Prove that every Smith-fair method is Condorcet-fair.

To prove that every Smith-fair method is Condorcet-fair, we need to understand and connect the definitions of Smith-fairness and Condorcet-fairness. Here’s a detailed explanation:

### Definitions

1. **Smith Set:**
   - The smallest set of candidates such that every candidate in the set can win a majority against any candidate outside the set in head-to-head matchups.

2. **Smith-Fair Method:**
   - A voting method that always selects a candidate from the Smith set as the winner.

3. **Condorcet Candidate:**
   - A candidate who wins in head-to-head matchups against every other candidate.

4. **Condorcet-Fair Method:**
   - A voting method that always selects the Condorcet candidate as the winner if one exists.

### Proof

To prove that every Smith-fair method is Condorcet-fair, we need to show that if a Condorcet candidate exists, a Smith-fair method will always select this candidate as the winner. The logical steps are as follows:

1. **Existence of a Condorcet Candidate:**
   - Assume a Condorcet candidate \( C \) exists in an election.

2. **Condorcet Candidate and Smith Set:**
   - Since \( C \) wins against every other candidate in head-to-head matchups, there is no candidate outside the set \(\{ C \}\) that can beat \( C \).
   - Therefore, \(\{ C \}\) must be the Smith set (or at least part of the Smith set).

3. **Smith-Fair Method's Selection:**
   - A Smith-fair method ensures that the winner is always from the Smith set.
   - Given that the Condorcet candidate \( C \) is either the sole member or a member of the Smith set, a Smith-fair method must consider \( C \).

4. **Condorcet-Fairness of Smith-Fair Methods:**
   - If a Smith-fair method always selects a candidate from the Smith set and the Condorcet candidate is within this set, the method will select the Condorcet candidate.
   - Hence, the Smith-fair method will select \( C \) as the winner.

Therefore, every Smith-fair method is Condorcet-fair because it guarantees that a Condorcet candidate, when one exists, is within the Smith set and thus will be selected as the winner by the method.

### Summary
- A Smith-fair method ensures the winner is always from the Smith set.
- Since the Condorcet candidate is within the Smith set, the method will select the Condorcet candidate if one exists.
- Thus, every Smith-fair method is inherently Condorcet-fair.

Would you like further details or have any questions?

### Related Questions

1. What are examples of voting methods that are Smith-fair?
2. How does the concept of Smith fairness handle ties?
3. Can a voting method be Condorcet-fair but not Smith-fair?
4. What is the significance of the Smith set in practical elections?
5. How does a Smith set handle scenarios with no Condorcet candidate?
6. What are some real-world applications of Condorcet-fair methods?
7. Are there any limitations to Smith-fair methods in complex elections?
8. How do different voting systems compare in terms of fairness criteria?

### Tip
Understanding the hierarchical relationship between fairness criteria in voting methods can help in designing systems that ensure the most broadly acceptable candidates are elected.

Explore the proof that every Smith-fair method is Condorcet-fair in voting theory. Understand the definitions of Smith-fairness and Condorcet-fairness, and how Smith sets and Condorcet candidates play a crucial role. Suitable for advanced mathematics enthusiasts and voting theory researchers.

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