Condorcet Tally

A common problem in a vote-counting rule is too many candidates dividing a group of voters.

We can solve that by asking each voter to rank the candidates. For a voter the solution is as easy as saying 1st choice, 2nd choice, 3rd choice. The people who count ballots can then use Condorcet's rule to elect the 1 candidate who can top each of the others in a series of 1 on 1 tests.

If more voters prefer (rank) A over B than vice versa, A passes that test. Each ballot's rank of A relative to B concerns us; the number of first-rank votes is not important. The winner of A versus B is tested with C.

This is sometimes called the "pairwise" voting rule or "tournament voting" because it is like a round-robin tournament during which every contestant must play every other contestant.

This example shows 7 voters choosing 1 winner from 4 candidates or proposals -- which are labeled A, B, C and D.

Table 1 a lists ballots from the 7 voters. Looking at Uri's ballot we see that he prefers A over the others; so in the 6 tests his ballot will count for A in A versus B, A versus C, or A versus D. It will count for B against C or D, and for C over D. You can highlight all the totals he adds to by clicking his preferences.

Table 1 a	7 Ballots
4 Ranks	Uri	Nic	Mo	Lil	Kit	Jo	Gil
1st 2nd 3rd 4th	A B C D	B C A D	B C D A	C B D A	D C B A	D C A B	D A C B

Table 2 a Pairwise Tests of 4 Candidates
	Votes Against
	A	B	C	D
for A	-	3	2	2	2 prefer A over D.
for B	4	-	3	4	4 prefer B over D.
for C	5	4	-	4	4 prefer C over D.
for D	5	3	3	-

Table 2 a tests all 4 candidates; each cell records 1 side of a 1 on 1 test. Its number tells how many voters preferred the name in the row heading over the name in the column heading. For example, 3 voters ranked A higher than B on their ballots; 4 ranked B above A. So B passes that test and A fails. Passing a test requires winning at least 4 of the 7 ballots.

Click a number in table 2 to check which ballots add to it. You can see there are many different majorities even in this group with only 7 voters. A candidate may say she won a majority; but she cannot honestly say she won the majority. The Condorcet winner got a different majority over each rival.

C can top any rival so C wins. Who wins by the plurality rule?
Click here to reset the ballots and the Condorcet pairwise table. We will see these 7 ballots again to show an Instant Runoff tally, a voting cycle, and suspended votes.

Sometimes no one passes all of her pairwise tests. Such ties can be broken by many rules including the Instant Runoff rule shown in the second page below. This type of tie seems to occur in about 1 out of 10 elections. Ties are more common in votes to set policies so the section on policy making will take a closer look at such "voting cycles".

The next short page of this chapter gives a sketch of the Marquis de Condorcet and some quotes about his election principle.

The software page has programs to tally Condorcet rules.

Accurate Democracy
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