Stable Voting

Stable Voting is a Condorcet method devised by Wesley H. Holliday and Eric Pacuit. It obeys the stability for winners with tiebreaking criterion:

If an alternative X wins after another alternative Y is eliminated from the election, and X beats Y pairwise, then X must still win with Y included unless there exists some other candidate X' with the same claim to being the winner. In such a case, a tiebreaker may choose between X and X'.

It also has a very low tie rate, making it useful for elections with a much larger number of candidates than voters.

The method works by calculating the margin of victory for every candidate against every other candidate. We say that the margin of victory for A vs. B is the number of votes for A minus the number of votes for B. In the case that B gets more votes than A, the margin is negative. Place every pairwise matchup of candidates in a list (A vs. B and B vs. A are separate and both on the list) and sort them from largest margin to smallest. Starting at the top pair, say X vs. Y, create a new election by removing Y from all the ballots and see who the Stable Voting winner is in that new election. If the winner in that election is X, then X also wins this election. Otherwise, move on to the next pair and repeat the process.

Technically, the above description is for "Simple Stable Voting." The official definition only considers negative margins for A vs. B only if there is a beatpath from A to B where every margin in the path is at least as large as the margin of B vs. A. However, absent ties in margins it remains an open question whether this restriction ever produces a different winner.

Example

Example (3 candidates):

30 ABD

25 DAB

23 DBA

21 BDA

The margin list is as follows:

D vs A: 39

A vs B: 13

B vs D: 3

D vs B: -3

B vs A: -13

A vs D: -39

The top entry is D vs. A. This means that if D would win the election were A not running, then D should also win this election. To check, let's remove A from all ballots and see the result:

51 BD

48 DB

D loses the majority head-to-head, so D does not win. Thus, we must move on to the next highest margin of victory. A vs B is now the top, so we must remove B and see if A wins:

30 AD

69 DA

A loses, so we move on to B vs D and see if B wins when D is removed. Removing D yields the following:

55 AB

44 BA

B loses. We now have to move on to the negative margins, which is what we expected since there was no Condorcet winner. In order for us to use D vs B, there must be a beatpath from D to B with a margin of more than 3. We see that D beats A by 39 votes and A beats B by 13 votes, so we have satisfied this rule. Now we must check if D will win when B is removed. As before, removing B yields the following:

30 AD

69 DA

D wins the election without B in it, and therefore D wins the election overall.

Criteria Compliance

It passes the Smith criterion and the Condorcet loser criterion. It fails the monotonicity criterion.

There is a web page available for running Stable Voting elections: https://stablevoting.org/.

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