Smith set facts for kids

The Smith set is a special group of candidates in an election. It helps us find the strongest candidates. Imagine an election where everyone votes for their favorite. The Smith set is the smallest group of candidates where everyone in that group is preferred over anyone outside the group, when you compare them head-to-head. It's like finding the "champions" who can beat all the "challengers."

This idea is also called the top cycle. If a voting system always picks a candidate from the Smith set, it's considered very fair. We say it passes the "Smith criterion."

Sometimes, a group of candidates where everyone inside the group beats everyone outside is called a dominating set. The Smith set is just the smallest possible dominating set.

What Makes the Smith Set Special?

• It Always Exists: No matter how many candidates or voters there are, you can always find a Smith set.
• It Can Have Many Candidates: Sometimes, the Smith set might have more than one person. This can happen if candidates tie, or if there's a "cycle" where A beats B, B beats C, but C beats A.
• The Strongest Winner is in It: If there's a candidate who can beat every other candidate in a head-to-head match (called a Condorcet winner), that person will always be the only one in the Smith set.
• Fairness: The Smith set only includes candidates who are truly preferred by the majority of voters.

How to Find the Smith Set

Finding the Smith set involves comparing every candidate against every other candidate. You can think of it like a tournament where each candidate plays a "match" against every other candidate.

One way to find it is to rank candidates by how many head-to-head matches they win. Then, you look for the smallest group of top-ranked candidates who can beat everyone not in their group.

Let's look at an example to make it clearer:

Imagine an election with candidates A, B, C, D, E, F, and G. We compare them head-to-head:

Let's figure out the Smith set step-by-step:

• First, we see that candidate A loses to C. This means A cannot be the only member of the Smith set. We need to include C. So, A, B, and C might be in the Smith set.
• Next, we check if C loses to anyone outside this group. Yes, C loses to D. So, D must also be included in our potential Smith set. Now we have A, B, C, and D.
• Then, we check if D loses or ties with anyone outside this group. Yes, D ties with E. So, E must also be included. Our group is now A, B, C, D, and E.
• Now, let's check this group (A, B, C, D, E). Do all of them beat everyone outside the group (F and G)? Yes, they do!
• Because A, B, C, D, and E all beat F and G in head-to-head matches, and this is the smallest group that does this, the Smith set is A, B, C, D, and E.

Related Ideas

• Condorcet method: This is another way to find a strong winner in an election.
• Condorcet criterion: A rule that says if a candidate can beat everyone else head-to-head, they should win the election.