EXAMPLE 10 Plurality Runoff

The plurality runoff method is somewhat similar in spirit to the Hare system. In fact, you might wonder if they aren’t just two different descriptions of the same voting system. That is, you might ask if the plurality runoff method and the Hare system always yield the same winner.

The answer is “no,” however, as we now demonstrate. Consider the following sequence of preference list ballots:

With the plurality runoff method, and initially tie with 4 first-place votes each, with 3 for and 2 for . In the runoff between and , the ballots are as follows:

424

With the plurality runoff method, is the winner, defeating in the runoff by a score of 9 to 4.

On the other hand, with the Hare system, we find that the only alternative deleted in the first round is , with only 2 first-place votes. With this deletion of , the ballots are as follows:

A and now have only 4 first-place votes compared to the 5 first-place votes that has. Hence, A and are now deleted, leaving as the winner with the Hare system.

Alas, the plurality runoff method also does not satisfy monotonicity. Exercise 25 (page 435) asks you to verify this.