§ 1.4 Plurality-with-Elimination (Instant Runoff Voting)

Runoff Voting In many elections, a candidate is required to get a majority of votes in order to be elected. If there are more than two candidates this often does not happen. The typical answer to this dilemma is to eliminate the candidate(s) with the fewest first-place votes and hold a runoff election. Since elections are expensive, one-way to possibly improve on the runoff is to utilize preference ballots. . .

Instant Runoff Voting . . .this method is referred to as instant runoff voting or the plurality-with-elimination method.

Instant Runoff Voting More formally, this method can be described as follows: Round 1. Count the first-place votes for each candidate. If a candidate has a majority of first-place votes then that candidate is the winner. Else, eliminate the candidate with the fewest first-place votes. Round 2. Cross out names of the candidates from the preference schedule and recount the first-place votes. If a candidate has a majority of first-place votes, declare that candidate the winner. Otherwise eliminate the candidate with the fewest first-place votes. Round 3, 4, etc. Repeat the process until there is a candidate with a majority of first-place votes.

Example: Let’s look at the Muppet example again--this time as an instant runoff election . Round 1. There are a total of 55 voters in this election. As seen yesterday, a total of 28 votes is needed for a majority. Since no Muppet has a majority of first-place votes and Kermit has the fewest first-place votes, he is eliminated from the election. Number of voters 21 15 12 7 1st Choice Piggy Gonzo Fozzie Kermit 2nd Choice 3rd Choice 4th Choice

Example: Let’s look at the Muppet example again--this time as an instant runoff election . Number of voters 21 15 12 7 1st Choice Piggy Gonzo Fozzie Kermit 2nd Choice 3rd Choice 4th Choice

Example: With Kermit eliminated the election now looks like this: Round 2. Again, no Muppet has the 28 first-place votes needed for a majority. In this round, Gonzo has the least number of first-place votes and is therefore eliminated from the election. Number of voters 21 15 12 7 1st Choice Piggy Gonzo Fozzie 2nd Choice 3rd Choice

Example: Now with Gonzo also eliminated we have: Number of voters 21 15 12 7 1st Choice Piggy Gonzo Fozzie 2nd Choice 3rd Choice

Example: Now with Gonzo also eliminated we have: Round 3. Now, in the third round, we see that Fozzie has a total of 34 first-place votes--therefore he would be elected as our new CEO under the Instant Runoff system. Number of voters 21 15 12 7 1st Choice Piggy Fozzie 2nd Choice

Example: The Springfield Republican primary (part II) Krusty the Clown, Sideshow Bob, Dracula and Mr. Burns are running in the primary to be Springfield’s congressional representative. Suppose the vote breaks down like this: Round 1. There are a total of 33 votes cast in this primary, so a total of 17 first-place are needed for victory. Krusty has a total of 18 first-place votes and is therefore the winner. This is an example of the fact that the plurality-with-elimination method satisfies the Majority Criterion. Number of voters 18 9 6 1st Choice Krusty Bob Dracula 2nd Choice Burns 3rd Choice 4th Choice

The Plurality-with-Elimination Method What’s wrong with this method?

Example: In the third century BCE the cities of Athens, Rome and Carthage are holding a vote to find out which is the most esteemed. The contest will be decided by a vote of 29 leaders, generals and philosophers. (This is basically a sillier version of Example 1.6 from the text.) Rome Athens Carthage

Example: Suppose a ‘straw poll’ is taken just prior to the vote with the following results: If we held a plurality-with-elimination election now, Rome would be eliminated in the first round. In round 2, Rome’s 8 votes would be allocated to Carthage which would then have a majority. Suppose the results of this poll were leaked and word goes out that Carthage is certain to win. As a result of this, the last four voters in the table shift their votes to Carthage. . . Number of voters 7 8 10 4 1st choice Athens Rome Carthage 2nd choice 3rd choice Rome Athens Carthage

Example: . . .the result of which can be seen in the preference schedule below. Number of voters 7 8 14 1st choice Athens Rome Carthage 2nd choice 3rd choice If we now hold a plurality-with-elimination election Athens will be eliminated in the first round. In the second round Rome will receive those 7 votes--enough for a majority and hence, victory. Rome Athens Carthage

The Monotonicity Criterion If choice X is a winner of an election and, in a reelection, the only changes in the ballots are ones that only favor X, then X should remain a winner of the election.

The Plurality-with-Elimination Method What’s wrong with this method? It violates the Monotonicity Criterion. . .

Example: In our Muppet example, is there a Condorcet candidate? Yes! In any head-to-head match-up, Kermit would have the majority of first-place votes. As we saw earlier, however, the plurality-with-elimination method yielded Fozzie as our winner. Number of voters 21 15 12 7 1st Choice Piggy Gonzo Fozzie Kermit 2nd Choice 3rd Choice 4th Choice

The Plurality-with-Elimination Method What’s wrong with this method? It violates the Monotonicity Criterion. . . . . .and the Condorcet Criterion.