# [EM] Does Bucklin 2-level satisfy Participation (mono-add-top)?

Ted Stern araucaria.araucana at gmail.com
Wed Jan 4 16:16:20 PST 2012

```On 03 Jan 2012 16:38:56 -0800, Jameson Quinn wrote:
>
> It depends on the "tiebreaker" used when there is are multiple
> majorities at second level. If the tiebreaker is that the most
> second-level votes wins, then I believe that the method meets
> participation. Otherwise, A>B votes can cause B &A (instead of just
> A) to pass the second-level threshold and trigger the tiebreaker;
> and B could win the tiebreaker.

I have never heard of an ER-Bucklin method that did not use highest
total threshold-level approval to pick the winner.

I.e., if there is more than one candidate that has a total
threshold-level approval above the quota, the highest total wins.

If A wins with the first N votes, A could win either in the first
level or second level round.

first round, A would still win.

If A had won the pre-x count only after dropping the threshold to the
second level, then the addition of x A>B votes would be equivalent to
adding the same number of A and B approvals to the second-level
approval totals.  Therefore if A had won pre-x, A would still win
post-x.

To answer Kristofer's point: in a two-level ER-Bucklin method,
mono-add-top is the same as Participation, because there is no way to
add A > B rankings without A having the maximum rating.

Okay, thanks to both of you!  That is encouraging ... that means that
2-level ER-Bucklin gets Steven Brams's seal of approval :-).

Ted

> Jameson
>
> 2012/1/3 Ted Stern <araucaria.araucana at gmail.com>
>
>     I've seen examples in which Bucklin (with equal ratings) fails the
>     Participation criterion, AKA Woodall's mono-add-top criterion for
>     deterministic methods:
>
>     ??"the participation criterion says that the addition of a ballot,
>     ??where candidate A is strictly preferred to candidate B, to an
>     ??existing tally of votes should not change the winner from candidate
>     ??A to candidate B." (from Wikipedia)
>
>     In a Bucklin single-winner election with 3 or more levels, it is
>     possible that in an election in which the quota is not met at the
>     first or second level threshold, candidate A may be selected after the
>     threshold has dropped to the third level, but after adding some number
>     of A > B ballots, B then has enough votes to exceed the quota at the
>     second threshold, thus failing Participation. ??So the extra A > B
>     voters might as well have not shown up.
>
>     However, if there are only two approval levels in the Bucklin
>     election, it appears that this problem could not occur, and the
>     no-show paradox would be avoided. ??The failure above hinges on the
>     fact that lower-ranked B fails to make quota at the 2nd level before
>     the new ballots are cast, but exceeds the quota afterward. ??With
>     levels compressed to two instead of three, B would exceed the quota at
>     the second level threshold initially.
>
>     [Chris Benham has made me aware that ER-Bucklin 2-level still fails
>     the Independence of Irrelevant Alternatives, but that is a different
>     situation.]
>
>     Does anyone know of any 2-level ER-Bucklin Participation failures?
>
>     Ted
>     --
>     araucaria dot araucana at gmail dot com
>
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>

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araucaria dot araucana at gmail dot com

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