tl;dr: Rather than eliminating the bottom-most candidate, instead eliminate the loser of a head-to-head matchup of the bottom-most two candidates, ensuring that the Condorcet winner (if one exists) is never eliminated.

In instant-runoff voting (IRV), each voter ranks the candidates in order of preference. The winner is decided by the following procedure:- Let
*X*be the candidate who is the first preference (among non-eliminated candidates) on the fewest ballots. Eliminate candidate*X*. - If only one candidate remains, elect this candidate and stop.
- Otherwise go to step 1.

A tweak to IRV to make it a Condorcet method is to change Step 1 of the above procedure to eliminate the loser of a head-to-head matchup of the bottom-most two candidates. The tweaked procedure is as follows:

- Let
*X*be the candidate who is the first preference (among non-eliminated candidates) on the fewest ballots. Let*Y*be the candidate who is the first preference (among non-eliminated candidates) on the second-fewest ballots. If a majority of voters prefer*X*over*Y*, then eliminate*Y*; otherwise, eliminate*X*. - If only one candidate remains, elect this candidate and stop.
- Otherwise go to step 1.

Example:

Percentage of voters | Ballot ranking |

38% | (1) Alice, (2) Bob, (3) Charles |

15% | (1) Bob, (2) Alice, (3) Charles |

15% | (1) Bob, (2) Charles, (3) Alice |

32% | (1) Charles, (2) Bob, (3) Alice |

- With IRV, Bob (with 15%+15%=30% of ballots ranking him the top choice) is eliminated in the first round.
- With tweaked-IRV, instead there is a head-to-head matchup of Bob against Charles. Bob is preferred to Charles by 38+15%+15%=68% of voters, so Charles is eliminated in the first round instead of Bob.

More info: https://electowiki.org/wiki/Bottom-Two-Runoff_IRV

Definitions:

- A
*Condorcet winner*is a candidate who beats every other candidate in a head-to-head election. I.e., if*X*is a Condorcet winner, then, for every other candidate*Y*, a majority of voters prefer*X*over*Y*. - An election method is a
*Condorcet method*iff it always elects a Condorcet winner if one exists.

**Proposition:** Assume that a Condorcet winner exists. Then the tweaked-IRV method elects the Condorcet winner.
**Proof:**

- The Condorcet winner is never eliminated in step 1 of the procedure:
- If
*X*is the Condorcet winner, then a majority of voters prefer*X*over*Y*, so*Y*is eliminated instead of*X*. - If
*Y*is the Condorcet winner, then there is no majority of voters who prefer*X*over*Y*, so*X*is eliminated, not*Y*.

- If
- When only one candidate remains, that candidate must be the Condorcet winner, because the Condorcet winner is never eliminated (as proven above).