intro to voting systems for single-winner elections; Condorcet rules.

Condorcet Tally

Introduction to Condorcet rules, chapter contents

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 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, C, wins a different majority over each rival.

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	-

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 described in a 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 page in this chapter gives a thumbnail sketch of the famous Marquis and some quotes about his election principle.

The software page has programs to tally Condorcet rules.

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