than can be achieved in
single-office elections, where about half the voters go
unrepresented regularly. The number 100, in
this example, is the quota. The quota Q can be defined
quite precisely as V+1 (one more than the number of
voters) divided by S+1 (one more than the number of
seats). In any fair system for electing a
representative body, any candidate who obtains Q or more
"1st-place" votes must be given a seat.
There are two main problems in designing such a
system.
One is that in general, few candidates will "achieve
quota"---especially when the number of candidates greatly exceeds
the number of seats. Solutions to this problem involve
ways of "winnowing the field": eliminating "irrelevant"
candidates. The Hare System and MORE solve the problem
by eliminating the candidate with the fewest 1st-place votes and
transferring any ballot on which that candidate is marked 1st to
the candidate marked 2nd among those candidates who are still
contenders (the two systems differing, however, in the definition
of "contender").
The other is that sooner or later, seats will be filled by
candidates who achieve quota, and invariably some of these
"winners" will exceed quota with a surplus of 1st-place
votes. If nothing is done with the ballots supplying
that surplus, it is possible that in our simplified example a
faction of 200 will be represented by one councilman instead of
two. The Hare System removes a winning candidate from
the list of contenders (and all ballots), then draws at random from
the ballots marking that candidate 1st only the surplus, and
transfers those ballots to the 2nd-place candidate, discarding the
other Q ballots. Suppose, for example, that candidate
Adams receives 200 1st-place votes on the first tally.
Adams is awarded one of the 9 seats, (a random selection of) 100 of
the 200 ballots on which Adams is marked 1st are redistributed, the
other 100 are discarded, and now there are only 19 contenders and
899 ballots for the remaining 8 seats.
My first (and obvious) improvement upon the Hare System was to
replace random drawing of surplus ballots with fractional values
for all ballots of winning candidates. Thus continuing
with our example, each of the 200 "Adams" ballots would be counted
as .50 vote for Adams, .50 vote for the 2nd-place
candidate. (If instead of 200 1st-place votes, Adams
had received only 125, then each of the 125 ballots for Adams would
be counted as .80 vote for Adams, .20 vote for the 2nd-place
candidate.)
My second (and subtle) improvement was to leave "winning"
candidates on the ballots and keep them classified as
contenders. To see the importance of this, suppose that
the ballots of the first two candidates to be eliminated, Taylor
with 20 1st-place votes and Smith with 30 1st-place votes, all have
Adams marked as 2nd and Clark marked as 3rd. Using the
Hare System, the 50 Taylor and Smith ballots would all go to Clark,
and the 50 voters for Taylor and Smith (two irrelevant candidates)
would not only see their most-preferred relevant candidate (Adams)
already elected but would cast the full weight of their ballots for
their next-most-preferred relevant candidate (Clark); meanwhile,
the 200 Adams supporters would cast their ballots for their
next-most-preferred relevant candidates at only half weight
(one-fifth weight if they numbered 125 instead of 200, no weight at
all if they numbers exactly 100). Using MORE, the 50
Taylor and Smith ballots are redistributed to Adams, bringing the
total for Adams to 250, with each counted as a recalculated .40
vote for Adams and .60 vote for the next-most-preferred relevant
candidate.
My third (and more subtle still) improvement was to use the
fractional votes only for purposes of elimination, and redistribute
them to next-in-line candidates once the 2nd-place candidate
achieves quota. Thus suppose the 200 Adams ballots all
listed Baker as the 2nd choice. Using the Hare System,
Baker would be declared a "winner" and if Baker were marked 1st on
no ballots at all, the 200 Adams ballots would be fully
exhausted. Using MORE, as soon as 100 full-weight
ballots for Baker trickle in from redistributed ballots of
eliminated candidates (e.g. Rosen, Queen and Price), the 250
(Adams, Taylor and Smith) ballots are counted at their current
(.60) weight for the next-in-line candidates. Why is
this an improvement? Because otherwise Adams and Baker
supporters have a strong incentive to misrepresent their
preferences. Figuring that Adams and Baker would
achieve quota without their 1st-place votes, they might mark their
(true) 3rd-place choice as 1st. Allowing the supporters
of very popular candidates to have their votes counted (albeit at
fractional values) wherever needed among their next-in-line choices
vitiates that incentive.
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