Explain why a Condorcet candidate is automatically a Smith candidate.

In voting theory, understanding the relationship between Condorcet and Smith candidates helps clarify the principles of fair and representative voting methods. Let’s explore why a Condorcet candidate is automatically a Smith candidate by defining the key terms and then logically connecting them.

### Key Definitions

1. **Condorcet Candidate:**
   - A candidate who, when compared one-on-one with each of the other candidates, is preferred over each of them by a majority of voters.

2. **Smith Set:**
   - The smallest set of candidates such that each member of the set can win a majority against any candidate outside the set in head-to-head contests.

### Logical Connection

To understand why a Condorcet candidate is automatically a Smith candidate, consider the following:

1. **Existence of a Condorcet Candidate:**
   - If a Condorcet candidate exists, this candidate beats every other candidate in one-on-one matchups.

2. **Smith Set Properties:**
   - The Smith set includes all candidates that are unbeatable by any candidate outside the set.
   - By definition, the Smith set is the smallest such set.

### Proof

1. **Condorcet Candidate's Dominance:**
   - By definition, a Condorcet candidate \( C \) wins against every other candidate in one-on-one matchups.

2. **Inclusion in Smith Set:**
   - Since \( C \) beats every other candidate, there is no candidate outside the singleton set \(\{ C \}\) that can beat \( C \).
   - Therefore, the smallest set that meets the Smith criterion must include \( C \).

3. **Smallest Set Verification:**
   - The smallest set containing \( C \) that satisfies the Smith criteria is \(\{ C \}\) itself.
   - Hence, \(\{ C \}\) is the Smith set, making \( C \) a Smith candidate.

Thus, if a Condorcet candidate exists, this candidate is the only member of the Smith set and hence a Smith candidate.

### Summary
- A Condorcet candidate, by virtue of defeating every other candidate in head-to-head comparisons, ensures no candidate outside the singleton set \(\{ C \}\) can win against them.
- This directly satisfies the conditions for being in the Smith set, making any Condorcet candidate automatically a Smith candidate.

Would you like further details or have any questions?

### Related Questions

1. What happens in the absence of a Condorcet candidate?
2. Can there be more than one Smith set in an election?
3. How does the Smith set relate to other voting methods like Borda count?
4. What is a Condorcet cycle and how does it affect the Smith set?
5. How do ranked-choice voting systems handle the Smith set?
6. What are some real-world examples where the Smith set has been applied?
7. How do different voting methods ensure the selection of a candidate from the Smith set?
8. What are the implications of a candidate being in the Smith set but not being a Condorcet candidate?

### Tip
Understanding the Smith set can help design more representative voting systems that better reflect the preferences of the electorate.

Explore why a Condorcet candidate is automatically a Smith candidate in voting theory. Understand the definitions of Condorcet candidates and Smith sets, and their logical connection. Ideal for advanced high school students or those interested in voting theory.

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