# Random Pair

Random Pair is a voting system where two candidates are drawn uniformly at random, and then the winner is the candidate who beats the other one pairwise.

If the two candidates are tied, the process is repeated unless every pairwise contest is a tie.

## Properties

Random Pair satisfies the monotonicity criterion, Condorcet loser criterion, and independence of irrelevant alternatives. It is strategy-proof, as shown by Gibbard's 1978 theorem.

However, it fails the majority criterion, mutual majority criterion, Condorcet criterion, and has a teaming incentive.

## 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

The results would be tabulated as follows:

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%
• [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
• [NP] indicates voters who expressed no preference between either candidate

Each cell in the matrix is equally likely to be chosen, and the winner of the two candidates of a cell wins the election if that cell is chosen. By labeling each cell with the winner, the table becomes:

A
Memphis Nashville Chattanooga Knoxville
B Memphis Nashville
Chattanooga
Knoxville
Nashville Nashville
Nashville
Nashville
Chattanooga Chattanooga
Nashville
Chattanooga
Knoxville Knoxville
Nashville
Chattanooga

Nashville is the winner in six cells, Chattanooga in four, Knoxville in two, and Memphis in none.

Thus Nashville wins with 50% probability, Chattanooga with 33.33% probability, and Knoxville with 16.67% probability. Memphis is the Condorcet loser and has no chance of winning.