Ranked Pairs Voting Explained Using The Ten Hundred Most-Used Words
Leaders have the power to make big, important changes that touch us all. To make sure the changes are good for everyone, we need to make sure we have the best leaders. To make sure we have the best leaders, we must use the best way to agree on who the leaders should be.
The way we choose leaders now is OK, but it is not the best. A much better way is used in games between teams. Say you have a group of teams. The best way to figure out which team is the best is to have every team play a game against every other team in the group. We can figure out who the best team is by looking at the history of games between each pair of teams. If one of the teams beats all of the other teams, then they are the best team.
To be sure, we can ask all the teams to play each other again, and again. The more rounds the teams play against each other, the more sure we can be about which team is best. Even the best teams have bad days, so the best team might not win every single game against every single team. However, the best team will win more often than they lose. We can figure out who the best team is by looking at the history of games between each pair of teams. We look for a team that wins more often than they lose in every pair. If we find a team like that, then they are the best team.
We can use the same way of choosing the best team to choose the best leaders. We only need to make two small changes. First, just change the word "team" to "leader" in the words above. Second, we think about games in a different way. Chance is very important in a normal game. Choosing a leader is more important than a normal game, so instead of leaving it up to chance, you get to decide which leaders win and lose games. You decide which leaders win and lose games by writing down the names of the leaders in order. First, write down the leader who you think is best. Write the next-best leader second. Keep writing down leaders like this until you have written down the worst leader last.
Your ordering of leaders has a lot of information -- it is like a whole round of games between all the leaders! The first leader you wrote down wins a game against all other leaders. The second leader you wrote down loses a game to the first leader you wrote down, but they win a game against all of the other leaders. Finally, the last leader you wrote down loses a game to all of the other leaders.
We look at the orderings of leaders from all the other people and add up how many times each leader won and lost. We figure out who the best leader is by looking at the history of games between each pair of leaders. We look for a leader that wins more games than they lose in every pair. If we find a leader like that, they are the best leader for the people.
Let's talk about teams again, but remember that the same steps below work for choosing leaders too.
You can almost always find the best team using this way, but there is a very small chance that it does not work. Sometimes the best team in the group loses more games than it wins against one team, even though it wins more games than it loses when you look at the team's history against every other team. There is still a great way to find the best team in this strange situation. It takes more work, but it is fun -- we get to draw a picture!
First we look at the history of games between each pair of teams. We pick a pair of teams and we figure out a number. We find the number of times that the winning team won, and take away the number of times that the winning team lost. We will use this number later to figure out how important the pair is. We add the pair to a table. Each row of the table has the pair's winning team name, the pair's losing team name, and the number we just figured out. When we are done adding all the pairs to the table, we sort the rows using the number. The pair with the biggest number should be on top of the table because it is the most important pair. The pair with the smallest number should be on the bottom of the table because it is the least important pair.
Next we draw a big circle. We write the names of the teams outside of the circle. Pretend you are drawing an old round time piece, but instead of writing numbers with an even distance between them, you are writing team names with an even distance between them. Now remove the circle from the drawing, but keep the team names.
Now we look at the pair on top of the sorted table. This is the most important pair. We find the pair's winning team in our drawing. We find the pair's losing team in the drawing. We draw a line from the winning team to the losing team. We give the line a pointy end so that it points at the losing team. We cross off the pair on top of the table.
Now we look at the next pair on top of the table. This is the second-most important pair. Once again, we draw a pointy line from the winning team the losing team. We cross off the pair on top of the table.
We repeat these steps for the third pair on top of the table. Now we need to add a step. We check to make sure that our pointy lines do not circle back on themselves. Follow the lines, starting with the ends that are not pointy and moving to the ends that are pointy. If you can follow the pointy lines from a team name and arrive back at the same team name, then the pointy lines are circling back on themselves. If that happens, then remove the line you just drew from the drawing. It is OK to remove this line because its pair is less important than the pairs whose lines we already drew.
Repeat these steps for all the pairs that remain in the table. When you are finished, there will be a team name that has no lines pointing at it. This team is the best, so it wins!
Explained in a normal way here.
Idea from XKCD Thing Explainer.
Written using this thing.
Tideman, T.N. Independence of clones as a criterion for voting rules. Soc Choice Welfare 4, 185–206 (1987). https://doi.org/10.1007/BF00433944
Zavist, T.M., Tideman, T.N. Complete independence of clones in the ranked pairs rule. Soc Choice Welfare 6, 167–173 (1989). https://doi.org/10.1007/BF00303170