[EM] Improved Instant Pairwise Elimination
Susan Simmons
suzerainsimmons at outlook.com
Thu Jul 8 18:04:26 PDT 2021
Simplified version:
At each elimination step eliminate from among the remaining candidates the pairwise loser between (1) the candidate whose maximum margin of support is minimal, and (2) the loser in the pairwise contest with fewest losing votes.
(1) is the Condorcet Loser if there is one, else (arguably) the candidate closest to that distinction in the sense that its max win margin is as close to zero as possible. [A candidate will have at least one positive margin of sjpport if and only if it is not the CL.] So let's call it the NCL, Nearest thing to Condorcet Loser.
(2) is the Gross Loser from Benham's version of Reynaud, BRGL.
So Improved Instant Pairwise Elimination eliminates (at each step) the pairwise loser between the NCL and the GL.
If the NCL and the GL are different, then the eliminated candidate is beaten by the other one. If they are the same, then the eliminated candidate is the GL, which is never a Condorcet candidate. In neither case is a Condorcet candidate eliminated ... so the method meets the Condorcet Criterion.
Note that the pairwise margins matrix is simply the pairwise support matrix minus its transpose, so the whole thing is efficiently precinct summable.
My preferred version of the pairwise support matrix is this: the (i, j) entry is the number of ballots on which candidate i is ranked strictly ahead of j, plus the number of ballots on which both are ranked Top, plus half the number of ballots on which both are ranked together (i.e. equal to each other) strictly between Bottom and Top.
This convention for equal rankings makes good sense, for example, when interpreting the diagonal elements of the matrix as implicit approvals, and in other similar contexts.
Thanks for listening!
Sent from my MetroPCS 4G LTE Android Device
-------- Original message --------
From: Susan Simmons <suzerainsimmons at outlook.com>
Date: 7/8/21 12:18 PM (GMT-08:00)
To: election-methods at lists.electorama.com
Subject: Improved Instant Pairwise Elimination
At each elimination step IPE eliminates the Condorcet Loser if there is one, otherwise it eliminates the loser of the pairwise contest with the most winning votes.
We propose eliminating (at each elimination step among those not eliminated previously) the Condorcet Loser if there is one, else the pairwise loser between (1) the candidate whose maximum support for any of its pairwise wins is minimal, and (2) the loser from the pairwise contest with the fewest losing votes (i.e. the Gross Loser).
In other words eliminate the CL when there is one, otherwise eliminate whichever is less preferred... the GL or the closest thing to a CL.
Is this better than BRGL which simply eliminates the GL at each step?
Yes and no. On the one hand it is more aggressive and thorough about getting rid of the least desireable remaining candidate as soon as possible. On the other hand it is probably a harder sell to a public lacking patience in these matters ... who tend to assume that the order of elimination of "losers" doesn't make much difference, if any.
Sent from my MetroPCS 4G LTE Android Device
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