[EM] Participation criterion and Condorcet rules
john.r.moser at gmail.com
Tue Aug 7 09:05:19 PDT 2018
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.
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
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
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
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
#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
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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