A tweak to IRV to make it a Condorcet method

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:

1. Let X be the candidate who is the first preference (among non-eliminated candidates) on the fewest ballots. Eliminate candidate X.
2. If only one candidate remains, elect this candidate and stop.
3. Otherwise go to step 1.

A major downside to IRV is that it isn't a Condorcet method; i.e., IRV can elect a candidate A even if a majority of voters prefer candidate B over candidate A.

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:

1. 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.
2. If only one candidate remains, elect this candidate and stop.
3. Otherwise go to step 1.

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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.

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

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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:

1. The Condorcet winner is never eliminated in step 1 of the procedure:
   1. If X is the Condorcet winner, then a majority of voters prefer X over Y, so Y is eliminated instead of X.
   2. If Y is the Condorcet winner, then there is no majority of voters who prefer X over Y, so X is eliminated, not Y.
2. When only one candidate remains, that candidate must be the Condorcet winner, because the Condorcet winner is never eliminated (as proven above).