# Sequential pairwise elimination

Sequential pairwise elimination is a class of voting methods devised by Forest Simmons. These methods elect from the Banks set and thus pass the Condorcet criterion and never elect covered candidates.

An SPE method, emulating legislative procedure, works by first determining a base social order by some method (e.g. Minmax or Range). Then starting with the loser of that order (or list), compare the current candidate at the end of the list with the candidate next to it. Replace the two adjacent candidates in the list with the candidate who beats the other pairwise, then repeat. The candidate who is left standing at the end is the winner.

In effect, each candidate can be considered a proposal (a bill or an amendment). Starting with the weakest proposal, the "legislature" repeatedly votes whether to keep the current bill or to replace it with an amended bill. At the end of the procedure, the last accepted amended bill wins.

## Example

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of Tennessee
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis
Pairwise Election Results
A
Memphis Nashville Chattanooga Knoxville
B Memphis [A] 58%
[B] 42%
[A] 58%
[B] 42%
[A] 58%
[B] 42%
Nashville [A] 42%
[B] 58%
[A] 32%
[B] 68%
[A] 32%
[B] 68%
Chattanooga [A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 17%
[B] 83%
Knoxville [A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 83%
[B] 17%

Suppose that the base order is according to first past the post. Its order is Memphis > Nashville > Knoxville > Chattanooga.

In the first round, starting at the end of the list, we compare Knoxville to Chattanooga. As the table on the right shows, more voters prefer Chattanooga to Knoxville than vice versa. So Chattanooga survives the comparison and Knoxville is eliminated.

In the second round, we compare Chattanooga to Nashville. As more voters prefer Nashville to Chattanooga, Chattanooga is eliminated in favor of Nashville.

Finally, we compare Nashville to Memphis. Memphis is eliminated.

Thus Nashville is the winner.

In this example, no matter what the order is, Nashville eventually becomes the incumbent and defeats all subsequent challengers. This happens because Nashville is the Condorcet winner and every SPE method passes the Condorcet criterion.