# [EM] Completion methods for Smith Sets

Michael Rouse mrouse at cdsnet.net
Sun Jun 17 21:40:29 PDT 2001

```While I was trying to figure out ways to use Approval voting in Smith sets
as a completion method, I came up with a completion method that has
probably already been independently discovered half a dozen times or so
(grin). Perhaps someone will recognize it by the following longwinded
description:

As a completion method for a single-member election: Choose the Condorcet
winner, if one exists. Otherwise, find the Smith set (which I usually see
defined as the smallest set where none of the candidates outside the set
beat any of the candidates within the set in a pairwise contest). Throw out
all candidates not in the Smith set.

Once you have the candidates in the Smith set, rank them, starting with the
candidate that has the highest number of first place votes (the plurality
winner), then the winner of the combined first and second place votes, then
the winner of the first, second, and third place votes together, continuing
until you have one fewer candidate on the list than you started with. Don't
count last place rankings; if there are remaining places on the "winner"
list, start with the second candidate on the "plurality" list and continue
down the line. Ties are ok unless there is one spot remaining; in that
case, choose the pairwise winner between the two candidates. Throw out the
loser, adjust the rankings, and begin the process again. Continue until you
have one candidate.

For a multi-candidate Condorcet completion method, find the largest inner
unbeaten set and the smallest unbeaten outer set that encloses the number
of positions available -- in other words, C<=N for the inner set (where C
is the number of candidates, and N is the number of positions in the winner
set) and C>=N for the outer set.

If the inner and outer set are the same, those candidates are elected, and
we are done. If they are different, the ones in the inner set are
automatically elected, and the ones between the inner and outer set are run
through the single-member method above, dropping one candidate at a time
until the number of candidates is reduced to its final number.

For now I won't worry about voting problems with truncated votes or tied
preferences. I really just wanted to see if someone knew what this method
is called. If anyone knows its name and what benefits/drawbacks it might
have, please let me know. I know it's probably not monotonic -- if someone
has an example, that would be great as well, as would any other info. Thanks!

Mike Rouse
mrouse at cdsnet.net

PS I'll write a bit more in another post where this is described as a
full-fledged method, and not just as a completion method as above. My
apologies to IRVing and Borda supporters, but I feel a voting method should
choose the Condorcet winner if it exists, and should choose from the Smith
set if it does not. If a method like Nanson's does so, maybe this one will
as well; if not, then maybe it can serve as a completion method.

```