Since I only get the digest version of the elections methods postings, I didn't
realize that Kevin had already suggested the sequential version of voter
cancellation in which (after the first) each successive voter cancellation
is decided by the most recently cancelled voter.
This method has
Form a matrix M whose entry in row i and column j is the percentage of
ballots on which alternative j is the highest ranked alternative that is not
majority defeated by alternative i.
[By definition no alternative is majority defeated by itself, so if every
alternative ranked higher
It's the transpose of M that is stochastic. So you can either postr multiply
by M or pre-multiply by the transpose of M.
Forest
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Juho wrote:
How about the smallest number of ballots on which some
alternative that beats A pairwise is ranked higher than A?
Juho
If I am not mistaken, this idea is equivalent to electing the alternative A
with the greatest number of ballots on which A is ranked higher than any
PROTECTED]
At 11:37 AM -0800 1/6/07, Simmons, Forest wrote:
In a small group the voters are asked to form clusters around their
favorite candidates. After five minutes of shuffling around
positions are frozen.
The two voters separated by the greatest distance are required to
sit down. This step
It has been pointed out from time to time that when the voters and candidates
all lie along the same one-dimensional spectrum of opinion, then the Condorcet
winner is the top preference of the medium voter. [This assumes that there are
an odd number of voters, or that the two voters next to
For every non-deterministic method M there is a deterministic method M' that
elects the candidate most likely to win under method M.
For example, if M is random ranked ballot, then M' is Plurality.
Suppose that M is random approval ballot. Then M' is the method that picks the
candidate most
In view of comments and suggestions from Chris, Warren, Markus, and all (for
which I thank you all warmly) I would like to suggest that this attempt at
clone proofing Copeland be used in a three slot setting.
I'll restate it for the record:
For each candidate X let p(X) be the probability
Here's an idea for clone proofing Copeland:
1. First (as in Copeland) compute the pairwise win/lose matrix, which has a
+1, -1, or zero in row i column j according as alternative i beats, loses
to, or ties with candidate j in the (i, j) pairwise comparison.
2. Then (unlike Copeland)
I haven't read all of the entries on this subject, so I don't know if this
suggestion has already been made:
Yee-Bolson diagrams can test for clone dependence by taking the candidate
corresponding to the green region and replacing it with a small triangle of
candidates assigned shades of
Warren suggested trying range ballots and seeking an approval equilbrium winner
X such that X is the approval winner when each candidate rated above X is
approved, each candidate rated below X is not approved, and each candidate
rated equal to X (including X itself) gets half approval.
This
Here's a parlour version of the rainbow lottery:
1. Go around the room and ask each player which of the alternatives is their
favorite.
2. Go back around in the reverse order and ask each player which of the
alternatives they approve of.
3. List the alternatives in order of approval
For your eyes only. Warning: do not proceed past this point if you don't
like lottery methods.
Ballots are approval style.
Ballots are counted in two different ways: (1) the approval count and (2) the
fractional (cumulative) count, which means the candidates marked on a ballot
are
Kevin,
I misread what you wrote, but now I see that you were indeed measuring which
method maximized expected range value for the voter.
However, there may be voters that wish to maximize the probability that their
ballot will be positively pivotal, i.e. they might wish to maximize their
In the zero information case (with many voters), above mean utility approval
strategy (Strategy E in Kevin's simulation) is optimal for maximinzing an
individual voters expected utility.
However, that's not what Kevin is using as a measure of success. If I
understand him correctly, a vote
How about having the voters rank the candidates with the option of skipping
numbers: the more numbers skipped the stronger the preference. Thus
1 Ann, 3 Jill, 4 Jack, 7 John
would translate to
AnnJillJackJohn
Forest
Juho wrote ...
Ranked preferences could be derived from range.
It seems to me that when estimating the strength of candidate X on the basis
of a pairwise comparison with candidate Y, we should take into account the
strength of candidate Y ; if Y is a weak candidate, then a large margin of
victory by X over Y may not be as significant as a small
It seems to me that if there is a majority winner, then she should at least
have a chance of winning. What if we chose by random ballot from among all of
the candidates that have a majority beat path to the Range winner (with a final
approval vote to ratify this choice)?
Forest
I think this style of ballot (with relative strengths of preferences indicated)
are a good compromise between range ballots and ordinary rankings.
We have just scratched the surface when it comes to their possible applications.
Forest
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election-methods mailing list - see
Let D1 be the candidate whose maximum pairwise opposition is minimum, in
other words the MMPO winner.
We could say that D1 is a good defensive candidate because she minimizes the
number of votes scored against her by any other candidate.
Similarly, let O1 be the candidate whose minimum
This method is based on rankings with truncations and approval cutoffs.
Let X be the candidate approved on the greatest number of ballots. Let Y be
the candidate ranked on the greatest number of ballots.
If X and Y are the same candidate, then this candidate wins.
Otherwise, a ballot is
Jobst,
how about this slight variation on your suggestion?
1. Ranked ballots with truncation.
2. Draw a ballot at random.
3. Draw additional ballots at random until there is one that has at least one
ranked candidate in common with the first ballot, or until all ballots are
exhausted.
I like RPR. The additional control that this gives to voters might make voters
less squimish about proxy.
Forest
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I like the idea of giving candidates the option of withdrawal before each
potential elimination.
It seems true that most candidates would be more reluctant to withdraw than
their supporters might want, but if no candidate withdraws, then IRV with
withdrawal reduces to plain IRV, no harm
Any time that IRV does not elect the sincere CW (when there is one) there is
going to be a strong incentive for order reversal under IRV, except under the
(non-existent) zero information case. [The only real life cases that exist in
hot elections are the positive information and positive
Jobst,
Great Idea!
It reminds me of Martin Harper's idea for recasting Approval as a vote
concentration method in order to appease the extreme one person one vote
people.
Martin suggested listing all of the candidates in order of approval count, and
then on each ballot circle the
This method makes use of ordinal information as well as approval information.
1. Eliminate each candidate X for which there is a candidate Y such that on
each of more than half of the ballots Y is approved and X is not.
2. Use random ballot among the remaining candidates to choose the
I agree with Brian Olson and RL Suter in the main point below that from the
voter point of view, for methods that rely on ranked ballots this common
feature looms larger than the difference in how the winner is determined once
those ranked ballots have been submitted by the voters.
As Steve
I like Raphael's idea of giving each voter three copies of the current lottery
in exchange for one lottery's worth of papers.
Let's call his idea the DLE(1/3) enhancement in contrast to my original
suggestion of DLE(1/2).
In general, suppose that n copies of the current lottery are
It's pretty obvious that under Approval you should approve your favorite
candidate, and that you should leave unapproved the candidate that you despise
the most. But it isn't always so obvious which of the remaining candidates to
approve.
A rule of thumb is to approve the candidate that you
Suppose that the approval results from a reliable poll are published as follows:
2 % approve A only.
25 % approve both A and B
23 % approve C only.
And now it's your turn to vote an approval ballot.
According to approval strategy A, you should put your approval cutoff on the B
side of A
Jobst,
Thanks for doing these simulations and getting us thinking along these lines.
I wonder how Bucklin would fare in your simulations? Or how about the quartile
variation of Bucklin in which the bar is lowered simultaneously on the range
style ballots until at least one candidate is
Let's consider the case of one candidate ordering (i.e. ranking) per candidate.
One way this can happen is by each voter supporting the ranking published by
his or her favorite candidate. Another way that this can be achieved is by
averaging together all of the ballots that rate a given
In some elections not all of the ballots are cast at the same time, and
furthermore, the partial results (from exit polls, say) may be available to
voters later in the sequence.
In some applications, like the US presidential election, geography roughly
determines the order of the ballots.
David Cary wrote
...
This example seems to contradict what I understood to be Forest's
earlier claim that a geometrically consistent election with 3
candidates always produced a Condorcet winner. Am I missing something?
I reply:
Your example is geometrically consistent, but it does not
Disclaimer: this method is for theoretical purposes only. Those who don't
believe in theoretical purposes should delete it immediately.
Somtimes PR STV is introduced by asking the reader to imagine a PR election in
which voters vote sequentially with knowledge of the current running subtotal
Antonio wrote
I happen to believe that adjusting a ballot to give each voter an ideally
strategic ballot will be the future of advanced voting system design, at least
where fairness is involved. It does take a lot of processing power, and I
believe it will usually take away the summability of
A simpler version of this DSV idea follows:
1. Initialize L(0) as the list of the ballots in random order. Let N be the
number of ballots. For each i between one and N, inclusive, let L(i) be the
tail of L(i-1). In other words, each list is obtaind from the preceding by
removing the first
Title: election-methods Digest, Vol 23, Issue 4
Paul,
Actually, I did not assume that there was
any linear or two dimensional relationship.You can use any measure of
closeness that you want, linear, non-linear, ten dimensional, or infinite
dimensional (for example the norm of the
Antonio Oneala lamented that proportional Condorcet methods tend to be
intractable. This is because if there are N candidates from which to choose K
winners, there are C(N,K)=N!/(K!*(N-K)!) subsets to be compared pairwise, for
a total of C(C(N,K),2) pairwise comparisons of subsets.
How can we best insert the selection of a published ordering option into the
following section of recently proposed statutory language that Rep. Toby Nixon
is spearheading in Washington State?
BALLOT
Rank-balloting definitions:
To rank a candidate means to assign to that candidate a postive
Steve Eppley suggested allowing voters to choose from published orderings and
then doing the tally by
... a good voting method, such as Maximize Affirmed Majorities (MAM).
Here's a suggestion for an easy-to-understand alternative to MAM that would be
adequate in this context:
In the case that
Nifty!
I think it should be called deconstruction rather than decomposition, but
it is a nifty procedure, especially for those of us who like to think up
election methods examples and counter-examples in tournament form.
Dave K. asked where one might encounter a condorcet matrix that wasn't
Suppose that after the ballots come in and after the pairwise matrix has been
published, any and all are allowed to submit single elimination tournament
schedules in the form of binary trees of minimum possible depth (approximately
log base two of the number of candidates).
All of these
Warren,
I should have been more clear:
I did not require complete rankings. Indeed, some of my examples incorporated
collapse of ranks.
However, I did assume a fair coin, since otherwise the method would not even be
monotonic.
With these two points clarified, let's prove that the method
I wrote ...
Disclaimer: Doubtless some situations require deterministic election methods.
This message has nothing to do with any such situations.
Simple Lottery Method:
If there is an alternative that pairwise defeats the approval winner A, then
toss a coin to decide between A and the
Paul Kislanko asked ...
Why introduce majority dense and not use that?
Forest answers:
1. Because it wasn't necessary for the purpose of my message, which was to
nudge readers out of their mental ruts.
2. Is the introducer the only one who can use an idea?
Paul went on to ask ...
Mike, you're right about the FBC. The same effect causes a monotonicity failure
in regular ER Bucklin (without the special delayed counting rule that you
proposed). But with your rule, in effect, the equally ranked candidates serve
as place holders to delay the compression, so that this cannot
I wrote ...
Ratings are a convenient way of providing for equal rankings and keeping the
ballots from becoming too unwieldy when there are large numbers of
candidates, as in a big election without primaries.
Mike replied:
But how is ratings more convenient than rankings? As long as the voters
The candidate with the maximum median rating is the ER Bucklin (whole) winner,
assuming that if two candidates have the same median rating R 0, then the one
at or above R on the most ballots is the winner.
Ratings are a convenient way of providing for equal rankings and keeping the
ballots
It seems to me that any method that uses number of ballots on which a
candidate is ranked as an important part of the procedure could substitute the
following device:
Let's say that a candidate has bottom status on a ballot if that candidate
is not ranked above any candidate on that ballot.
Dear EM aficionados,
Here's a method that elects the candidate with the best ratio of offensive
strength to defensive weakness. Until a better name comes up, call it
Offense/Defense or O/D.
For each pair of candidates X and Y, let F(X,Y) be the number of ballots on
which X is ranked equal
From: [EMAIL PROTECTED] on behalf of [EMAIL PROTECTED]
Sent: Sun 9/25/2005 12:00 PM
To: election-methods-electorama.com@electorama.com
Subject: Election-methods Digest, Vol 15, Issue 53
I had written ..
Note that in ordinary Bucklin the ordinal
In the recent message quted below there are two questions.
1. What should we call the Approval method that allows an extra mark to
identfy the favorite candidate, thus satisfying the Approval voter's urge to
give more moal support to Favorite than to Compromise?
I suggest Approval Plus or
Adam wrote:
I haven't been following this line of threads terribly closely, so I just
want to be clear that I understand. The way I think about Bucklin is an
approval election where the approval cutoff bar on everyone's ballot keeps
getting lowered until we have a majority approved winner. It
Title: Election-methods Digest, Vol 15, Issue 50
Someone wrote:
I think the "+" to show "I like B better than A even though
I ranked A=B" disingenuous and unnecessary. If you prefer one of the equally
ranked alterntatives more than the other, just don't rank them equally.
Forest
I like the modified ER Bucklin Whole version that Kevin and Mike have been
considering.
I have two suggestions that might make it more viable as a public proposal:
1. Keep the number of possible distinct ranks down to seven or eight, for
ballot simplicity.
2. Allow a special mark + to be
Here's an interesting example with four candidates, in which (under Shulze) an
order reversal between Favorite and Compromise would give the win to Compromise
(instead of a third candidate D) even though Compromise already beats Favorite.
In other words, there seems to be incentive to betray
Mike suggest that the best public proposals are ...
Best: MDDA, or maybe MDDB, which combine FBC with SFC, thereby accomodating
the needs of different kinds of voters. It now seems to me that MDDA is
better than MDDB. I'd said that SR would be a good proposal under certain
conditions, when
Kevin wrote () in response to my ():
Also, there's a second election at this convention! Supposing the method used
at
this convention satisfies weak FBC, what have we gained (in terms of FBC
compliance)
by voting for delegates first?
Or are you just saying that although this method might come
about the most.
From: [EMAIL PROTECTED] on behalf of
Simmons, Forest Sent: Sat 9/17/2005 3:00 PMTo:
election-methods-electorama.com@electorama.comCc:
[EMAIL PROTECTED]Subject: [Condorcet] Favorite Betrayal in
DMC
One of the nice things about DMC is that it is easy to pinpoint
the precise
Forest claimed:
That said, an even simpler method comes closer to satisfying the Strong FBC
than any of these
other more complicated methods: Asset Voting:
Voters vote for their favorite, who represents them by proxy in an election
completion
convention. Write-ins are allowed. In
Perhaps we could distinguish sincere approval strategy from merely
consistent approval strategy according to whether both or only the first of
Chris' conditions are satisfied.
I agree strongly with Chris' remark concerning the advantage of DMC zero info
strategy over Shulze(WV) zero info
Here's the weakness of SR:
60 ABCDEF
40 BCDEFA
Here A should be the winner, but B has by far the best SR score of only 60
versus A's lousy 200.
I believe that this defect is called teaming.
Forest
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I sympathize with Rob's complaint about the meaning of approval versus
disapproval in Approval.
There is a trade-off, a price for the simplicity of Approval.
However, DMC takes the pressure off this question because in DMC, approval is
only used to eliminate enough of the Smith set for an
One of the nice things about DMC is that it is easy to pinpoint the precise
circumstances in which there is a Favorite Betrayal incentive, i.e. where
Favorite Betrayal is more likely to payoff than not. It seems to be much
harder to pin this down in Schulze.
Here are the conditions that
Recently Rob Lanphier asked how to determine the strategically optimal approval
cutoff when voting under DMC.
Here are my suggestions:
(1) For the case where you have enough polling information to discern the Smith
set:
First identify (with the letter C) your favorite member of the
Title: Election-methods Digest, Vol 15, Issue 33
Here's the corrected strategy for the case where you
have enough polling information to discern the Smith set:
Firstidentify(with the letter
C)your favoritemember of the Smith set (or, if possible, the
uncovered set).
Put your approval
, and truncate all others. I'm not sure.
Forest
Kevin replied to the following:
Forest,
--- Simmons, Forest [EMAIL PROTECTED] a écrit :
Kevin, your ICA method interests me. In particular, your creative use of
equal ranked top
might be called power top analogous
Title: [Condorcet] Re: Voting as duty (was ties truncation)
Somebody thought that the
candidates would frequently fail to be ratified by the electorate.
Well, if the statisticians
take this into account properly, and submit for ratification only those
candidates that have a 99 percent
Title: RE: [Condorcet] Can we come to consensus? (ICA)
Kevin, your ICA method interests me. In
particular, your creative use of "equal ranked top" might be called "power top"
analogous to what Mike Ossipoff recently called "power truncation" for equal
(non)ranking at the bottom.
I
Another measure of logical complexity of a method is the number of alternations
between quantifier types, from universal to existential and back, in the
complete definition of the winner.
For example MinMax is more complex in this regard than DMC, because the MinMax
winner is the candidate C
Title: Re: [Condorcet] Ties Truncation: Information Loss
The main reason that "lazy"
voters don't take the time to study up on and carefully rank all of the
candidates is that they know that in these large scale elections the chance that
their vote will be pivotal is practically nil.
Title: [Condorcet] Plain English description of Schulze(wv)
The plain English description
of Shulze is pretty good except for the last step (5), which is incorrect.
In Shulze you nullify the weakest defeat in a cycle. The "in a
cycle" part is extremely important.
To see this pointsuppose
I meant to say that you could use matrices M and PM (not CM) to formulate ICA,
in the last paragraph quoted below.
Sorry for the confusion.
Forest
I suggest that we consider methods that sum two modified pairwise matrices in
addition to the basic pairwise matrix:
(This description is at
Woops, now I see that step 3 puts all defeats in cycles, so disregard my
objection.
Forest
I had written:
The plain English description of Shulze is pretty good except for the last step
(5), which is incorrect. In Shulze you nullify the weakest defeat in a cycle.
The in a cycle part
Title: Re: [Condorcet] Why Schulze is Better than DMC
There are other
objectivemeasures of complexity besides the ones mentioned by Jobst. (He
mentioned computational complexity from the point of view of the smallest
accurate description of the method or algorithm, as well as the minimal
, Beatpath, and Ranked Pairs all agree
on C as winner.
I hope that this earns some respect for DMC's Favorite Betrayal
resistance :-)
My Best,
Forest
From: Adam Tarr
[mailto:[EMAIL PROTECTED]Sent: Fri 9/9/2005 7:18 PMTo:
Simmons, Forest Cc: election-methods-electorama.com@electorama.com;
[EMAIL
I believe that most voters would feel worse about having to rank Favorite equal
with Compromise than they would about ranking Favorite ahead of Compromise
while approving both.
If you collapse the order, then nobody can tell from your ballot which you
prefer, unless (somehow) approval can
Kevin has written (in response to)
--- Simmons, Forest [EMAIL PROTECTED] a écrit :
This brings up a question. How good is MinMax(Approval Against)?
By Approval Against I mean number of ballots on which X is approved and Y
is not.
I call this approval opposition.
Maybe I'm crazy
Title: Re: [Condorcet] Can we come to consensus?
Jeff Fisher recently opined that DMC voters would
likely adopt the strategy of approving all candidates that theyconsidered
certainto be beatenpairwise by their Favorite. This
would put these candidatesin a better position to doubly defeat
I wrote ...
Yes, this is the measure of defeat strength used in AWP, but here's the
question: Is AWP based
on River the same as AWP based on MinMax?
Somewhat surprisingly, the answer to the analogous question for winning
approval as a measure
of defeat strength is affirmative:
Dear James,
as I said in a recent message, I also think that AWP is more resistant to
burying than DMC. But until there is a simpler description of AWP, I will
support DMC over AWP in the category of Condorcet Public Proposals.
This brings up a question. How good is MinMax(Approval
Title: Re: [Condorcet] Comment on DMC
Here is an "ABC" example that illustrates
Jeff Fisher's concern as I understand it (see below):
Sincere zero info ballots:
45 ACB
20CBA
35BAC
Pairwise cycle is
ACBA.
Approval order is ABC.
So the sincere DMC winner is B.
Butin thenear perfect
Title: Re: A class of ballot set with unbeaten in mean lotteries.
The following lottery method is easier to explain in
terms of ratings (range ballots), but can (and should) be adapted to rankings
(ordinal ballots) by modifying the following definition.
Definition 1: Lottery L1beats
://electorama.com/em for list
info
--
Message: 6
Date: Sun, 04 Sep 2005 02:26:03 +0930
From: Chris Benham [EMAIL PROTECTED]
Subject: [EM] Approval variants of MinMax
To: Simmons, Forest [EMAIL PROTECTED],
election-methods-electorama.com@electorama.com
Message-ID
My two cents worth on utility:
1. Utility can be a useful concept for an individual to use in making a
decision, even though it may be impossible to calculate. For example, if
candidates A, B, and C have equal priors of winning, and my preference order is
ABC, then I might decide to approve
Title: Re: [Condorcet] Comment on DMC
Here's something I posted
today on the Condorcet list.
Forest
From: Simmons, Forest Sent: Tue
8/30/2005 1:36 PMTo: [EMAIL PROTECTED]Subject:
Recent History Perspective on Condorcet Methods
As most of you know, the
Election Methods group has
To the message from T.S. copied below, I would like to give a little more
background.
A few months after I came up with the idea of bubble sorting the approval
order, I came across an article at
http://www10.org/cdrom/papers/577/
in which the authors suggested bubble sorting the Borda order
Warren (wds) asked if I could be more precise about reason #17.
17. It is resistant to the burying strategy that plagues some
Condorcet methods. This is related to reason number 9.
The expert on burying in Condorcet methods is James Green-Armytage, who
invented a method called Cardinal
Three more reasons:
16. Like any method that makes germane use of both ordinal and approval
information it is well adapted to three-slot ballots, i.e. voters that don't
want to submit complete rankings can opt to have their approval order extended
by the order of their favorite.
17. It is
Suppose that ...
1. there are three candidates A, B, and C,
2. ballot rankings are strict,
3. in each ordinal faction second ranked candidates are distributed uniformly
between the other two,
and
4. there is a beat cycle ABCA .
Let (alpha, beta, gamma) equal
Under the heading
4 - DISCUSSION OF ALTERNATE NON-GRAPH-BASED ALGORITHMS
Adam said ...
Warren Smith has proposed an alternate solution, which has been
brought up before on the EM list, of simply dragging a cutting edge
across the state to make a division at the proper population ratio,
and
Title: Re: A class of ballot set with unbeaten in mean lotteries.
Jobst and All:
I did make one
(inconsequential) boo boo. The normalization factor for the weights x+y-z,
y+z-x, and z+x-y should be 1/N, not 1/(2N), since the sum of these weights is
x+y+z=N.
To physically carry out the
In a recent message, partly quoted below, Adam Tarr outlined an NP hard
optimization approach to redistricting. He suggested that a genetic
optimization algorithm might be used for practical purposes.
It has been suggested before that in such cases anybody with a proposal found
by any means
Mr. Lomax wrote:
Asset Voting, per se, was invented by Warren Smith, though it strongly
resembles delegable proxy (which seems to have been invented independently
by a number of people, I know I did not get it from anyone else).
I reply:
It's true that the name Asset Voting was invented by
In his response under this subject heading Mr. Lomax seemd to think that I was
advocating Cumulative Voting, then he offered what amounted to a plausibility
argument for my assertion that Cumulative Voting is strategically equivalent to
Plurality. I'm slightly miffed that he would imply that
Title: Re: [EM] 0-info approval voting, repeated polling, and adjusting priors
Let x, y, and z be positive integers such that
x+y+z=N, and max(x,y,z)N/2, where N is the number of some large population
of voters, and the ordinal preferences are divided into three
factions:
x: ABC
y: BCA
Abd ulRahman Lomax proposed:
The proposal is that the ballots might be counted first as ordinary
approval. If a majority appears from this process for a given candidate in
a single-winner election, the candidate would be elected. If not, then the
ballots would be retabulated as fractional
The approval strategy that maximizes voting power (thus minimizing the
probability of an approval voter's regret) in a close three way race is this:
First decide your preference order among the three major candidates, say ABC.
Of course you should approve A and leave C unapproved. Approve B
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