At 02:28 PM 4/24/2010, Markus Schulze wrote:
Hallo,
Bucklin violates mono-add-top. See:
http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-April/012752.html
The criteria failures of Bucklin don't apply to all Bucklin methods.
Woodall's definition of mono-add-top:
Monotonicity. A candidate x should not be harmed if:
* (mono-add-top) further ballots are added that have x top (and
are otherwise arbitrary);
The example given showing mono-add-top failure for Bucklin used deep
(full) ranking for six candidates, with no equal ranking allowed.
Bucklin in original usage was a three-rank system, with equal ranking
allowed in third rank. (Duluth Bucklin.) It's an Approval voting
system, except that it phases in approvals according to ranks, and it
is clearly improved by allowing equal ranking in all ranks, I
wouldn't even consider tossing overvoted ballots (i.e, that rank more
than one candidate in first or second rank.)
I'd like to look at this example, because it shows how preposterous
assumptions about voting systems can lead to noise about how they perform.
12: A>B>C>D>E>F
11: C>A>B>D>E>F
10: B>C>A>D>E>F
27: D>E>F
This doesn't resemble real elections with Bucklin at all, but let's
set that aside. This is what is common with the use of voting systems
criteria to study methods. Scenarios are created, sometimes cleverly,
to cause a failure of a criterion. Does it matter if those conditions
never exist? It should. The only way to really study this is through
simulations that could give some measure of how often a criterion
failure might happen. And without utility information that would
explain the preference profiles given, we have no idea of the damage
done by criterion failure. It is entirely possible that, from a
utility point of view, the election was improved by criterion failure!
I'll arrange this as Bucklin was usually reported:
60 Ballots, majority is 31.
A: 12 + 11 = 23, + 10 = 33
B: 10 + 12 = 22, + 11 = 33
C: 11 + 10 = 21, + 12 = 33
D: 27
E: 00 + 27
F: 00 + 00 + 27
A, B, and C are tied. They resolve the tie by looking back to the
previous round and A is stronger. Maybe. Not sure I like that. What
would range analysis say? Bucklin ballots, I've been claiming, are
Range 4 ballots, with rating of 1 missing.
12: A = 4: total 48; B = 3: total 36; C = 2: total 24
11: A = 3: total 36; B = 2: total 24; C = 4: total 48
10: A = 2: total 20; B = 4: total 40; C = 3: total 30
27: D = 4: total 108
(E and F have
total scores A: 104 B: 100; C: 102 D: 108
I want to point this out: this is a very unusually close election.
It's really close to a six-way tie. Note that with FPTP, the ABC trio
have no chance at all, not with top two runoff or IRV. But D wins
this election by Range, assuming that the ballot is a Range 4 ballot
with the rating of 1 missing.
Now, comes 6 more ballots:
12: A>B>C>D>E>F
06:A>D
11: C>A>B>D>E>F
10: B>C>A>D>E>F
27: D>E>F
The ballots are very strange. If we go to the fourth round, the A, B,
and C voters are all approving of D. Certainly, what makes the
difference here is that the counting can go into the fourth round.
And now we come to my objection to Woodall's "harm" criteria. The
consideration is whether a vote "harms a candidate," not whether or
not it harms the *election,* i.e., the *electorate.*
This really isn't a known Bucklin form any more, for had it been, and
those A, B, and C voters would, if they approved of D, voted for D in
the third rank (equally with their other third choice), we will
assume that the ballot does have four ranks. We must *still* assume
that these are all approvals. In Bucklin, by voting for a candidate,
they are approving of the election of the candidate.
Just looking at A and D.
66 ballots, majority is 34
A: 18 + 11 = 29, + 10 = 39
D: 27 + 06 = 33, + 00 = 33
A has a majority, A still wins, contrary to what was said. But I bet
you could figure out an arrangement that causes this to count into
the fourth round. But try to find one that uses three-round
Bucklin-ER, and that isn't some insane 6-way election where a
butterfly running into a window in China couldn't knock the election
in some different direction.
I trust this analysis not at all.
Note that in real elections, far more ballots are truncated, getting
a majority is harder, and multiple majorities would be rare. In
rank-order systems such as Condorcet methods, deep ranking makes
sense. In Bucklin, ranking below the approval cutoff makes no sense
if these votes are going to be used to determine a winner.
Buckling can be made to have more than three ranks, sure, and the
analysis of the ballot can be more sophisticated than simply
sequential approval with a lowering approval cutoff, but using
Bucklin in a runoff system is probably where it will shine the most,
for in that environment, there is a stop-loss for ranking at a very
particular level of approval: one will not vote for a candidate,
sensibly, if one would prefer to see a runoff election than see this
candidate win. And it's possible to add a disapproved rank to a
Bucklin ballot and not use it to determine the winner, but to detect
certain anomalies (basically loss of utility from declaring a winner
other than the Range winner, or Condorcet winner different from
Bucklin or Range winner). This all makes a lot of practical sense if
there is to be a runoff if there is majority failure, and it *might*
be considered majority failure when there are multiple majorities but
some discrepancy with range analysis of the ballots or condorcet failure.
The majority has not made an explicit choice, perhaps.
----
Election-Methods mailing list - see http://electorama.com/em for list info
