27  Comparing Voting Methods

We have introduced seven different voting methods:

1. Plurality (Chapter 22)
2. Borda (Chapter 22)
3. Instant Runoff (Chapter 23)
4. Coombs (Chapter 23)
5. Copeland (Section 26.1)
6. Minimax (Section 26.2)
7. Split Cycle (Section 26.3)

These methods share a common feature: they require the same input from voters—a ranking of the candidates (except Plurality, which only requires voters to select a single candidate). Later in this book, we will explore other types of voting methods, such as approval voting and score voting, which use different types of ballots.

As we have seen, different voting methods can produce different winners in the same election. For instance, Plurality, Borda, Instant Runoff, and Coombs may fail to elect a Condorcet winner even when one exists. On the other hand, Copeland, Minimax, and Split Cycle are Condorcet consistent, meaning they always elect the Condorcet winner whenever one exists.

This illustrates that the choice of voting method can significantly impact the outcome of an election. A natural question arises: How often do voting methods produce different results? More precisely, if an election is randomly generated, what is the probability that the winner under one method differs from the winner under another?

To address this question, a simulation was conducted in which 50,000 random elections were generated for each combination of the number of candidates and voters. The reported value represents the percentage of elections where the winning set under one voting method differs from the winning set under another method.

3 Candidates, 10 Voters

3 Candidates, 11 Voters

3 Candidates, 1000 Voters

3 Candidates, 1001 Voters

4 Candidates, 10 Voters

4 Candidates, 11 Voters

4 Candidates, 1000 Voters

4 Candidates, 1001 Voters

5 Candidates, 10 Voters

5 Candidates, 11 Voters

5 Candidates, 1000 Voters

5 Candidates, 1001 Voters

So, which voting method is the best? Which method should be used in large-scale political elections? What about smaller-scale elections? There is no one-size-fits-all answer to these questions. When choosing a voting method, several factors come into play:

• Pragmatic considerations: Is the procedure easy to use? Is it legally permissible? Ease of use is particularly important. Despite its flaws, Plurality rule remains the most widely used method due to its simplicity.

• Information required from voters: What type of information does the voting method require from the ballots? For instance, does it ask voters to select a single candidate, rank all candidates, or provide information about the intensity of their preferences?

• Axiomatic principles: How do the different voting methods align with normative principles of group decision-making?

For the remainder of this text, we focus on the normative principles of group decision-making. Examining these principles provides insight into the trade-offs inherent in each voting method.

27.1 Exercises

The interactive app below allows you to explore different voting methods, calculate margins, identify the Condorcet winner and loser, and visualize the margin graph. The app generates random elections with a specified number of candidates and voters. You can use the sliders (accessible by selecting the triangle button at the top left) to adjust the number of candidates and voters and calculate the winning sets for different voting methods.