23 Iterative Methods
An iterative method is a voting method where each voter submits a ranking of the candidates. In these methods, poorly performing candidates are removed in successive rounds until one candidate emerges as the absolute majority winner (in the reduced profile). Iterative methods differ in how they decide which candidates to eliminate at each step. Below are two well-known examples of iterative methods:
Instant Runoff (aka Ranked Choice): Iteratively remove all candidates with the fewest number of voters who rank them first, until there is a candidate with a majority of first-place votes. If, at some stage of the removal process, all remaining candidates have the same number of voters who rank them first (so all candidates would be removed), then all remaining candidates are selected as winners.
Coombs: Iteratively remove all candidates with the most number of voters who rank them last, until there is a candidate with a majority of first-place votes. If, at some stage of the removal process, all remaining candidates have the same number of voters who rank them last (so all candidates would be removed), then all remaining candidates are selected as winners.
We will illustrate these two methods on the following profile:
\[\begin{array}{cccc} 7 & 5 & 4 & 3 \\\hline a & b & d & c \\ b & c & b & d \\ c & d & c & a \\ d & a & a & b \\ \end{array}\]There are 21 voters in this election. To be the absolute majority winner, a candidate must be ranked first by at least 11 voters. The first step in applying any iterative method is to determine how many voters rank each candidate in first place. In other words, we calculate the Plurality scores of the candidates:
\[\begin{array}{rcl} PluralityScore(a) & = & 7\\ PluralityScore(b) & = & 5\\ PluralityScore(c) & = & 3\\ PluralityScore(d) & = & 4\\ \end{array}\]Since no candidate is ranked first by more than half of the voters, we proceed by iteratively removing candidates based on their performance.
We begin with Instant Runoff Voting. Candidate \(c\) has the fewest first-place votes, so \(c\) is removed from the profile. The reduced profile, after \(c\) has been eliminated, is as follows:
\[\begin{array}{cccc} 7 & 5 & 4 & 3 \\\hline a & b & d & d \\ b & d & b & a \\ d & a & a & b \\ \end{array}\]In this reduced profile, the Plurality scores are:
\[\begin{array}{rcl} PluralityScore(a) & = & 7\\ PluralityScore(b) & = & 5\\ PluralityScore(d) & = & 7\\ \end{array}\]Again, none of the candidates have more than half of the first-place votes. Therefore, we remove the candidate with the fewest first-place votes. In this case, candidate \(b\) is eliminated. The reduced profile, after \(b\) has been removed, is as follows:
\[\begin{array}{cccc} 7 & 5 & 4 & 3 \\\hline a & d & d & d \\ d & a & a & a \\ \end{array}\]The Plurality scores in this reduced profile are:
\[\begin{array}{rcl} PluralityScore(a) & = & 7\\ PluralityScore(d) & = & 14\\ \end{array}\]Since \(d\) has more than half of the first-place votes in the final round, \(d\) is the winner of the original profile according to Instant Runoff Voting.
Next, we apply Coombs’ method to the original profile. To iteratively remove candidates according to Coombs, we must find the number of voters that rank each candidate in last place. In the original profile, the number of last-place votes for each candidate is as follows:
\[\begin{array}{rcl} \text{Number of voters that rank } a \text{ in last place }& = & 9\\ \text{Number of voters that rank } b \text{ in last place } & = & 3\\ \text{Number of voters that rank } c \text{ in last place } & = & 0\\ \text{Number of voters that rank } d \text{ in last place } & = & 7\\ \end{array}\]Candidate \(a\) has the most last-place votes, so \(a\) is removed from the profile. The reduced profile, after \(a\) has been eliminated, is as follows:
\[\begin{array}{cccc} 7 & 5 & 4 & 3 \\\hline b & b & d & c \\ c & c & b & d \\ d & d & c & b \\ \end{array}\]In this reduced profile, the Plurality scores are:
\[\begin{array}{rcl} PluralityScore(b) & = & 12\\ PluralityScore(c) & = & 3\\ PluralityScore(d) & = & 4\\ \end{array}\]Since \(b\) has more than half of the first-place votes in the reduced profile, \(b\) is the winner of the original profile according to Coombs’ method.
23.1 Exercises
- Determined the Instant Runoff and Coombs winner(s) in the following profile:
- Determined the Instant Runoff and Coombs winner(s) in the following profile:
- True or False: There is always a unique winner according to the Instant Runoff voting method.
- True or False: There is always a unique winner according to the Coombs voting method.
- True or False: In any profile, if a candidate \(a\) is the absolute majority winner, then \(a\) is the unique Instant Runoff winner.
- True or False: In any profile, if a candidate \(a\) is the absolute majority winner, then \(a\) is the unique Coombs winner.