John

2018-08-07 16:05:19 UTC

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.

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.