[EM] Participation criterion and Condorcet rules

[Kristofer Munsterhjelm]

On 2018-08-09 21:28, John wrote:
> 
> 
> On Thu, Aug 9, 2018 at 2:29 PM Kristofer Munsterhjelm 
> <km_elmet at t-online.de <mailto:km_elmet at t-online.de>> wrote:
> 
>     On 2018-08-07 18:05, John wrote:
>      > 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.
> 
>     Not all Condorcet methods are Smith-efficient. For instance, Minmax
>     is not.
> 
> 
> True.  Most methods attempt to resolve a Condorcet cycle, but must be 
> Smith-efficient for the above to be true.  Most methods people talk 
> about (Schulze, Ranked Pairs) when advocating Condorcet over IRV in 
> public discourse are Smith-efficient.
> 
>      > 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.
> 
>     I think that's more true of methods that go "If the CW exists, elect
>     him, otherwise...". 
> 
> 
> Not really.  If a method provably always elects from a particular subset 
> (Smith, Schwartz) which can be identified by some algorithm, then that 
> method essentially elects from a pool of suitable candidates and 
> excludes other candidates identified as not-suitable.  The decision to 
> use such a method inherently assumes that this subset is suitable and 
> those outside this subset are non-suitable.
> 
> 
>      > 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.
> 
>     James Green-Armytage's paper on strategy resistance,
>     http://jamesgreenarmytage.com/strategy-utility.pdf , gives some proofs
>     as to when "Condorcetifying" a method only improves its strategic
>     resistance. If I recall correctly, making a method Condorcet-compliant
>     usually doesn't alter its susceptibility to burial while it improves
>     its
>     resistance to compromising.
> 
> 
> I intended that a set of ballots which produces a Condorcet cycle is 
> more-vulnerable to manipulation than a set of ballots which produces a 
> single Condorcet winner.
> 
> If the winner is just A (Condorcet winner), you have to defeat A before 
> the outcome can change.  If B defeats A and is undefeated by all other 
> candidates, B becomes the Condorcet winner.  If B defeats A but C (also 
> undefeated by all except A) defeats B, you have a Condorcet cycle:  the 
> Smith Set is A B C.
> 
> In that situation, you want to rank B C A, because you need B to defeat 
> C or C to defeat A to change the winner from A, and you want B to defeat 
> C.  If you rank B A C, you strengthen A's win over C.

If your setting automatically excludes ballot sets that have no
Condorcet cycles (thus ignoring universal domain), then Condorcet passes
all of Arrow's other criteria.

It passes Pareto dominance since if every voter prefers A to B, then A
pairwise beats B, and so A comes before B in the social ordering.

It passes monotonicity because if some voters decide to rank A higher,
then that can never make A lose to some other candidate B that A used to
beat pairwise.

It passes non-imposition because for any ordering of the candidates,
there exists a ballot set that would lead Condorcet to give this ordering.

And it passes IIA because A>B changing to B>A can't introduce a cycle
(if it could, it would be forbidden). Thus the order stays transitive
and whether A ranks ahead of C depends only on whether A beats C pairwise.

But introducing cycles provides a contradictory region that can't be
evened out, which makes Condorcet methods fail IIA. By Condorcetifying a
method (like Smith,IRV), you replace the contradictions of IRV with a
well-behaved Condorcet region and some other contradictions in the cycle
region and the border between the two.

Green-Armytage's paper that the changes usually reduce strategy.
However, they can introduce criterion failures where none existed before
(e.g. IRV meets LNHarm; Smith,IRV does not).

> Alternative Smith seems to me as if a burying strategy would only work 
> if you elected a less-desirable candidate.  That is:  if you like the 
> Republican (Hogan) more than the Libertarian, and you like the 
> Libertarian more than the Democrat (Ben), any attempt to bury potential 
> winner Ben will only succeed if you rank the Green (Ian) FIRST, then 
> Hogan; and Ben is between Ian and Hogan.  Why?
> 
> You need Ian to defeat Ben and knock him out of the Condorcet Cycle, so 
> you need to rank Ian above Ben.
> 
> If there is a Condorcet cycle, then a third candidate must be ranked 
> above Hogan—likely Ian.

It might be possible to use burial to exploit the boundary between
Condorcet winner and cycle. E.g. the Condorcet winner is a centrist
which IRV doesn't like; then some voters bury the centrist to create a
cycle for the benefit of the IRV winner. Something like a modified LCR:

43: L>C>R
35: R>C>L
10: C>R>L

Some R-voters then bury C under L:

43: L>C>R
33: R>C>L
 2: R>L>C
10: C>R>L

Now the Smith set is {C, L, R} and so Smith,IRV chooses R.

Of course, I would still prefer Smith,IRV (or Alternative Smith; they're
very similar) to plain old IRV, because while the Condorcetified method
provides a bad result given strategy, IRV provides a bad result without
strategy.

>>     But I imagine Participation is more a paradox-avoidance criterion than
>>     it is a strategic criterion, similar to monotonicity. (Again in my
>>     opinion,) IRV's monotonicity failure isn't something that can be
>>     exploited in strategy as much as it is evidence of the method "getting
>>     it wrong". You have two ballot sets where going from the first to the
>>     second only improves candidate A's situation, but A wins according to
>>     the first ballot set yet loses in the second.
> 
> 
> Yes.  Voters need confidence that their vote does what they want.  I 
> think the best we can do is say it usually does what they want.
> 
> IRV's failure is that the candidate elected seems to not be one favored 
> by anyone:  you can have a Condorcet Winner, a Plurality Winner, and an 
> IRV Winner in a race between three candidates, and each candidate can be 
> the winner of each of these three methods.  If the Condorcet Winner 
> beats the other two head-to-head and the Plurality winner just bluntly 
> gets the most votes when everyone is asked to pick one of the three, 
> what in the heck is the IRV winner?

In a simple election, I think the easiest explanation is that the IRV
winner is "the strongest candidate of the strongest wing". E.g. the LCR
from above:

43: L>C>R
35: R>C>L
10: C>R>L

We have two "wings" (left and right) and a moderate position (center).
Condorcet tries to find the median, so C is the CW. L wins a Plurality
election due to C splitting the vote. But the right wing is stronger
than the left if the center is removed, which is what IRV does. It
successively removes weaker factions until it's a contest between two of
them, and then the stronger faction wins.

In more complex settings, IRV can get quite chaotic since it's
path-dependent. The fragmented Yee diagrams make this pretty clear.

> With a Smith-efficient method, every candidate not in the Smith set 
> would lose one-on-one to any and all candidates in the Smith set.  We 
> can tell the voters, with absolute authority, that those candidates 
> excluded are candidates who cannot win against any of the candidates we 
> might elect.  You have that confidence that we'll elect someone 
> meaningful to the expressed will of the voters.
> 
> As for what the method does besides that, well, it may do something 
> strange.  It follows a mathematical rule and will not give anyone the 
> power to dictate the winner; the precise outcome may have a confusing 
> relationship with the ballots cast, even though the group from which we 
> elect a winner has a very clear relationship.
> 
> If nothing else, it's less about ground game and more about the voters 
> as a whole finding the candidate acceptable.  Everyone's vote matters.  
> Under plurality (and with sufficiently-few candidates), your vote 
> doesn't matter if your district is 60% Democrat or 60% Republican.


>> 
>> Since Smith//Plurality passes ISDA, that should answer your five
>> questions in the negative.
>> 

> Interesting.  I wonder if close victories are a solvable election 
> problem or if that's an error that simply will never go away.  We just 
> had an election here where a candidate won by about 20 votes, and three 
> candidates all received nearly the same number of votes.  Two hundred 
> votes change and the third-place winner is the first-place winner—out of 
> 80,000 votes.
> 
> If fifteen voters had voted Brochin instead of Olszewski, the results 
> would have reversed.

In every voting method, a candidate is either elected or he isn't;
there's no such thing as "winning slightly". Similar to how rounding
must introduce a discontinuity on the output side (e.g. rounding to the
nearest integer, 0.5-epsilon is mapped to 0 and 0.5+epsilon to 1), a
voting method would also introduce a discontinuity. So you can't get
around close contests in full generality.

Stochastic methods like random ballot can get around this problem
because a continuous parameter is mapped to another continuous
parameter. You can "win slightly"; it just means that your probability
of being elected, while being low, is nonzero.

> Now you tell me:  what's the difference between that and the election 
> you describe above?  I've been approaching this from a technical 
> standpoint; I'm starting to think there's a philosophical problem 
> here—one that can't be solved. 

The difference is that there isn't just a discontinuity, but that the
change itself is wrong.

If you had a two-candidate election like this:

49: A
50: B

and then two delayed ballots for A arrive, then that would flip the
winner from B to A. That's the discontinuity: a slight change in the
vote counts for A relative to B makes for a great change on the output
side. For something like Random Ballot, there would be no great change;
the probability of B winning would simply go from 50.5% to 49.5%.

However, the problem with a monotonicity failure is that something that
favors B happens, and it makes B lose. It's more like having the
two-candidate election above (where B wins), then two delayed ballots
voting for B (instead of A) arrive, and *then* the winner switches from
B to A. That would just be bizarre.

Since deterministic ranked voting methods are subject to Arrow's
theorem, we have to live with some measure of bizarre. Just what kinds
of bizarre we can banish entirely is a good question, but very hard to
answer.

> Condorcet tries—it achieves something akin to mutual majority
> consensus—but if the needle only has to move a few fractions of a
> millimeter to change the winner, have we elected someone or did they
> simply win a contest?

There's probably a statistical point to be made here, too. If you have
an election with a close margin like the two-candidate election above,
then it's possible that the winner just got lucky: a few ballots got
miscounted, or some of the voters just had a bad day and crossed off the
wrong candidate by accident.

If we have a model of how often those things happen, then we could
statistically determine whether the election is too close to call - at
least for Plurality elections. I don't know how to do it for more
complex ones like Condorcet.

More complex voting implementations (verifiable ballots, recounts, etc)
can decrease the noise in the process, making closer elections more
certain, but they can't get around any indecision in the voters' minds
themselves. On the other hand, if the result isn't close, it would be
sufficient to ask just a small sample (e.g. as
https://rsvoting.org/whitepaper/white_paper.pdf suggests). Perhaps one
could use deliberative polling if voter ignorance is a big problem, but
it could also have legitimacy and corruption problems if it were
directly used to decide winners rather than just to inform the public.