Questions: If a candidate wins an election and, in a recount, the only changes are that one or more of the other candidates are removed from the ballot, then that candidate should still win the election. This criterion is called the criterion.

The answer is: irrelevant alternatives criterion.

Explanation for each option:

1. Head-to-head: This criterion, also known as the Condorcet criterion, states that if a candidate would win a one-on-one election against every other candidate, then that candidate should win the overall election. This is not related to the removal of other candidates from the ballot.

2. Monotonicity: The monotonicity criterion states that if a candidate wins an election, and then in a re-election, the only changes are that some voters rank this candidate higher, then the candidate should still win. This criterion does not involve the removal of candidates.

3. Majority: The majority criterion states that if a candidate has more than half of the first-place votes, then that candidate should win the election. This criterion is not about the removal of candidates from the ballot.

4. Irrelevant alternatives: The irrelevant alternatives criterion, also known as the independence of irrelevant alternatives (IIA), states that if a candidate wins an election, and one or more of the other candidates are removed from the ballot, then the original winner should still win. This is the correct criterion for the situation described in the question.