Everything about The Kemeny-young Method totally explained
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's a Condorcet winner, it'll 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 don't 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.
Satisfied 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's 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 can't 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 doesn't 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 isn't in the
Smith set, adding or withdrawing choice X doesn't 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 don't apply to the Kemeny-Young method):
» ;
Independence of irrelevant alternatives
:Adding or withdrawing choice X doesn't change a result in which choice Y is identified as most popular.
» ;
Invulnerability to burying
:A voter can't displace a choice from most popular by giving the choice an insincerely low ranking.
» ;
Invulnerability to compromising
:A voter can't 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) can't 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 don't apply to the Kemeny-Young method):
» ;
Independence of clones
:Offering a larger number of similar choices, instead of offering only a single such choice, doesn't change the probability that one of these choices is identified as most popular.
» ;
Invulnerability to push-over
:A voter can't 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
:An algorithm is known to determine the winner using this method in a runtime that's 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.
History
The Kemeny-Young method was developed by
John Kemeny in 1959.
In 1978
Peyton Young and Arthur Levenglick showed that this method was the unique neutral method satisfying reinforcement and the Condorcet criterion. In another paper, Young argued that the Kemeny-Young method was a possible interpretation of one of
Condorcet's proposals.
Since 1991 the method has been promoted under the name "VoteFair popularity ranking" by Richard Fobes.
Further Information
Get more info on 'Kemeny-young Method'.
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