2. 1 The Mathematics of Voting
1.1 Preference Ballots and Preference Schedules
1.2 The Plurality Method
1.3 The Borda Count Method
1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)
1.5 The Method of Pairwise Comparisons
1.6 Rankings

3. Extended Rankings Method
Quite often it is important not only to know who wins the election but also to know who comes in second, third, and so on. We need a voting method that gives us not just a winner but also a second place, a third place, and so on - in other words, a ranking of the candidates.
Each of the four voting methods we discussed earlier in this chapter has a natural extension that can be used to produce a ranking of the candidates.

4. Extended Rankings Method
A summary of the results of the Math Club election using the different extended ranking methods is shown in Table The most striking thing about Table 1-21 is the wide discrepancy of results.

5. Extended Rankings Method
While it is somewhat frustrating to see this much equivocation, it is important to keep things in context: This is the exception rather than the rule.
One purpose of the Math Club example is to illustrate how crazy things can get in some elections, but in most real-life elections there tends to be much more consistency among the various methods.

6. Recursive Rankings Method
The idea behind a recursive process is similar to that of a feedback loop: At each step of the process the output of the process determines the input to the next step of the process.
In the case of an election the process is to find the winner and remove the winner’s name from the preference schedule, thus creating a new preference schedule. We then start the process all over again.

7. Recursive Rankings Method
The recursive ranking approach follows a consistent strategy, but the details depend on the voting method we want to use. Let’s say we are going to use the recursive version of some generic voting method X to rank the candidates in an election. We first use method X to find the winner of the election. So far, so good. We then remove the name of the winner on the preference schedule and obtain a new preference schedule with one less candidate on it.

8. Recursive Rankings Method
We apply method X once again to find the “winner” based on this new preference schedule, and this candidate is ranked second. We can continue this process to rank as many of the candidates as we need to.
We will illustrate recursive ranking with a couple of examples, both based on the Math Club election.

9. Example 1.21 The Math Club Election (Recursive Plurality)
Here is how we rank the four candidates in the Math Club election using the recursive plurality method.
Step 1. (Choose the winner and remove.) Table 1-22 shows the original input (the original preference schedule). We already know the winner is A with 14 first-place votes.

10. Example 1.21 The Math Club Election (Recursive Plurality)
Table 1-23 shows the output of this step - the preference schedule when A is removed. This will be the input to the next step.

11. Example 1.21 The Math Club Election (Recursive Plurality)
Step 2. The “winner” of this election is B with 18 votes. Thus, second place goes to B. Table 1-24 shows the result of removing B from Table This will be the input to the next step.

12. Example 1.21 The Math Club Election (Recursive Plurality)
Step 3. (Choose third and fourth places.) The last step is to find the results of the two-candidate election shown in Table Clearly, C wins with 25 votes. Thus, third place goes to C and last place goes to D.

13. Example 1.21 The Math Club Election (Recursive Plurality)
The final ranking of the candidates under the recursive plurality method is shown in Table It is worth noting how different this rank- ing is from the ranking based on the extended plurality method. In fact, except for first place (which will always be the same), all the other positions turned out to be different.

14. Example 1.22 The Math Club Election (Recursive Plurality-with-Elimination)
Be careful to distinguish the two different types of “elimination”. Under the plurality-with-elimination method, candidates are eliminated in rounds until there is a winner left. Under the recursive approach, the winner at each step of the recursion is removed from the preference schedule so that we can move on to the next step. Thus, each step consists of several rounds of elimination and at the end, the removal of the winner. Here is how it works:

15. Example 1.22 The Math Club Election (Recursive Plurality-with-Elimination)
Step 1. (Choose the winner and remove.) Table 1-26 shows the original input (the original preference schedule). After several rounds of elimination, the winner is D.

16. Example 1.22 The Math Club Election (Recursive Plurality-with-Elimination)
Step 1. Table 1-27 shows the output of this step - the preference schedule when D is removed. This will be the input to Step 2.

17. Example 1.22 The Math Club Election (Recursive Plurality-with-Elimination)
Step 2. (Choose second place and remove.) In this election no rounds of elimination are needed, since C has 19 votes (a majority). Thus, second place goes to C.

18. Example 1.22 The Math Club Election (Recursive Plurality-with-Elimination)
Step 2. We now remove C from the preference schedule and get the preference schedule shown in Table 1-28 after like columns are combined. This will be the input to the last step.

19. Example 1.22 The Math Club Election (Recursive Plurality-with-Elimination)
Step 3. (Choose third and fourth places.) The last step is to find the results of the two-candidate election shown in Table Clearly, B wins with 23 votes. Thus, third place goes to B and last place goes to A.

20. Example 1.22 The Math Club Election (Recursive Plurality-with-Elimination)
The final ranking of the candidates under the recursive plurality-with-elimination method is shown in Table Once again, if we compare this ranking with the ranking under the extended plurality-with-elimination method, the differences in how second, third, and fourth places are ranked are striking.

21. Which Produces Better Rankings?
It is clear from the last two examples that other than first place, recursive ranking methods and extended ranking methods can produce very different results. Which one produces better rankings? As with everything else in election theory, there is no simple answer to this question. It is true, however, that in real-life elections, extended ranking methods are almost always used. Recursive ranking methods, while mathematically interesting, are of little practical use.