[EM] [CES #8845] Re: MAV on electowiki

[Jameson Quinn]

My general response to Abd: a number of good points, and a number where I'd
disagree. Tough to respond to a kitchen-sink list, so I'll try to
prioritize. Unfortunately that means that things I don't respond to could
be either "you're right" or "I disagree but don't think it's a productive
argument".

2013/6/19 Abd ul-Rahman Lomax <abd at lomaxdesign.com>

> ...
>
>  If one or more candidate has a majority, then the highest majority wins.
>>
>
> Arrghh. That is ordinary Bucklin.
>

Just for the first rank, because at that point there's no higher rank to
fall back to.

>
> At some point it must be specified how "rank overvotes" are handled. There
> are possible ways:
>
> 1. Ballot is voided. We don't like that!
> 2. Vote for candidate is voided. Also that is not great.
> 3. Highest rating marked counts. (This may have been traditional Bucklin.
> The instructions said "don't do it.")
> 4. Lowest rating marked counts.
> 5. Top and bottom ratings are averaged. (So a rating of A and B would give
> a rating that could be called A- or B+. GPA contribution 3.5.)
>

6. Vote counts for forming a majority at top rating, but doesn't count as
"higher than" until you pass bottom rating. This is actually strategically
attractive (if you were in a chicken dilemma, you could give the other side
an honest B, but also a chicken-conservative D or even F, and that would
basically be saying "I think playing chicken is silly but if it turns out
I'm in the minority then game on"), in keeping with the spirit of the
method, and in effect very similar to 5.

But I think that pre-specifying this is premature. These are implementation
details, not part of the method.

>
> I like the fifth option because it actually would allow voters to give an
> intermediate rating, thus providing some additional range resolution with
> no additional ballot complexity, but some cost in canvassing. That
> additional cost would be small, because most voters would not use it, most
> voters not needing it.
>
> In amalgamation, then, as to rank, 3.5 would be added in after A but
> before B.
>
> This would convert a Range 4 ballot to Range 8, with the penultimate
> bottom rating being missing, unless the bottom rating is explicit.
>
>
>  If not, votes at next grade down ("B") are added to each candidate's
>> approval scores. If there are one or more candidates with a majority, the
>> winner is whichever of those had more votes at higher grades (the previous
>> stage). If there were no majorities, then the next grade down ("C") is
>> added and the process repeats; and so on.
>>
>> Note that if this process continues without a majority until the last
>> grade ("F") is added, no new rules are needed.
>>
>
> This is technically correct, because the lowest grade amalgamation *must*
> show 100%. This assumes an F default, so marking of F is *irrelevant*
> except as a confirmation.

If Fs are amalgamated with blanks not being treated as F, then F becomes a
> vote for the bottom-rating candidate. We don't want to create that
> confusion for a moment, even.
>

I agree that I should be more explicit about F default.

>
> For simplicity, dump the F rating entirely. It is assumed, an unexpressed
> default rank.
>

I strongly disagree that there should be no explicit F. As redundant as it
seems to you, there are definitely voters who get satisfaction from
explicitly voting against someone, and I would never design a survey where
an explicit answer was always considered to connote more support than an
implicit "no".

>
> MAV as a method should not specifically depend on the number of ratings.


Absolutely agree. I'll make that even more explicit.


> MAV outside of a runoff system is a plurality method (as are all voting
> systems that can complete without an *explicit* approval of a majority of
> voters. In treating a midrange vote ("C", 2/4) as an approval, we have
> already stretched *slightly* beyond that. Taking this down more deeply into
> D/F waters is pretending a majority that doesn't exist.
>

OK, but in a non-runoff method, sometimes you'll elect without majority
support, and it's still fair to say that D is better than F.

>
> I can see more complex amalgamation rules that could use the D ratings.
> The simplest rule, attempted here, doesn't cut it.


Any of those more complex rules is also trying to infer a majority from a
plurality. They might sometimes do a slightly better job of that, but my
feeling is that it's a false hope to think they'll always succeed, and so
KISS.


>


>   Since by that point all grades will have been counted, all candidate
>> tallies will reach 100%.
>>
>
> In other words, amalgamating the F ratings is *useless*. This treats F as
> if it were a majority, it's a multiple majority, so *unconditionally* the
> previous ratings will be used and the F amalgamation will be ignored. So
> why do we even count it?
>
> Just to make the system seem uniform all the way down?
>
> As I've mentioned, the system should exist in two forms: plurality result,
> or runoff feeder (which, then, is also in two forms: conditional runoff and
> unconditional runoff. Unconditional runoff may be better handled with a
> different system, an explicitly multiwinner one, which has not yet been
> examined in detail.)
>
>
>   The process above then naturally elects the candidate with the most
>> approvals at the higher grades (D or above); that is, whichever has the
>> fewest F's.
>>
>
> And I would say, fewest disapprovals, and disapproval includes D and F. It
> also includes blanks.


Great. Make your own, slightly more-complex system, and give it a different
name.


>
>
>   This is the best way to resolve such an election using only the
>> information on the given ballots.
>>
>
> "Best" is a red flag word when it has not been clearly defined. You'd
> never be allowed to say "best" on Wikipedia, without that, and maybe even
> with it. That's called a "peacock word."
>

OK. A way.

>
> Why would MAV be any "better" than Bucklin?


Wait a minute. In this particular case, MAV and "Bucklin" (you mean, Grand
Junction) do exactly the same thing: use second-to-bottom rank if there are
no majorities.

>From the information on the ballots, the plurality winner in the
> majority-finding stage is arguably the "best winner." No, we support MAV
> because of the effect on strategic voting. If we have a Bucklin system with
> greater ballot resolution, MAV backup may be unnecessary. If voters are
> more informed, it may be unnecessary.
>

You seem to be under the impression that there's a difference between MAV
and "Bucklin" in this case.


>
> I am not satisfied that we have adequately examined the issues, to, yet,
> proclaim MAV as the "best" system up from pure Approval.
>

Nope. I am not sure it's "best", and don't really care. I know that the
differences between Bucklin methods are tiny, yet we'll make no progress
until we settle on one, simple method.


>
> This point should be clearly made, though: MAV is Bucklin, with an
> exception only appearing in the handling of multiple majorities. Those will
> tend to be rare, because, more common, we can expect in some kinds of
> elections, will be majority failure.
>
> How majority failure is handled is a classic problem. Robert's Rules'
> recommendation to organizations: Don't Do It! *Never* elect with a mere
> plurality. The version of IRV that they described *requires* a true
> majority (and even then, they point out the center squeeze problem.)
>
> The backup concept, treating multiple majorities as possibly indicating
> over-rapid addition of approvals, is interesting, but looking at this is
> causing me to warm to a graduated median method.... plus pushing toward
> finer resolution, which will reduce the number of multiple majorities.
> *None of this is going to be fully satisfactory, because making a decision
> with a mere plurality is intrinsically problematic.*
>

Right. And the differences are slim. So let's keep debating these technical
points among ourselves, but use common language in our activism. That's
what this is about, principally.


>
> What I *don't* like with MAV is that a large majority can be ignored vs a
> bare majority, based on the votes at the higher rating. Graduated median
> amalgamation can balance this.


Right. For honest votes, GMJ is clearly superior. See above for why I don't
care.


> But I have not seen a detailed examination of this issue, and it's
> important.


No, it isn't.


> The MAV backup may be just a bit *too* simple. There is an alternative, as
> well, that would pick the candidate with the highest GPA. That would
> consider *all* the ratings. It's theoretically superior, because it would
> be Range. The D ratings would count.
>
> Another approach would use pairwise comparison. We *do* have explicit
> approvals of a majority, if the median vote is C or higher. So, then the
> ballots would be recanvassed to see how the majority-approved candidates
> fare against each other. The collapsed approvals are separate again. This
> would make the method Condorcet-compliant, if I'm correct. And that would
> be a plus. I'm on board the concept that the Condorcet criterion is not
> absolute, because of social utility, but without explicit majority
> confirmation, I dislike finding *against* the preference of a majority.
>
>
>   However, in this and other cases of multiple majorities, a runoff, if
>> feasible, would be a better way to ensure a clean majority win.
>>
>
> It's a *way*. Not simply a "better" one. The alternative presented here is
> *not* a way to "ensure a clean majority win."


In my book, not putting your hand in a band saw is a "better" way not to
cut it off. That is, "a way" is better than "not a way".


>
>  This system was promoted and named due to the confusing array of Bucklin
>> and Median proposals.
>>
>
> Premature. It has not been promoted. It has been *tentatively* named.
> This, by the way, would be in a History section of an article, and it only
> begins to tell the history.
>
>   It is intended to be a relatively generic, simple Bucklin option with
>> good resistance to the <http://wiki.electorama.com/**wiki/Chicken_dilemma<http://wiki.electorama.com/wiki/Chicken_dilemma>>chicken
>> dilemma. It was named by a <http://lists.electorama.com/**
>> pipermail/election-methods-**electorama.com/2013-June/**031938.html<http://lists.electorama.com/pipermail/election-methods-electorama.com/2013-June/031938.html>>poll
>> on the electorama mailing list in June 2013.
>>
>
> Geez, *recentism* to the 9s.
>
> You know what I think about the "chicken dilemma." (Voting is not a game
> of chicken, and "dilemma" is merely an ordinary problem in strategy, which
> only afflicts a small segment of the population, the rest will simply
> *vote.*) What MAV will do is to, I expect, *slightly increase* the number
> of additional approvals, because they become *safer*, i.e., less likely to
> cause the loss of the favorite to the lower ranked candidate.
>
> I prefer, Jameson, that the issue of the exact definition of MAV be *left
> open* for the time being. Let "Majority Approval Voting" actually be
> Supermajority Approval Voting," i.e, ultimately approved by a supermajority
> of voting systems activists.
>
> Not as the "ideal voting system," which it is not, but as the "ideal next
> step beyond ordinary Approval, in a plurality context." It introduces a
> ranked/rated ballot. It's simple to amalgamate and understand. In most
> elections, the "tiebreaker" procedure is not activated. Indeed, in most
> elections under some fairly common conditions, there will be majority
> *failure*, not the multiple majorities that the backup procedure handles.
>
>  The grades or ranks for this system could be numbers instead of letter
>> grades. Terms such as "graded MAV" or "rated MAV" can be used to
>> distinguish these possibilities if necessary. In either case, descriptive
>> labels for the ratings or grades are recommended. For instance, for the
>> letter grades:
>>     * A: Unconditional support
>>    * B: Support if there are no other majorities above "C"
>>    * C: Support if there are no other majorities above "D"
>>    * D: Oppose unless there are no other majorities at all.
>>    * F: Unconditional opposition.
>>
> This treats D as an approval, so it divides the approved categories into
> four rather than the three from Bucklin. We will need, ultimately, a
> single, coherent, simple system to propose and try. Notice that the
> definitions treat not-oppose as support. In my analysis of the traditional
> Bucklin ballot, it's clear that the 3rd rank was *minimum approval.* The
> ordinary meaning of a grade of "D" is a kind of failure.
>

In most schools, you can pass with a "D".


>
> This is a *really complicated* explanation of the rating levels.
>

It's also accurate.

>
>  As the above labels indicate, support at the middle grades or ratings is
>> not partial, as in <http://wiki.electorama.com/**wiki/Score_voting<http://wiki.electorama.com/wiki/Score_voting>>Score
>> voting, but conditional.
>>
>
> That is correct. If the method terminates above the rating, it is not
> used. That, by the way, is a flaw, from a social utility point of view. I
> don't like that voters may be casting votes that are not counted. I'm
> starting to think toward lines of using range amalgamation in the
> "tiebreaker." And of suggesting that all votes be counted, even if only for
> reporting. Incubator effect.


Count and report all grades: of course.
Votes counting fully is a flaw: It gives less social utility for honest
votes, but by doing less to encourage exaggerated votes, it could result in
better information and thus better social utility overall. That is the
basic premise of median versus average. Like it or not, that is what MAV
gives you. And MAV can do a lot better at giving that than "maybe we should
do this, or maybe that, no wait I just thought of another thing..."


>
>
>   That is, the typical ballot will still count fully for or against a
>> given candidate.
>>
>
> Yeah. But I want to see a much more detailed analysis of this.
>
>
>  The different grade levels are a way to help the voting system figure out
>> how far to extend that support so that some candidate gets a majority.
>>
>
> Uh, really? It's more like "so that majority support is found if possible
> from the votes."


OK, sure.


> And a D vote is *not* "support." It might be a weak stand-aside. "Okay, if
> the rest of you insist, this one is better than that one." A strong
> stand-aside would be a C vote. In consider C as neutral, as to preference
> strength. So-so. Not good, not bad. "Passing," but barely. No honors. But
> ... might be serviceable.
>
>
>  For a strategic voter, the most important ratings are the top ("A"),
>> second-to-bottom ("D"), and bottom ("F").
>>
>
> "Most important" is, again, Peacock. Here, Jameson, you are giving your
> own analysis, not the community's analysis.


OK. Section name: "Preliminary and tentative strategic analysis".


> This is not a signed article, even if your name is in the History. If you
> want to present your own analysis, *attribute it.* Or attribute analysis to
> others. I suggest not presenting your own opinions as if they were fact.
> Sometimes in writing Wikipedia articles, I'd use "weasel words." This is
> weak language, like, "Some say that ...." or "It could be claimed that
> ...." or "According to some sources, ...." I really only did that to find
> quick consensus, not to propose weasel language as stable or desirable. It
> was typically replacing *strong language* that did not actually reflect
> consensus.
>
>
>  A typical zero-knowledge strategy would be to give the best 30% of
>> candidates an "A", the next 25% a "D", and the bottom 45% an "F".
>>
>
> "Typical" according to what standard? What population? I would *not* think
> this way *at all.*


"Strategic" means at a minimum "game-theoretic equilibrium". Your
suggestion is certainly not that, so it is not "strategic", it is "honest".


> I would cast a Range ballot, period, using the same strategy, i.e.,
> considering what I know of the preferences of others. With Bucklin, I can
> vote sincerely for my favorite under almost all conditions, and the loss of
> voting power is miniscule. If it's a zero-knowledge situation, I'd vote
> simple, normalized Range. I'm really not worried about the multiple
> majority problem. My votes will represent *true preference strength*. That
> only shifts with knowledge of the electorate, where I will then vote Von
> Neumann - Morgenstern utilities. I.e., to put it simply, which will
> powerfully choose between realistic possibilities, reserving *some voting
> power* for the expression of true preference, which is an independent
> value. (And where partisan elections are involved, that has a long-term
> effect. I am not just concerned about the *present election.*)
>

Great analysis of why an honest vote is a good idea in this system.

>
> Some voting systems activists seem to think that voters are obsessed with
> "winning," but that all that "winning" means to them is getting their
> favorite elected. No, many voters consider it a "win" if the winner has
> broad support, *even if they were personally opposed.*
>
>
>  If the typical "honest" voter roughly calibrates their grades to an
>> academic curve, with a median vote at "B" or "C", then strategic and honest
>> votes will mesh well.
>>
>
> Perhaps. All this is vague, ungrounded argument. We don't have the
> simulations yet.
>
>
>  For instance, if candidates can differ on two dimensions, ideology and
>> quality, and voters are normally distributed along the one dimension of
>> ideology (with all voters preferring highest quality), then this system
>> will tend to elect the candidate preferred by the median voter, that is,
>> the one with the smallest sum of quality deficit plus ideological skew; and
>> this tendency will hold for any unbiased combination of "honest" and
>> "strategic" voters as defined above.
>>
>
> Jameson, you are presenting as if it were fact, your "back of the
> envelope" conclusions. Please don't do that!
>
>
>  The assertions in the strategic paragraph are based on some
>> back-of-the-envelope diagrams; that is, I consider them likely to be true,
>> but I have not run simulations to prove them. I think it would be
>> interesting to do so. Would others be as interested as I would in such
>> results; that is,
>>
>
> "In the opinion of Jameson Quinn ...."
>
> Or, "As shown in a poll conduced by the Center for Election Science....."
>
>
>  1. Finding an equilibrium zero-knowledge strategy (percentile-grade
>> correspondence) in impartial culture. (I think this would be an exciting
>> new direction for simulation research.)
>>
>
> I want to see Bucklin applied again, because the ballots will be
> collecting Range data, not only for its value as a voting system.
>
> The true revolutionary system is Asset, of course. It sweeps all these
> concerns aside, makes them moot.
>
>
>  2. Finding how broad the strategic conditions are (testing different
>> "honest" grade distributions, unbiased strat/hon mixes, and strategic
>> biases) in which MAV elects the median voter's favorite in the 2D/1D model
>> sketched above? If my intuition is right, this model (unlike sparse or
>> impartial models as criticized by Regenwetter) will allow good systems to
>> show near-optimal BR; so MAV and Score will be have nearly the same (and
>> nearly 0) honest BR, and the differences will be in that BR's robustness to
>> different strategic profiles.
>>
>
> I'm suspecting that a different tiebreaker rule can do better. A Range
> rule would *explicitly* maximize BR, as to what was expressed. Runoff
> systems, of course, are known to take us beyond single-ballot Range as to
> BR optimization with realistic votes.
>
>
>  Jameson
>>
>> 2013/6/18 Jameson Quinn <<mailto:jameson.quinn at gmail.**com<jameson.quinn at gmail.com>
>> >jameson.quinn at gmail.com>
>>
>> I've reworked the description. See what you think.
>>
>>
>> 2013/6/18 Abd ul-Rahman Lomax <<mailto:abd at lomaxdesign.com>a**
>> bd at lomaxdesign.com <abd at lomaxdesign.com>>
>>
>> At 04:25 PM 6/18/2013, Juho Laatu wrote:
>> I quickly read the article. Here are some observations.
>>
>> - Term "Bucklin system" has not been defined. I can guess that it
>> probably refers to Bucklin style stepwise addition of new approvals, but
>> that may not be as obvious to all readers. If there is no definition of
>> "Bucklin system", maybe one could say "As in Bucklin" instead of "As with
>> any Bucklin system".
>>
>>
>> There is a link to Bucklin voting in the article.
>>
>>
>> - Sentence "if there are more than one with a majority, the "B" votes are
>> removed and the highest sub-majority wins" is ambigious in the sense that
>> it is not clear if "highest sub-majority" refers to all candidates or to
>> candidates that had majority after adding the "B" votes.
>>
>>
>> It's poorly worded, all right. Minor point: "There are more than one"
>> grates. (I find the use of the singular or plural with "more" to be
>> ambiguous. I'd avoid it.)
>>
>> An example is given when the principle has not been stated.
>>
>> The method does not make sense as stated. The "back-up" is a tie-breaker,
>> considering multiple majorities as if they were ties. They *are* ties in
>> median vote. The tie-breaker only selects a member of the tied set.
>>
>> Something went south. What was proposed was a Bucklin system. Bucklin
>> does use, I've suggested, a range ballot, but the way that it does this is
>> with a ranked structure. I ran into this when trying to design a set of
>> votes to show a problem that I have not seen examined.
>>
>> The description on the wiki page makes the system seem more complex than
>> it is.
>>
>> It's been designed to be five-rank, with explicit F. That's a fish
>> bicycle. "No support" means merely "no support." No vote. Introducing the D
>> vote is a later possible reform, it is an unapproved category. It makes the
>> ballot considerably more complex, and the explanation is more complex.
>>
>> *D: Oppose unless there are no other majorities at all.
>>
>>
>> Is that clear? I don't think so. Bucklin as Approval Voting doesn't have
>> a "disapproved rank." All blanks are disapproved.
>>
>>
>> - It is not quite clear what happens and if it is possible that there is
>> no majority after the "F" votes have been counted.
>>
>>
>> The F votes are never counted, first of all. Listing them is a mistake.
>> (If the F votes continued the amalgamation, then someone would be voting
>> *for* a candidate rated F. That was the intention for the D rating.
>>
>> It is far better, however, to introduce a D rating in combination with a
>> runoff system, where the D rating could improve runoff candidate selection.
>>
>> When a voter rates a candidate as "D", they are opposing the election of
>> that candidate.
>>
>> The Bucklin system required amalgamating three ranks. It's looking like
>> MAV requires five, but that could be reduced to four, but the whole idea
>> here was to have a *simple* next step beyond basic Approval Voting, and, as
>> well, a clear similar method for use in a runoff system.
>>
>> (We basically need a step up from approval as a plurality method, and
>> from approval as a primary method in a runoff system.)
>>
>>
>> - The grades could be letters or numbers, but they could also be e.g.
>> columns without any letter or number. This part of text discusses what the
>> ballots might look like. I'm not sure if ballot different ballot formats
>> should be seen as an essential part of the method definition, or if the
>> method should be defined abstractly without referring to what the actual
>> ballots might look like. I tend to define the methods abstractly without
>> assuming anything on the ballots, and then discuss possible ballot formats
>> as a separate topic, but I'm not saying that's the only and best approach.
>> The current text is thus ok. I just first read the grades of the definition
>> as abstract grades, not as definitions on what would be written in the
>> ballots.
>>
>>
>> *Something* should be on the ballot that expresses the *function* of a
>> vote. Jameson took this concept from me. A voter should be able to see the
>> ballot and have a reasonably clear idea, just from it, what the vote
>> *means* ... and the meaning is the *effect* that the vote causes.
>>
>> The original Bucklin ballot, however, simply instructed voters to mark
>> "1st choice," "2nd choice," or '3rd choice." The googlebooks copy is
>> unclear, <http://books.google.com/**books?id=QcIqAAAAYAAJ&pg=RA1-**
>> PA757&dq=The+Grand+Junction+**plan+of+city+government+and+**
>> its+results&hl=en&ei=**uOTdS7aFKMKclgfq9739Cg&sa=X&**
>> oi=book_result&ct=result&**resnum=3&ved=0CDsQ6AEwAg#v=**
>> onepage&q=The%20Grand%**20Junction%20plan%20of%20city%**
>> 20government%20and%20its%**20results&f=false<http://books.google.com/books?id=QcIqAAAAYAAJ&pg=RA1-PA757&dq=The+Grand+Junction+plan+of+city+government+and+its+results&hl=en&ei=uOTdS7aFKMKclgfq9739Cg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDsQ6AEwAg#v=onepage&q=The%20Grand%20Junction%20plan%20of%20city%20government%20and%20its%20results&f=false>
>> >http://**books.google.com/books?id=**QcIqAAAAYAAJ&pg=RA1-PA757&dq=**
>> The+Grand+Junction+plan+of+**city+government+and+its+**results&hl=en&ei=*
>> *uOTdS7aFKMKclgfq9739Cg&sa=X&**oi=book_result&ct=result&**
>> resnum=3&ved=0CDsQ6AEwAg#v=**onepage&q=The%20Grand%**
>> 20Junction%20plan%20of%20city%**20government%20and%20its%**
>> 20results&f=false<http://books.google.com/books?id=QcIqAAAAYAAJ&pg=RA1-PA757&dq=The+Grand+Junction+plan+of+city+government+and+its+results&hl=en&ei=uOTdS7aFKMKclgfq9739Cg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDsQ6AEwAg#v=onepage&q=The%20Grand%20Junction%20plan%20of%20city%20government%20and%20its%20results&f=false>
>>
>>
>> Page 95. It looks like they actually instructed people to vote for all
>> but one. But that part is quite unclear. In the first set of instructions,
>> at the top of the ballot, they did suggest not voting for one candidate.
>> There may be another copy of this ballot somewhere. Bucklin was widely
>> covered.
>>
>> MAV *assumes that voters err if they approve two candidates by a
>> majority.* That's why it backs up. But what, indeed, if it backs up and the
>> multiple majority candidates are not the plurality winner in the previous
>> round? What if there are *no* votes for those candidates in that round, or
>> the vote is small.
>>
>>
>> It said: "
>>
>>
>> - The linked definition of "evaluatve" says that ranked systems can not
>> give same ratings to two candidates. I think that's confusing and wrong.
>>
>> Juho
>>
>>
>> Well, that's a common assumption of "ranked systems." It's essentially a
>> definition, which is why we have said that Bucklin is *not* a ranked
>> system. But, really, it's a ranked system that allows equal ranking.
>> (Original Bucklin allowed equal ranking in the third rank only. We have
>> simple expanded the approval principle to all ranks. *That is a convenience
>> to voters.*)
>>
>>
>> ----
>> Election-Methods mailing list - see <http://electorama.com/em>http**
>> ://electorama.com/em <http://electorama.com/em> for list info
>>
>>
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "The Center for Election Science" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> email to electionscience+unsubscribe@**googlegroups.com<electionscience%2Bunsubscribe at googlegroups.com>
>> .
>> For more options, visit <https://groups.google.com/**groups/opt_out<https://groups.google.com/groups/opt_out>
>> >https://groups.**google.com/groups/opt_out<https://groups.google.com/groups/opt_out>
>> .
>>
>>
>>
> --
> You received this message because you are subscribed to the Google Groups
> "The Center for Election Science" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to electionscience+unsubscribe@**googlegroups.com<electionscience%2Bunsubscribe at googlegroups.com>
> .
> For more options, visit https://groups.google.com/**groups/opt_out<https://groups.google.com/groups/opt_out>
> .
>
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20130619/533dc0df/attachment-0004.htm>