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 winner of that order (or list), compare the current candidate at the head 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). 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) |
|---|---|---|---|
|
|
|
|
| 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, we compare Memphis to Nashville. As the table on the right shows, more voters prefer Nashville to Memphis than vice versa. So Nashville survives the comparison and Memphis is eliminated.
In the second round, we compare Nashville to Knoxville. As more voters prefer Nashville to Knoxville, Knoxville is eliminated and Nashville stays in the game.
Finally, we compare Nashville to Chattanooga. Chattanooga 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 in the order. This happens because Nashville is the Condorcet winner and every SPE method passes the Condorcet criterion.