[EM] Participation criterion and Condorcet rules

[John]

Current theory suggests Condorcet methods are incompatible with the
Participation criterion:  a set of ballots can exist such that a Condorcet
method elects candidate X, and a single additional ballot ranking X ahead
of Y will change the winner from X to Y.

https://en.wikipedia.org/wiki/Participation_criterion

This criterion seems ill-fitted, and I feel needs clarification.

First, so-called Condorcet methods are simply Smith-efficient (some are
Schwartz-efficient, which is a subset):  they elect a candidate from the
Smith set.  If the Smith set is one candidate, that is the Condorcet
candidate, and all methods elect that candidate.

>From that standpoint, each Condorcet method represents an arbitrary
selection of a candidate from a pool of identified suitable candidates.
Ranked Pairs elects the candidate with the strongest rankings; Schulze
elects a more-suitable candidate with less voter regret (eliminates
candidates with relatively large pairwise losses); Tideman's Alternative
methods resist tactical voting and elect some candidate or another.

Given that Tideman's Alternative methods resist tactical voting, one might
suggest a bona fide Condorcet candidate is automatically resistant to
tactical voting and thus unlikely to be impacted by the no-show paradox.

I ask if the following hold true in Condorcet methods where tied rankings
are disallowed:

   1. In methods independent of Smith-dominated alternatives (ISDA),
   ranking X above Y will not change the winner from X to Y *unless* Y is
   already in the Smith Set prior to casting the ballot.
   2. In ISDA methods, ranking X above Y will not change the winner from X
   to Y *unless* some candidate Z both precedes X and is in the Smith set
   prior to casting the ballot.
   3. In ISDA methods, ranking X above Y will not change the winner from X
   *unless* some candidate Z both precedes X and is in the Smith set
*after* casting
   the ballot.
   4. In ISDA methods, ranking X above Y and ranking Z above X will either
   not change the winner from X *or* will change the winner from X to Z if
   Z is not in the Smith Set prior to casting the ballot and is in the Smith
   Set after casting the ballot.
   5. in ISDA methods, ranking X above Y will not change the winner from X
   to Y *unless* Y precedes Z in a cycle after casting the ballot *and* Z
   precedes X on the ballot.

I have not validated these mathematically.

#1 stands out to me because ranking ZXY can cause Y to beat W.  If W is in
the Smith Set, this will bring Y into the Smith Set; it will also
strengthen both Z and X over W.  Z and X beat Y, as well.

This is trivially valid for Ranked Pairs; I am uncertain of Schulze or
Tideman's Alternative.  Schulze should elect Z or X.

In Tideman's Alternative, X can't win without being first-ranked more
frequently than Z and W; bringing Y into the Smith Set removes all of X's
first-ranked votes where Y was ranked above X (X* becomes YX*).  Y cannot
suddenly dominate all candidates in this way, and should quickly lose
ground:  X might go first, but that just turns XZ* and XW* votes into Z and
W votes, and Z and W previously dominated Y and so Y will be the
*second* eliminated
if not the *first*.

#2 is similar.  If you rank X first, Ranked Pairs will tend to get to X
sooner, possibly moving it ahead of a prior pairwise lock-in of Y, but not
behind.  The losses for X get weaker and the wins get stronger.  X also
necessarily cannot be the plurality loser in Tideman's Alternative, and
will not change its position relative to Y.  X must be preceded by a
candidate already in the Smith Set prior to casting the ballot for the
winner to change from X to Y.

#3 suggests similar:  if a candidate Z precedes X and is not in the Smith
set after casting the ballot, X is the first candidate, and #2 holds (this
is ISDA).

#4 might be wrong:  pulling Z into the Smith set by ZXY might not be able
to change the winner from X.

#5 suggests you can't switch from X to Y unless the ballot ranks Z over X
*and* Y has a beatpath that reaches X through Z.

I haven't tested or evaluated any of these; I suspect some of these are
true, some are false, and some are weaker statements than what does hold
true.

The fact that Condorcet methods fail participation is fairly immaterial.  I
want to know WHEN they fail participation.  I suspect, to be short, that a
Condorcet method exists (e.g. any ISDA method) which can only fail
participation when the winner is not the first Smith-set candidate ranked
on the ballot.  Likewise, I suspect that the probability of such failure is
vanishingly-small for some methods, and relies on particular and uncommon
conditions in the graph.
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