Kemeny-Young method

The Kemeny-Young method is a voting system that uses preferential ballots, pairwise comparison counts, and sequence scores to identify the most popular choice, and also identify the second-most popular choice, the third-most popular choice, and so on down to the least-popular choice. This is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.

The Kemeny-Young method is also known as the maximum likelihood method, the linear ordering problem, VoteFair popularity ranking, and the median relation.

Description

The Kemeny-Young method uses preferential ballots on which a voter ranks the choices according to their order of preference. A voter is allowed to rank more than one choice at the same preference level. Unranked choices are usually interpreted as least-preferred.

Kemeny-Young calculations are usually done in two steps. The first step is to create a matrix or table that counts pairwise voter preferences. The second step is to test all possible order-of-preference sequences, calculate a sequence score for each sequence, and compare the scores. Each sequence score equals the sum of the pairwise counts that apply to the sequence, and the sequence with the highest score is identified as the overall ranking, from most popular to least popular.

Example

This matrix summarizes the corresponding pairwise comparison counts:

	Memphis	Nashville	Chattanooga	Knoxville
Memphis	-	42%	42%	42%
Nashville	58%	-	68%	68%
Chattanooga	58%	32%	-	83%
Knoxville	58%	32%	17%	-

The Kemeny-Young method arranges the pairwise comparison counts in the following tally table:

All possible pairs of choice names	Number of votes with indicated preference
	Prefer X over Y	Equal preference	Prefer Y over X
X = Memphis Y = Nashville	42%	0	58%
X = Memphis Y = Chattanooga	42%	0	58%
X = Memphis Y = Knoxville	42%	0	58%
X = Nashville Y = Chattanooga	68%	0	32%
X = Nashville Y = Knoxville	68%	0	32%
X = Chattanooga Y = Knoxville	83%	0	17%

The Kemeny sequence score for the sequence Memphis first, Nashville second, Chattanooga third, and Knoxville fourth equals (the unit-less number) 345, which is the sum of the following annotated numbers.

:42% (of the voters) prefer Memphis over Nashville:42% prefer Memphis over Chattanooga :42% prefer Memphis over Knoxville:68% prefer Nashville over Chattanooga :68% prefer Nashville over Knoxville:83% prefer Chattanooga over Knoxville

This table lists all the sequence scores.

First choice	Second choice	Third choice	Fourth choice	Sequence score
Memphis	Nashville	Chattanooga	Knoxville	345
Memphis	Nashville	Knoxville	Chattanooga	279
Memphis	Chattanooga	Nashville	Knoxville	309
Memphis	Chattanooga	Knoxville	Nashville	273
Memphis	Knoxville	Nashville	Chattanooga	243
Memphis	Knoxville	Chattanooga	Nashville	207
Nashville	Memphis	Chattanooga	Knoxville	361
Nashville	Memphis	Knoxville	Chattanooga	295
Nashville	Chattanooga	Memphis	Knoxville	377
Nashville	Chattanooga	Knoxville	Memphis	393
Nashville	Knoxville	Memphis	Chattanooga	311
Nashville	Knoxville	Chattanooga	Memphis	327
Chattanooga	Memphis	Nashville	Knoxville	325
Chattanooga	Memphis	Knoxville	Nashville	289
Chattanooga	Nashville	Memphis	Knoxville	341
Chattanooga	Nashville	Knoxville	Memphis	357
Chattanooga	Knoxville	Memphis	Nashville	305
Chattanooga	Knoxville	Nashville	Memphis	321
Knoxville	Memphis	Nashville	Chattanooga	259
Knoxville	Memphis	Chattanooga	Nashville	223
Knoxville	Nashville	Memphis	Chattanooga	275
Knoxville	Nashville	Chattanooga	Memphis	291
Knoxville	Chattanooga	Memphis	Nashville	239
Knoxville	Chattanooga	Nashville	Memphis	255

The highest sequence score is 393, and this score is associated with the following sequence, so this is the winning preference order.

Preference order	Choice
First	Nashville
Second	Chattanooga
Third	Knoxville
Fourth	Memphis

If a single winner is needed, the first choice, Nashville, is chosen. (In this example Nashville is the Condorcet winner.)

Characteristics

In all cases that do not result in an exact tie, the Kemeny-Young method identifies a most-popular choice, second-most popular choice, and so on.

A tie can occur at any preference level. Except in some cases where circular ambiguities are involved, the Kemeny-Young method only produces a tie at a preference level when the number of voters with one preference exactly matches the number of voters with the opposite preference.

atisfied criteria for all Condorcet methods

All Condorcet methods, including the Kemeny-Young method, satisfy these criteria:

:;Non-imposition::There are voter preferences that can yield every possible overall order-of-preference result, including ties at any combination of preference levels.

:;Condorcet criterion::If there is a choice that wins all pairwise contests, then this choice wins.

:;Majority criterion::If a majority of voters strictly prefer choice X to every other choice, then choice X is identified as the most popular.

:;Pareto efficiency::Any pairwise preference expressed by every voter results in the preferred choice being ranked higher than the less-preferred choice.

:;Non-dictatorship::A single voter cannot control the outcome in all cases.

Additional satisfied criteria

The Kemeny-Young method also satisfies these criteria:

:;Universality::Identifies the overall order of preference for all the choices. The method does this for all possible sets of voter preferences and always produces the same result for the same set of voter preferences.

:;Monotonicity::If voters increase a choice's preference level, the ranking result either does not change or the promoted choice increases in overall popularity.

:;Smith criterion::The most popular choice is a member of the Smith set, which is the smallest set of choices such that every member of the set is pairwise preferred to every choice not in the Smith set.

:;Local independence of irrelevant alternatives::If choice X is not in the Smith set, adding or withdrawing choice X does not change a result in which choice Y is identified as most popular.

:;Reinforcement::If all the ballots are divided into separate races and the overall ranking for the separate races are the same, then the same ranking occurs when all the ballots are combined.

:;Reversal symmetry::If the preferences on every ballot are inverted, then the previously most popular choice must not remain the most popular choice.

Failed criteria for all Condorcet methods

In common with all Condorcet methods, the Kemeny-Young method fails these criteria (which means the described criteria do not apply to the Kemeny-Young method):

:;Independence of irrelevant alternatives::Adding or withdrawing choice X does not change a result in which choice Y is identified as most popular.

:;Invulnerability to burying::A voter cannot displace a choice from most popular by giving the choice an insincerely low ranking.

:;Invulnerability to compromising::A voter cannot cause a choice to become the most popular by giving the choice an insincerely high ranking.

:;Participation::Adding ballots that rank choice X over choice Y never cause choice Y, instead of choice X, to become most popular.

:;Later-no-harm::Ranking an additional choice (that was otherwise unranked) cannot displace a choice from being identified as the most popular.

:;Consistency::If all the ballots are divided into separate races and choice X is identified as the most popular in every such race, then choice X is the most popular when all the ballots are combined.

Additional failed criteria

The Kemeny-Young method also fails these criteria (which means the described criteria do not apply to the Kemeny-Young method):

:;Independence of clones::Offering a larger number of similar choices, instead of offering only a single such choice, does not change the probability that one of these choices is identified as most popular.

:;Invulnerability to push-over::A voter cannot cause choice X to become the most popular by giving choice Y an insincerely high ranking.

:;Schwartz::The choice identified as most popular is a member of the Schwartz set.

:;Polynomial runtime [J. Bartholdi III, C. A. Tovey, and M. A. Trick, "Voting schemes for which it can be difficult to tell who won the election", "Social Choice and Welfare", Vol. 6, No. 2 (1989), pp. 157–165.] ::An algorithm is known to determine the winner using this method in a runtime that is polynomial in the number of choices.

Calculation methods

Calculating all sequence scores requires time proportional to N!, where N is the number of choices. Although one need not compute scores to find the winner, any algorithm finding the winner requires superpolynomial time (unless P=NP). Nevertheless, fast calculation methods based on linear programming allow the computation of full rankings for votes with as many as 40 choices.Vincent Conitzer, Andrew Davenport, and Jayant Kalagnanam, " [http://www.cs.cmu.edu/~conitzer/kemenyAAAI06.pdf Improved bounds for computing Kemeny rankings] " (2006).]

History

The Kemeny-Young method was developed by John Kemeny in 1959.John Kemeny, "Mathematics without numbers", "Daedalus" 88 (1959), pp. 577–591.]

In 1978 Peyton Young and Arthur Levenglick showedH. P. Young and A. Levenglick, " [http://www.econ.jhu.edu/People/Young/scans/VR16.pdf A Consistent Extension of Condorcet's Election Principle] ", "SIAM Journal on Applied Mathematics" 35, no. 2 (1978), pp. 285–300.] that this method was the unique neutral method satisfying reinforcement and the Condorcet criterion. In another paper [ H. P. Young, "Condorcet's Theory of Voting", "American Political Science Review" 82, no. 2 (1988), pp. 1231-1244.] , Young argued that the Kemeny-Young method was a possible interpretation of one of Condorcet's proposals.

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

Since 1991 the method has been promoted under the name "VoteFair popularity ranking" by Richard Fobes. [Richard Fobes, "The Creative Problem Solver's Toolbox", (ISBN 0-9632-2210-4), 1993, pp. 223-225.]

Notes

External links

* [http://www.VoteFair.org VoteFair.org] — A website that calculates Kemeny-Young results. For comparison, it also calculates the winner according to plurality, Condorcet, Borda count, and other voting methods.