Skip to main content
\(\require{cancel}\newcommand\degree[0]{^{\circ}} \newcommand\Ccancel[2][black]{\renewcommand\CancelColor{\color{#1}}\cancel{#2}} \newcommand{\alert}[1]{\boldsymbol{\color{magenta}{#1}}} \newcommand{\blert}[1]{\boldsymbol{\color{blue}{#1}}} \newcommand{\bluetext}[1]{\color{blue}{#1}} \delimitershortfall-1sp \newcommand\abs[1]{\left|#1\right|} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section5.4Pairwise Comparisons and the Condorcet Criterion

SubsectionThe Method of Pairwise Comparisons

The Method of Pairwise Comparisons

The Method of Pairwise Comparisons is like a round robin tournament: we compare how candidates perform one-on-one, as we've done above. It has the following steps:

  1. List all possible pairs of candidates.

  2. For each pair, determine who would win if the election were only between those two candidates.

    1. To do so, we must look at all the voters. Each voter casts their ballot for whichever candidate they ranked higher of the pair being considered (even if they didn't rank that candidate first).
    2. The winner of each pair is given 1 point, and the loser, 0 points. If it is a tie, they both get \(\frac{1}{2}\) point.
  3. After finding each pairwise winner, add up the points for each candidate. The candidate with the most points is the winner.

Note that this method results in ties a little more often than some of the other methods.

Example5.29The 2009 mayoral election in Burlington, VT

In 2009, the city of Burlington, Vermont used IRV to elect their mayor. The results of this election are shown below, in a simplified form to include only the three main candidates. 5More complete details can be found at the Wikipedia article. These three candidates were Bob Kiss, Kurt Wright, and Andy Montroll.

Percent of voters
Rankings 34 37 15 9 5
1st choice K W M M M
2nd choice M M K W
3rd choice W K W K
Table5.30

Using this information, we can find the winner of this election using the method of pairwise comparisons as follows.

Pair Percent of votes Winner/points
1 Kiss 34+15=49 Win=1
versus Wright 37+9=46 Lose=0
2 Kiss 34 Lose=0
versus Montroll 37+15+9+5=66 Win=1
3 Wright 37 Lose=0
versus Montroll 34+15+9+5=63 Win=1

We see that Montroll wins two pairs for 2 points, while Kiss and Wright each won one pair for 1 point each. Thus, Montroll would win this election using the method of pairwise comparisons.

SubsectionThe Condorcet Criterion

In Example5.29, we saw that Montroll was the pairwise winner because he won the most pairs of any candidate. We can say more, however: not only did Montroll win the most pairs, he actually won every pair in which he was one of the options. In other words, Montroll would never lose a one-on-one contest against any of his opponents. A candidate that satisfies this property is called a Condorcet candidate, named after Nicolas, Marquis de Condorcet. 6Condorcet was a French philosopher and mathematician who lived during the 18th century. His ideas were influencial during the earlier part of the French Revolution, and he was a major author of a proposed constitution. However, his ideas were not adopted, and he became a target during the Reign of Terror. After unsuccessfully attempting to flee, Condorcet was arrested and died in prison in 1794. It is still unknown whether his death was suicide or murder. You can read more about his interesting life at the MacTutor History of Mathematics and, of course, Wikipedia. A Condorcet candidate would seem to be strongly supported by voters; this leads to our next fairness criterion:

The Condorcet Criterion
  • A candidate who would defeat any other candidate in a one-on-one contest is called a Condorcet winner (also called a Condorcet candidate), pronounced condor-say.
  • The Condorcet criterion states that, if a Condorcet winner exists, then that candidate should be the winner of the election.
  • Note that not every election may have a Condorcet winner! If there is no Condorcet winner, then the Condorcet criterion does not apply; that is, it does not state that any candidate ought to win.