Sequential Pairwise Elimination
What is the best generic agenda for SPE or Agenda Based Landau (ABL)?
I am assuming that (inclusive) pairwise information is available in the
form of a matrix that has a row for each alternative... the j_th entry in
the row for alternative i is the number of ballots on which alternative i
is ranked preferable or equal to alternative j. In particular the k_th
entry in row k is the number of ballots on which alternative k is ranked
equal to itself, since it cannot be preferred ahead of itself. This number
is sometimes referred to as the implicit approval for alternative k. (It
must have some acceptability or why bother ranking it at all?)
Now within each row, sort the entries left to right from smallest to
largest.
Now sort the rows among themselves by the value in the first entry of each
row ... smaller to larger, top to bottom, and resolve ties by comparing
entries further to the right as in dictionary order when they start the
same way.
The more candidates, the more tie-breaking entries!
The method is "nothing, if not decisive!"
Now that the rows are in order, if you can remember which row goes with
which alternative, you have your agenda!
This method can be adapted for round robin tournaments starting with the
pairwise "score against" matrix.
The j_th entry in the i_th row is the score of team i in its match against
tem j. The diagonal elements will all be zeros.
So once the entries within the rows are sorted from from smallest to
greatest, every row will start with a zero. That's a good reason to go into
overtime if necessary to resolve at least some of the ties.
Message: 1
Date: Tue, 2 Mar 2021
Dear EM-ers,
I wish I knew more about the current practice of SPE. Who uses it the
most? What are the customary procedures for setting the elimination
agenda?
When the status quo is on the agenda is it generally first on the list or
last? Are the elimination votes mostly done one-by-one (roll call, show
of
hands, acclimation, etc. before going on to the next alternative on the
agenda as in the old days? How large or dispersed does the assembly have
to
be before requiring rankings of all the options for use in an instant
simulation of the method?
For any body that routinely makes use of the instant version of SPE that
deduces the pairwise eliminations from a pairwise win/loss/tie matrix
derived from ranked preference ballots, I urge you to consider applying the
following procedure (based on exactly the same inputs used by the instant
version of SPE) instead of SPE:
Initialize the variable X with the name of the last agenda item, assuming
the agenda order is the customary order of least promising to most
favorable.
Then ...
... while X is covered by some other agenda item Y, reassign to X the name
of the most favorable such Y.
The final value of X is the name of the winner.
[Y covers X iff Y beats or ties every Z that X beats or ties and strictly
surpasses X in at least one case ... possibly beating a Z that X only ties
or tying a Z that beats X, i.e. Y's row weakly dominates X's row in the
win-tie-loss matrix.]
The set of uncovered alternatives is called the Landau set. So this method
is Landau compliant. Copeland is also Landau compliant, but is less
decisive and fails to preserve clone independence.
The prevous messages in this threadfollow:
This is a continuation of the above named topic/subject discussion
starting
with an example showing Chicken Defense compliance ...
Here's the scenario with sincere preferences in brackets:
49 C
26 A>B
25 B [sincere B>A]
Implicit Approval plus least minimum support:
A 26 + 26
B 51 + 25 (B against A)
C 49 + 49
So the seed/agenda order is A(52), B(76), C(98)
The pairwise defeat cyclic order is A beats B beats C beats A.
So the sequential pairwise single elimination tournament summary is
B is eliminated by A, which in turn is beaten by C, the Sequential
Pairwise
Elimination winner.
As we now show, the B faction would have done better by voting sincerely:
In that case the only change in seed score would be A's, from 52 to 77,
due
to an increase in A's approval, so the line up would be
B, A, C
But this time in the second contest A beats C, 51 to 49, so the sincere
CW
wins which is indeed better for B.
What if, the sincere preferences [in brackets] were ...
49 C [C>B]
26 A>B
25 B [B>C]
In this case B is the sincere CW: B beats A, 74 to 26, and B beats C, 51
to
Evidently C has attacked B by truncation (partial burial ... full burial
would play out the same way), while the B faction has employed the
standard
CW prudent defense of truncation below the sincere CW.
We have just seen that the Chicken Defense Criterion requires C or A to
win
in this configuration, but the Plurality criterion requires A to lose. It
seems that the same ballot configuration requires C to win to satisfy
Chicken Defense and Plurality, and for C to lose in order to defend
against
(i.e. not reward) a truncation or burial attack by C.
Fortunately, the natural way out of this dilemma is for the A faction to
pursue its best interest by raising its compromise B to equal first.
Then B
becomes the ballot Condorcet winner, the implicit winner, and the
candidate
whose min pairwise support is greatest, all at once!
In sum, in the face of rational voters pursuing their own interests,
voters
can defend against burial of the sincere CW without any ballot rank
reversals.
Contrast this with IRV where voters can save their compromise fall back
sincere CW candidate from the center squeeze effect (so prominently
ubiquitous in IRV Yee diagrams) only by blatant insincere order reversals
that painfully violate the conscience of the honest voter!
Or for that matter, how would the sincere CW candidate B fare with
sincere
ballots in the 2nd scenario above? She would be eliminated in the first
round ... with no attack from the C faction!
. under zero information conditions! No way to defend!
Under perfect information the A faction could save its compromise B by
insincerdly burying favorite A under B... ouch!!!
Isn't election reform supposed to save us from lesser-evilism? (as much
as
possible)
I hope you find this explanation to be helpful. It is intended to help
leaders understand the stakes, and give them enough understanding to see
why zero strategy rankings are optimal for SPE (as with IRV) and that SPE
defense against complete information manipulative attacks are thwarted
with
greater finesse (with less violence to the conscience) compared to Voting
under IRV.
True, no method is perfect, but some methods are much better than others
on
the qualities that matter most!
Is there any good property of IRV that is more important to you than the
properties we have discussed so far? How about Later No Help/Harm? Are
you
willing to trade monotonicity and the Condorcet Criterion for them?
Let's get all these concerns out in the open!
I have suggested that the compliances of SPE are the most important ones
... sincere strategy optimality under zero info conditions including no
accidental eliminatiln of the Condorcet Candidate, ease of defending the
CC when under strategic attack, efficient precinct summability and
transparency of the count, Clone Independence, Monotonicity, Plurality,
and
Immunity to Chicken Attack.
The most important missing criterion is the FBC, which seems to be
incompatible wth the Condorcet Criterion....but our SPE has a base method
for the agenda/seeding that satsfies the FBC .. giving max FBC compliance
for a Condorcet method. In fact, if you skip the elimination steps and
jump
straight to the last (i.e. most promising) candidate in the lineup/agenda
you get a perfectly good FBC compliant method ... the Approval Stable
Winner ... the candidate whose sum of Implicit Approval and
Minimum Support against any other candidate is maximal.
I'll take a break here ...
---------- Forwarded message ----------
From: Forest Simmons <fsimmons@for pcc.edu>
Date: Saturday, February 27, 2021
Subject: Condorcet method - Wikipedia
The Wikipedia Article [link below] on Condorcet Voting describes
Sequential
Pairwise Elimination as the most studied version of Condorcet voting in
the
literature because it is the one most widely used in deliberative
assemblies as recommended in Robert's Rules of Order.
It is important to distinguish Sequential Pairwise Elimination from
(mere)
Sequential Loser-Elimination methods like Raynaud, IPE, IRV, etc. that
may
or may not use pairwise information in determining what kind of loser to
eliminate at each step. The crucial difference between the two kinds of
elimination (SPE and SLE) in the case where they both require a pairwise
loss in the elimination step is that the former follows a predetermined
schedule or agenda (the seeding order in a single elimination tournament)
of whom to compare next with the winner of the previous round (the
champion
so far) while the latter determines the next pair of candidates to be
compared pairwise as though starting from scratch, with all remaining
(uneliminated) candidates on an equal footing.
The difference might seem unimportant but it is essential for
monotonicity
criterion compliance, which elimination methods invariably fail when not
working off a monotonic agenda. This compliance is a clear advantage over
IRV that we cannot afford to throw away.
For example, some have suggested that Borda be used as the base method
for
SLE ...at each step eliminate the remaininng candidate with the lowest
Borda score (Baldwin's method)... if we wanted to reinforce Condorcet
compliance we could change that to eliminating the pairwise loser between
the two lowest Borda score candidates ... but are we talking Borda scores
as determined at the beginning? ...or newly calculated Borda scores
reflecting only the rankings relative to the remaining candidates?
Of these options only the SPE version (the one with the Borda agenda
adhered to throughout) is monotone. I use it only as a example to clarify
the difference between SPE and SLE.
Why not propose it? It beats IRV on four important counts ...
monotonicity,
Condorcet compliance, efficient precinct summability, and transparency.
It
is equally easy to vote since it makes use of the same ranked preference
ballots. Why not propose it?
Because IRV is better in the single most important way ... IRV is clone
independent. Clone Idependence (with sincere zero information voting) is
the only justification for proposing any method requiring a ranked
preference ballot.... IRV and various other methods pass on this score,
but
all Borda based methods fail ... it's a deal breaker.
So what should we use as an agenda setter (seeding order) if not Borda?
Why not Approval? Approval would provide a monotone, clone free, agenda
order, so why not propose SPE Approval?
One tiny objection ... how to decide approval cutoffs ... an additional
burden on the voter (frown emoji).
Suggested solutions include....
... making approval marks optional with default approval just above
truncation or just below equal top, or the average of the two... not a
bad
idea, though slightly complicating the count and worrying the voters
about
the option ... (If I don't make best use of the option, am I failing my
civic duty?)
....Candidate Proxy where voters mark the approvals and disapprovals that
they feel sure about and delegate the rest to their favorite or other
candidates to decide.... great idea, but not worth the miniscule increase
in complexity in the minds of some.
.... DSV Approval. A good DSV Approval cutoff would be the lowest
candidate
on the ballot that pairwise beats all candidates ranked above it. But
this
complicates the count ... one pass to determine the pairwise matrix, and
another to determine the approvals for the agenda/seed order.
...Imlicit Approval would not work in Australia where complete ballot
rankings are mandatory (no truncations or equal rankings allowed).
How about seeding the candidates from least minimum pairwise support to
greaest minimum pairwise support? This works best when the pairwise
support
of X against Y is defiined as the number of ballots on which X is ranked
greater than or equal to Y.
Or how about the "stable approval" order, a synergistic sum of these
last
two suggestions that optimizes zero information strategy ... the best
strategy when no reliable horse race information is available is to vote
sincere ranked preferences as you would under IRV under the same zero
info
conditions.
With this seeding of the agenda, our SPE method also satisfies Plurality
and Immunity to "Chicken" threats, two important criteria that IRV
supporters could embarrass us with if we failed them.
When I get another free moment I will give a couple of examples ... for
now
I conclude with the following observation ...
Remember, the support for X against Y is the number of ballots on which X
is voted above or equal to Y ... which is identical to X's approval
score,
no matter the choice of Y, whenever all voters vote only at the extremes
... no intermediate ranks. So in that case there is no difference in
seeding by Implicit Approval (least to most) or seeding by minimum
pairwise
support (least to most). Our proposed SPE method reduces to approval
whenever no intermediate ranks are occuppied on the ballot: This is a
longstanding tradition for acceptable EM list methods ...
In particular the following example that shows Plurality failure for the
MMPO (MaxMin Paiirwise Opposition) method elects the approval winner:
44 A=C
13 C
43 B=C
awards the win to the Approval winner A:
The seeding order is C, B, A while the pairwise beat order is A beats B
beats C, so C loses to B who in turn loses to A ... the undisputed champ!
To be continued ....
---------- Forwarded message ----------
From: Susan Simmons suzerainsimmons@outlook.com
Date: Saturday, February 27, 2021
Subject: Condorcet method - Wikipedia
To: Forest Simmons fsimmons@pcc.edu
https://en.m.wikipedia.org/wiki/Condorcet_method
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