Section 8.2: Voting Systems - How Do We Choose a Winner?
Welcome back to Professor Baker's Math Class! Today, we're diving into the fascinating world of voting systems. Specifically, we will explore different methods for determining a winner in an election and discuss some of their properties and potential pitfalls. Let's get started!
Learning Objectives
- Understand that with three or more candidates, there is no perfect voting system.
- Become familiar with different voting systems used in practice.
- Grasp key concepts like Instant Runoff Voting, the Spoiler Effect, Plurality Voting, Condorcet Winner, and Independence of Irrelevant Alternatives.
Key Concepts
- Voting System: A set of rules under which a winner in an election is determined.
- Plurality Voting: The candidate that receives more votes than any other candidate is the winner. It's simple, but not always the fairest. For example, if four candidates run and 100 people vote, a candidate could win with as few as 26 votes!
The Spoiler Effect
A spoiler candidate is someone who has no realistic chance of winning but whose presence in the election affects the outcome. This can happen when a candidate draws votes away from a similar candidate, leading to a different result.
Example: The 2000 US Presidential Election in Florida. Here's a simplified view:
- George W. Bush: 2,912,790 votes
- Al Gore: 2,912,253 votes
- Ralph Nader: 97,488 votes
Even a small shift of Nader's votes could have changed the outcome, leading to debates about whether he acted as a spoiler.
Preferential Voting Systems
These systems involve voters ranking candidates in order of preference. This can help avoid the spoiler effect and ensure a more representative outcome.
- Top-Two Runoff System: If no candidate receives a majority, a second election is held with only the two highest vote-getters.
- Elimination Runoff System: If no candidate receives a majority, the lowest vote-getter is eliminated, and the votes are redistributed based on the voters' next preference. This repeats until a majority is reached.
Let's say we have 10 voters and three candidates: Alfred, Gabby, and Betty. Here's how they ranked their choices:
| Rank | 4 voters | 4 voters | 2 voters |
|---|---|---|---|
| First Choice | Alfred | Gabby | Betty |
| Second Choice | Betty | Alfred | Gabby |
| Third Choice | Gabby | Betty | Alfred |
Using the elimination runoff system, Betty is eliminated first. The 2 voters who initially chose Betty now have Gabby as their top choice. Gabby now has 6 votes (4+2) and Alfred has 4, so Gabby wins!
Borda Count
The Borda count is a method of ranked balloting that assigns points to each candidate based on their ranking on each ballot. The candidate with the highest total points wins.
How it works: If there are $n$ candidates, the first-place choice receives $n-1$ points, the second-place choice receives $n-2$ points, and so on, with the last-place choice receiving 0 points.
Example: Imagine five friends are deciding what food to order. Their ranked preferences are:
| Ballot | Pizza | Tacos | Burgers |
|---|---|---|---|
| 1 | 2 | 1 | 0 |
| 2 | 2 | 1 | 0 |
| 3 | 2 | 1 | 0 |
| 4 | 0 | 2 | 1 |
| 5 | 0 | 2 | 1 |
The Borda count is calculated as follows:
- Pizza: $2+2+2+0+0 = 6$
- Tacos: $1+1+1+2+2 = 7$
- Burgers: $0+0+0+1+1 = 2$
Even though pizza received a majority of first-place votes, tacos win based on the Borda count.
The Condorcet Winner
A Condorcet winner is a candidate who would beat each of the other candidates in a one-on-one election. However, a Condorcet winner doesn't always exist!
Arrow's Impossibility Theorem
This theorem states that with three or more candidates, there is no voting system (other than a dictatorship) that satisfies both the Condorcet winner criterion and the Independence of Irrelevant Alternatives.
Keep exploring these concepts and remember that understanding voting systems is key to making informed decisions in social choices. Good luck!