Kemeny–Young method

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results The Kemeny–Young method is an electoral system that uses ranked ballots and pairwise comparison counts to identify the most popular choices in an election.

The Kemeny–Young method uses preferential ballots on which voters rank choices according to their order of preference.

The first step is to create a matrix or table that counts pairwise voter preferences.

The above preference order can be expressed as the following tally table:[citation needed] Now suppose that multiple voters had voted on those four candidates.

After all ballots have been counted, the same type of tally table can be used to summarize all the preferences of all the voters.

[clarification needed] After the overall ranking has been calculated, the pairwise comparison counts can be arranged in a summary matrix, as shown below, in which the choices appear in the winning order from most popular (top and left) to least popular (bottom and right).

No other possible ranking can have a summary matrix that yields a higher sum of numbers in the upper-right, triangular half.

The academic papers by John Kemeny and Peyton Young[2][3] refer to finding this minimum sum, which is called the Kemeny score, and which is based on how many voters oppose (rather than support) each pairwise order:

In this arrangement the largest ranking score (393) equals the sum of the counts in bold, which are in the upper-right, triangular half of the matrix (with a green background).

[8] It has been reported[9] that calculation methods based on integer programming sometimes allowed the computation of full rankings for votes on as many as 40 candidates in seconds.

However, certain 40-candidate 5-voter Kemeny elections generated at random were not solvable on a 3 GHz Pentium computer in a useful time bound in 2006.

[9] The Kemeny–Young method can be formulated as an instance of a more abstract problem, of finding weighted feedback arc sets in tournament graphs.

Using a simple probabilistic model for these noisy signals, Young showed that the Kemeny–Young method was the maximum likelihood estimator of the true preference order.

Young further argues that Condorcet himself was aware of the Kemeny-Young rule and its maximum-likelihood interpretation, but was unable to clearly express his ideas.

In the papers by John Kemeny and Peyton Young, the Kemeny scores use counts of how many voters oppose, rather than support, each pairwise preference,[2][3] but the smallest such score identifies the same overall ranking.