The Coombs system, also known as the **Coombs rule** or **Coombs method**, is a voting system where voters rank candidates in order of preference. In this system, the candidate with the most "last-place" votes is successively eliminated until one candidate remains. 

To determine if the Coombs system satisfies **Independence of Irrelevant Alternatives (IIA)**, we need to understand what IIA entails:

### Independence of Irrelevant Alternatives (IIA)
The IIA criterion states that if a candidate \( A \) is preferred to candidate \( B \) in a vote, then introducing or removing another candidate \( C \) should not change the outcome of \( A \) being preferred to \( B \). In other words, the relative ranking between \( A \) and \( B \) should remain unaffected by the presence or absence of other candidates.

### Coombs System and IIA
The Coombs system **does not satisfy IIA**. Here's why:
- In the Coombs method, candidates are eliminated based on their number of last-place votes. If a new candidate is introduced or an existing candidate is removed, the distribution of last-place votes can change significantly, potentially altering which candidate is eliminated in each round. This can lead to different final results.
- As a result, the presence or absence of irrelevant candidates can indeed affect the outcome, violating the IIA criterion.

So, to summarize: **The Coombs system does not satisfy IIA** because the elimination process is sensitive to changes in the pool of candidates.

Would you like more details on how this criterion is evaluated in other voting systems?

### Follow-up Questions:
1. What other voting systems satisfy or fail the IIA criterion?
2. How does the Coombs system compare with the Borda count in terms of IIA?
3. What are other criteria that the Coombs system fails or satisfies?
4. Can you provide a detailed example where the Coombs system violates IIA?
5. How does the elimination process in the Coombs system influence strategic voting?

### Tip:
In voting theory, a system that satisfies many fairness criteria often struggles to achieve majority support in real-world applications due to trade-offs between competing principles.

Explore whether the Coombs system satisfies the Independence of Irrelevant Alternatives (IIA) criterion in voting theory. Understand how the elimination process impacts the IIA and compare it with other voting systems. Suitable for undergraduate students studying social choice theory.

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