[EM] Pinning down the meaning of "IRV without elimination"

[Kevin Venzke]

Hello,

Occasionally someone has the idea: Why don't we keep IRV but just not eliminate the
candidates? The idea seems like it could have obvious advantages, but I'm not sure I've
ever seen anyone define how it would work. Have I just missed it?

Here's a definition I think works best. I call it "NEIRV type 1":

1. Each voter starts at the top of their ballot, and goes down, casting a vote for each
candidate, but stopping once a vote has been cast for a candidate who has not been
"eliminated" (which is in quotes because candidates are not really eliminated).
2. If the vote leader has votes from a full majority of the voters (remembering there is no
exhaustion here), then they are elected. End of method.
3. If all candidates have been "eliminated," then also elect the vote leader.
4. If only one candidate has not been "eliminated," and that candidate is the vote leader,
then also elect the vote leader.
5. "Eliminate" the vote loser among non-eliminated candidates and go to step 1.

Step 4 might be controversial. In this case, and only here, some voters are allowed to say
in effect, "we don't want to use our preferences below the vote leader." But presumably,
earlier, other voters might have wished to say the same. (It's harder to calculate, earlier
in the process, whether this is desirable. And it probably makes a mess of the whole
method to try to allow it in earlier rounds.)

Simply deleting step 4 results in, let's say, "type 2." But in simulations type 2 seems to
have worse Condorcet efficiency than IRV, and truncation incentive nearly as bad as Bucklin.

A few notes on "type 1" based on simulations using four candidates and five blocs:

Better Condorcet efficiency than IRV, but slightly worse than a simple approval-elimination
runoff (AER).

Seems to fail Condorcet Loser but satisfy mutual majority.

Monotonicity (Mono-raise) failures are about twice as frequent as in IRV.

Compromise incentive: Better than IRV, but not as good as Bucklin, and further still from
matching AER's low incentive.

Truncation incentive: Half as bad as Bucklin, but not as good as AER.

Burial incentive: It's low. Much much lower than AER.

Likelihood that the winner has a majority against them: Better than IRV and even Bucklin,
but twice as bad as AER.

Minimal defense: Seems to satisfy it (as do Bucklin and AER). (This is the property that a
majority with a preference in common can defeat a candidate simply by not ranking them over
any other candidate.)

Anyway, do tell if you think there's a different way of doing IRV without eliminations.

Kevin
votingmethods.net