Electer

This web page has a utility for deciding a winner in an election where each voter may rank entrants in order of preference. A voter is allowed to give some entrants equal rankings. A voter needn't rank all the entrants; any entrants his vote doesn't mention are deemed to have equal bottom rank.

Type the candidate list into the left-hand text area. There should be one line for each candidate. This line consists of a key string, a comma, and then his name. The votes will use the key string to identify the candidate. The names will be used when the results are reported; they are not used in the voting.

Type the votes into the right-hand text area. Each vote should be on its own line. It should consist of a sequence of candidate-keys, arranged from best to worst, with ">" between candidates where there is a preference, and "=" between candidates which this vote ranks equally. You may optionally precede this sequence with a number and a colon, to indicate that number of identical votes.

Then click on "Elect".

Some examples are shown below the form.

This electer shows the results of using the following methods:

1. the Condorcet method
2. the Condorcet method with Schwartz sequential dropping
3. the Copeland method

There is information about the Condorcet method here, here and here. This page also discusses Schwartz sequential dropping.

These and other electoral methods have been discussed by Mike Ossipoff.

Briefly, these methods work as follows.

In the Condorcet method, for each pair of candidates, a pairwise contest is held between them. In this contest, each vote is counted as a vote for whichever of the these two candidates the voter prefers (votes which rank them equal are ignored). For each candidate, the numbers of pairwise contests he wins and draws are found. If a candidate wins each pairwise contest that involved him, he is the outright winner.

Otherwise, for each candidate, his worst result (over all pairwise contests that involved him) is found. The winners are those candidates whose worst results are best (i.e. least bad) over all candidates. I implement Condorcet and two variants. The variation is in the way that "worst result" is defined. The ways are that a candidate's worst result is defined as the one

1. with the most votes cast against that candidate. In effect, votes that expressed no preference are counted with votes for that candidate. This is the Condorcet method.
2. with the greatest margin by which the candidate was defeated. In effect, votes that expressed no preference are ignored.
3. with the fewest votes cast for that candidate. In effect, votes that expressed no preference are counted with votes against that candidate.

Often, the variants give the same result. However, sometimes, if one variant declares a dead heat, another can break it.

Condorcet's method rewards a candidate for doing consistently well against all opposition. It has the nice property that there is nothing particularly meritorious about getting that one vote that puts you ahead of the other candidate in a pairwise contest.

Schwartz sequential dropping works by first finding the Schwartz set. The idea is to isolate the set of best candidates from which the Condorcet method cannot pick a winner. The Schwartz set is defined as follows. An unbeaten set is a set C1 such that no candidate outside C1 beats any in C1. A minimal unbeaten set is one that doesn't contain any unbeaten sets except (trivially) itself and the null set. The Schwartz set is the union of all the minimal unbeaten sets.

A Condorcet election is then held on just the Schwartz set.

In the Copeland method, for each candidate, a score is calculated: this is twice the number of pairwise contests he won plus the number of pairwise contests he drew. The winners are those with the highest score.

Copeland's method counts victories, not votes. Thus, if there are, say, 9 votes, getting 4 votes is useless; the 5th vote is all-important, and more decisive victories are not rewarded any more than a victory by one vote is.

Condorcet seems to be better than Copeland in another way. Consider a candidate B who defeated most opponents but who was in turn beaten by A by a large margin. Condorcet punishes B for doing badly against A. This makes sense; after all, the object is to find a winner. A is probably better, so B probably doesn't deserve to be elected. By contrast, Copeland rewards B for defeating the small fry, which is not what the election was designed to find out.

Some examples:

1,Paul Berriman
2,Richard Sabey
3,Paddy Grove
4,Mark Iredell
5,Mel Hochster
6,Danny Kodicek
7,Paul Coulter
8,Ray Steiner

6>1>7>8>3>5>4>2
3>1>4>2>5>7>6>8
6>7>5>3>4>8>1>2
2>5>7>1=3=4>8>6
5>2>4>7>3>1>8
8>1>6>3>5>7>4>2
4>3>8>5>7>2>1

1,Jonathan May
2,Paul Coulter
3,Mark Iredell
4,Jim Sherman
5,Mel Hochster
6,Luciano Ward
7,Richard Sabey

2>1>4>3>5>6>7
2>3>5>1>4>7>6
4>7>3>5>6>2>1
7>3=5>6>1>2>4
5>6>7>3>2>4>1
3>5>4>6>1>7=2
6>5>7>4=3>2>1
1>3>7>2=4=5>6

a,Ann
b,Bob
c,Carl
d,Dee

a=b>c>d
a>c>d>b
a>d>b>c
b>c>d>a
c>d>b>a
d>b>c>a

a,Ann
b,Bob
c,Carl

1:a>b=c
1:a>b>c
3:a=b>c
3:c>a=b
2:b=c>a