Condorcet method

Condorcet is partly an electoral system, and partly a way of

thinking about preference electoral systems that elect one candidate.

The Condorcet winner is the candidate who, when compared in turn

with each of the other candidates, is preferred to them. It is not

guaranteed that there will be any candidate to whom this applies, so

any Condorcet electoral system must have a way of resolving such

results.

Each voter ranks all candidates in order of preference.

For each pair of candidates, it is determined how many voters

preferred each candidate by counting whether they were higher-ranked

on the ballot. If any candidate is preferred to all other candidates,

they are declared the winner.

If there is no initial winner, the winner must be determined some

other way. One way is to define the 'top cycle' to mean that

candidates are said to be in the 'top cycle' if each of them will beat

all candidates outside the top cycle in pairwise competition but not

all the candidates inside the top cycle.

Then the winner can be chosen by having an

Alternate Vote (AV) election between the top cycle

candidates. Or, Another way would be to choose the candidate in the

top cycle who, in the pairwise contest that they do worst in, they

lose by the least amount.

In an election, there are 3 'top cycle' candidatess. Considering only

preferences between these candidates,

  41 voters voted X 1st, Y 2nd, Z 3rd

  33 voters voted Y 1st, Z 2nd, X 3rd

  22 voters voted Z 1st, X 2nd, Y 3rd

In pairwise comparisons:

  X: against Y = 41+22-33 = +40 (ie X won by 40 votes)

     against Z = 41-33-22 = -14 (ie X lost by 14 votes)

  Y: against X = -40

     against Z = 52

  Z: against X = 14

     against Y = -52

So X wins, because his worst result (-14) is less bad than Y's

or Z's worst results (-40 and -52 respectively).

The choice of a winner from any candidate within the top cycle is

to some extent arbitrary, in the sense that there are good reasons

for picking any of them.

There are good reasons to regard the Condorcet criterion, when

fulfilled, as the best test of who should win: so if there is a

Condorcet winner, then a system for selecting one winner ought to win

it. On this view, AV is not as good as the Condorcet scheme, because

there are circumstances in which it will fail to pick the Condorcet

winner.