[EM] (3) Best Single-Winner Method

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[EM] (3) Best Single-Winner Method

steve bosworth

Hi Ted,

Thank you also for discussing MJ.  I will respond inline below but firstly I want to make it clear that in the end, I would like answers or explanations for the following two questions:

  1. Does your preferred voting method guarantee that its single-winner is supported by a majority of all the votes cast?
  2. If not, why is this method still favored by you?


From: Ted Stern <[hidden email]>
Sent: Friday, June 14, 2019 12:33 AM
To: steve bosworth
Cc: [hidden email]; EM list
Subject: Re: [EM] (2) Best Single-Winner Method-IBIFA vs. MJ

 

Hi Steve,

T:  I have developed a slight modification of Chris Benham's IBIFA that is more similar to Majority Judgment.  I've started editing a page describing the method at

https://electowiki.org/wiki/Relevant_rating

Balinski and Laraki have an alternate form of Majority Judgment in which each candidate receives what they call a Majority Grade.

The Majority Grade has several parts.  The primary portion is the candidate's median rating.  That is, the rating MR at which the number of ballots approving candidate X at rating MR exceeds the total number of ballots rating X below MR.

S:  Yes.

S: It is not particularly important for our purposes, but my recollection is that B&L’s term for this three part summary of each candidates majority-grade is its “majority-gauge”.

T:  The secondary portion of the Majority Grade is a measure of what median rating the candidate would get if median ballots are removed until the rating changes.  But that rating is simply the rating that is "closer" to the median ballot. 

S:  More exactly, the median-gauge starts with the verbal expression which contains all the grades that are equal in value to that candidate’s median-grade(e.g. Good).  This grade is in the middles of a complete list of all the grades received by a given candidate, e.g. in the middle of a list of all the grades received, listed highest to lowest, top to bottom (i.e. the middle grade if the number of voters is odd, the lower middle if the number is even).

T:  In other words, if the total number of ballots rating X below the MR is less than the total number of ballots rating X above the MR, then the secondary rating is MR+1, followed by the above MR total.  Otherwise, the secondary portion of the Majority Grade is MR minus one, followed by the at-and-above-MR total.

S:  Instead, according to my recollection of B&L, an example of a majority-gauge about which you are referring above is the follow: 42% Good+31%.

This would mean that 25% of each of all the grades given to this candidate are of a value equal to this candidate’s median-grade (i.e. Good).  Perhaps this way of reporting the grades received by each candidate helps some readers better to understand how MJ works.  However, I see the steps described earlier on pp. 5&17 of Majority Judgment as providing the simplest way to find the winner when there is a tie, i.e. one-by-one, remove one grade from the list of each tied candidate equal to the value of their shared highest median-grade.  Do this repeatedly until only one has the highest median-grade.

T:  So B&L are choosing to see the tie breaking as removing median ballots, but an alternative viewpoint would be to interpret the tie-breaking as comparing the above-median-rating strength to the below-median-rating-preference.

S: Yes.

T:  If you're talking about an election, I think the latter viewpoint is more meaningful.

S:  No, again I see the removal of ballots one-by-one currently reporting the same highest median-grade from each of the tied candidates as the simplest and most exact way of discovering the one candidate who continues to have the highest median grade.

T:  So what Chris and I are saying is that the meaningful part of that comparison is not simply the total number of ballots expressing a preference other than X, but whether there is a relevant alternative candidate Y whose total approval is comparable to that of X at that rating level. ….

S:  In some vague sense there could be some other “comparable” candidates but none of these would have this highest median grade at the end of he above tie breaking process.

T: If additional ballots don't contribute to the approval of a meaningful alternative candidate, then we experience a spoiler effect, a tyranny of the majority.

S:  Any election by any voting method might be changed by adding more ballots containing certain or preferences.  MJ is no exception.  As I see it, the great advantage that MJ offers is firstly that it allows each voters to express themselves fully and most meaningfully when grading the suitability of each candidate for office.  This means that the public will know exactly how many of each of the grades from Excellent to Reject each candidate received from all the voter after the count of the election, i.e. the public will be more informed by MJ’s results than by any alternative election. 

Secondly, it guarantees that each of all the votes will be counted equally to help determine the median-grade of each candidate.  This allows the most discerning and meaningful discovery of the one candidate who has finally received the highest median-grade (i.e. received at least 50% plus one of all the grades which have a value at least as high as their highest median-grade).  This second benefit makes it clear that no citizens vote can be discarded as “irrelevant”.

T:  I mean, isn't the whole point of using a median rating in the first place an attempt to avoid the sensitivity to outliers inherent in the mean? ….

S:  Yes, it tends to moderate the effect of outliers but most importantly, it guarantees that the winner is explicitly supported by a majority.

T: …. If your answer experiences large changes in response to small changes in input, those of us in the numerical analysis community would call that ill-conditioned, something to be avoided. 

S:  No mater what counting rules might be used, small changes in input can produce large changes.

T:  Personally, what I want to achieve with a voting method is to find the candidate closest to the center of mass of the population, but that's hard to do when each individual is making their own best guess to how near they are in preference space to each candidates.

S:  Since B&L show how MJ is the method most likely to prompt the honest grading of the candidates, doesn’t the fact that MJ also elects the candidate most highly supported by a majority of the citizens make it most probable that both this majority and the winner will be “closest to the center of the mass of the population”?

Also, recall that MJ allows each citizen to guarantee that they will have up to four sets of opportunities for one of their highly graded candidates to be the one elected.  Firstly, a candidate whom they give an Excellent will be elected by an absolute majority, provided a sufficient number of other citizens see that candidate as at least Acceptable.  If not, they have a second opportunity for any of the candidates they grade as Very Good to receive such a instead.  Similarly, they have a third and fourth opportunity for candidates they might have graded as Good or Acceptable.

T:  For all practical purposes, my relevant ratings method would give the same answer as MJ anyway.

S:  Perhaps, but to the extent that this is not guaranteed, RR would be flawed.  In fact, according to my current understanding of RR (as presented in your recently edited ElecoWiki article), it is flawed. RR prevents some approvals for a potential rival candidate from being counted for that rival, i.e. any approval for the rival which is on the same ballot that has already been added to the candidate who currently has the largest number of Top-ratings must not be added to the rival’s potential total.

Is my interpretation correct?  What do you think?

I look forward to the next step in our dialogue.

Steve



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Re: [EM] (3) Best Single-Winner Method

Ted Stern
See inserted answers

On Mon, Jun 17, 2019 at 4:28 PM steve bosworth <[hidden email]> wrote:

Hi Ted,

Thank you also for discussing MJ.  I will respond inline below but firstly I want to make it clear that in the end, I would like answers or explanations for the following two questions:

  1. Does your preferred voting method guarantee that its single-winner is supported by a majority of all the votes cast?

Hi Steve,

You seem to have not processed the response from Chris Benham which was posted earlier, with the example I proposed 18 months ago:
46: A
03: A>B
25: C>B
23: D>B

97 ballots  (majority threshold = 48)
with 3 ballots supporting E added later.

Here we can clearly see that Relevant Ratings will find A as the winner, as if the ballots by E were never cast.  So A is clearly not supported by a majority.

What RR / IBIFA does do is guarantee that there is no other candidate approved by more people rating A below top rating (in this case) than those who rate A at top rating.
 
  1. If not, why is this method still favored by you?

I prefer methods that satisfy the favorite betrayal criterion and later-no-help (for burial resistance), while coming close to Condorcet compliance, because that tends to ensure that the winner will be close to the centroid of the voters.  As we can see by the example above, A is the Condorcet Winner both before and after the addition of "irrelevant ballots".

B loses pairwise to A.  If there were a run-off election, B would lose, at least based on stated preferences.

If you added 3 B top-rated votes to the example above, instead of 3 E top-rated votes, that would be a significant change, since B is the strongest complementary preference to A.  That would also switch the Condorcet Winner from A to B.  So Relevant Ratings / IBIFA gives the B faction an incentive to try to persuade new voters to support B.  But under MJ, all the B block has to do is persuade a sufficient number of protest voters to participate to deny A the victory, even though those voters are effectively abstaining from the A vs. B election.

So while there is very little difference between Relevant Ratings and Majority Judgment in practice, there could still be the possibility of a preference failure of this type if the electorate is somewhat splintered.

I speculate that if there is ever a difference between the RR and MJ winners, that the RR winner would be preferred pairwise.

I would certainly favor Majority Judgment over many other options, if there were no other choice, though I'd prefer if it included a runoff against the Condorcet Winner if one exists.
 


From: Ted Stern <[hidden email]>
Sent: Friday, June 14, 2019 12:33 AM
To: steve bosworth
Cc: [hidden email]; EM list
Subject: Re: [EM] (2) Best Single-Winner Method-IBIFA vs. MJ

 

Hi Steve,

T:  I have developed a slight modification of Chris Benham's IBIFA that is more similar to Majority Judgment.  I've started editing a page describing the method at

https://electowiki.org/wiki/Relevant_rating

Balinski and Laraki have an alternate form of Majority Judgment in which each candidate receives what they call a Majority Grade.

The Majority Grade has several parts.  The primary portion is the candidate's median rating.  That is, the rating MR at which the number of ballots approving candidate X at rating MR exceeds the total number of ballots rating X below MR.

S:  Yes.

S: It is not particularly important for our purposes, but my recollection is that B&L’s term for this three part summary of each candidates majority-grade is its “majority-gauge”.

T:  The secondary portion of the Majority Grade is a measure of what median rating the candidate would get if median ballots are removed until the rating changes.  But that rating is simply the rating that is "closer" to the median ballot. 

S:  More exactly, the median-gauge starts with the verbal expression which contains all the grades that are equal in value to that candidate’s median-grade(e.g. Good).  This grade is in the middles of a complete list of all the grades received by a given candidate, e.g. in the middle of a list of all the grades received, listed highest to lowest, top to bottom (i.e. the middle grade if the number of voters is odd, the lower middle if the number is even).

T:  In other words, if the total number of ballots rating X below the MR is less than the total number of ballots rating X above the MR, then the secondary rating is MR+1, followed by the above MR total.  Otherwise, the secondary portion of the Majority Grade is MR minus one, followed by the at-and-above-MR total.

S:  Instead, according to my recollection of B&L, an example of a majority-gauge about which you are referring above is the follow: 42% Good+31%.

This would mean that 25% of each of all the grades given to this candidate are of a value equal to this candidate’s median-grade (i.e. Good).  Perhaps this way of reporting the grades received by each candidate helps some readers better to understand how MJ works.  However, I see the steps described earlier on pp. 5&17 of Majority Judgment as providing the simplest way to find the winner when there is a tie, i.e. one-by-one, remove one grade from the list of each tied candidate equal to the value of their shared highest median-grade.  Do this repeatedly until only one has the highest median-grade.

T:  So B&L are choosing to see the tie breaking as removing median ballots, but an alternative viewpoint would be to interpret the tie-breaking as comparing the above-median-rating strength to the below-median-rating-preference.

S: Yes.

T:  If you're talking about an election, I think the latter viewpoint is more meaningful.

S:  No, again I see the removal of ballots one-by-one currently reporting the same highest median-grade from each of the tied candidates as the simplest and most exact way of discovering the one candidate who continues to have the highest median grade.

T:  So what Chris and I are saying is that the meaningful part of that comparison is not simply the total number of ballots expressing a preference other than X, but whether there is a relevant alternative candidate Y whose total approval is comparable to that of X at that rating level. ….

S:  In some vague sense there could be some other “comparable” candidates but none of these would have this highest median grade at the end of he above tie breaking process.

T: If additional ballots don't contribute to the approval of a meaningful alternative candidate, then we experience a spoiler effect, a tyranny of the majority.

S:  Any election by any voting method might be changed by adding more ballots containing certain or preferences.  MJ is no exception.  As I see it, the great advantage that MJ offers is firstly that it allows each voters to express themselves fully and most meaningfully when grading the suitability of each candidate for office.  This means that the public will know exactly how many of each of the grades from Excellent to Reject each candidate received from all the voter after the count of the election, i.e. the public will be more informed by MJ’s results than by any alternative election. 

Secondly, it guarantees that each of all the votes will be counted equally to help determine the median-grade of each candidate.  This allows the most discerning and meaningful discovery of the one candidate who has finally received the highest median-grade (i.e. received at least 50% plus one of all the grades which have a value at least as high as their highest median-grade).  This second benefit makes it clear that no citizens vote can be discarded as “irrelevant”.

T:  I mean, isn't the whole point of using a median rating in the first place an attempt to avoid the sensitivity to outliers inherent in the mean? ….

S:  Yes, it tends to moderate the effect of outliers but most importantly, it guarantees that the winner is explicitly supported by a majority.

T: …. If your answer experiences large changes in response to small changes in input, those of us in the numerical analysis community would call that ill-conditioned, something to be avoided. 

S:  No mater what counting rules might be used, small changes in input can produce large changes.

T:  Personally, what I want to achieve with a voting method is to find the candidate closest to the center of mass of the population, but that's hard to do when each individual is making their own best guess to how near they are in preference space to each candidates.

S:  Since B&L show how MJ is the method most likely to prompt the honest grading of the candidates, doesn’t the fact that MJ also elects the candidate most highly supported by a majority of the citizens make it most probable that both this majority and the winner will be “closest to the center of the mass of the population”?

Also, recall that MJ allows each citizen to guarantee that they will have up to four sets of opportunities for one of their highly graded candidates to be the one elected.  Firstly, a candidate whom they give an Excellent will be elected by an absolute majority, provided a sufficient number of other citizens see that candidate as at least Acceptable.  If not, they have a second opportunity for any of the candidates they grade as Very Good to receive such a instead.  Similarly, they have a third and fourth opportunity for candidates they might have graded as Good or Acceptable.

T:  For all practical purposes, my relevant ratings method would give the same answer as MJ anyway.

S:  Perhaps, but to the extent that this is not guaranteed, RR would be flawed.  In fact, according to my current understanding of RR (as presented in your recently edited ElecoWiki article), it is flawed. RR prevents some approvals for a potential rival candidate from being counted for that rival, i.e. any approval for the rival which is on the same ballot that has already been added to the candidate who currently has the largest number of Top-ratings must not be added to the rival’s potential total.

Is my interpretation correct?  What do you think?

I look forward to the next step in our dialogue.

Steve



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Re: [EM] (3) Best Single-Winner Method

robert bristow-johnson



---------------------------- Original Message ----------------------------
Subject: Re: [EM] (3) Best Single-Winner Method
From: "Ted Stern" <[hidden email]>
Date: Mon, June 17, 2019 5:24 pm
To: "steve bosworth" <[hidden email]>
Cc: "[hidden email]" <[hidden email]>
--------------------------------------------------------------------------

> I would certainly favor Majority Judgment over many other options, if there
> were no other choice, though I'd prefer if it included a runoff against the
> Condorcet Winner if one exists.
 

I know that i am just stuck in Condorcet land and I haven't completely groked MJ (because i don't like the ballot), but can you guys help me understand how **any** candidate beats the Condorcet Winner in a runoff?

If you have an election method and subject that winner of that method to the CW (whom must have been chosen from ranked-ballot results) in a runoff, does not always the CW win?  If so, then why not just elect the CW?


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Re: [EM] (3) Best Single-Winner Method

Ted Stern


On Mon, Jun 17, 2019, 18:51 robert bristow-johnson <[hidden email]> wrote:



---------------------------- Original Message ----------------------------
Subject: Re: [EM] (3) Best Single-Winner Method
From: "Ted Stern" <[hidden email]>
Date: Mon, June 17, 2019 5:24 pm
To: "steve bosworth" <[hidden email]>
Cc: "[hidden email]" <[hidden email]>
--------------------------------------------------------------------------

> I would certainly favor Majority Judgment over many other options, if there
> were no other choice, though I'd prefer if it included a runoff against the
> Condorcet Winner if one exists.
 

I know that i am just stuck in Condorcet land and I haven't completely groked MJ (because i don't like the ballot), but can you guys help me understand how **any** candidate beats the Condorcet Winner in a runoff?


Voters are not rational. Also, in a runoff, they are not necessarily the same voters. 

If you have an election method and subject that winner of that method to the CW (whom must have been chosen from ranked-ballot results) in a runoff, does not always the CW win?  If so, then why not just elect the CW?

Indeed, that would be the case. But I think voters may not realize the ramifications of their selections in early use of any new method, so an option to reconsider might be beneficial. 


--

r b-j                         [hidden email]

"Imagination is more important than knowledge."
 

 

 

 


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Re: [EM] (3) Best Single-Winner Method

robert bristow-johnson



---------------------------- Original Message ----------------------------
Subject: Re: [EM] (3) Best Single-Winner Method
From: "Ted Stern" <[hidden email]>
Date: Mon, June 17, 2019 10:24 pm
To: [hidden email]
Cc: "steve bosworth" <[hidden email]>
"EM" <[hidden email]>
--------------------------------------------------------------------------

> On Mon, Jun 17, 2019, 18:51 robert bristow-johnson <[hidden email]>
> wrote:
>
>>
>>
>> ---------------------------- Original Message ----------------------------
>> Subject: Re: [EM] (3) Best Single-Winner Method
>>
From: "Ted Stern" <[hidden email]>
>> Date: Mon, June 17, 2019 5:24 pm
>> To: "steve bosworth" <[hidden email]>
>> Cc: "[hidden email]" <
>> [hidden email]>
>> --------------------------------------------------------------------------
>>
>> > I would certainly favor Majority Judgment over many other options, if there
>> > were no other choice, though I'd prefer if it included a runoff against the
>> > Condorcet Winner if one exists.
>>
>>
>> I know that i am just stuck in Condorcet Land and I haven't completely
>> groked MJ (because i don't like the ballot), but can you guys help me
>> understand how **any** candidate beats the Condorcet Winner in a runoff?
>>
>
> Voters are not rational.

Yeah, but I wouldn't wanna base a governmental election system on that assumption.

> Also, in a runoff, they are not necessarily the same voters.

But one of the purposes of voting with ranked ballots is so that they **are** the same voters.  This is why we collect contingency preferences from the *same* voters on the *same* election day.

>> If you have an election method and subject that winner of that method to
>> the CW (whom must have been chosen from ranked-ballot results) in a runoff,
>> does not always the CW win? If so, then why not just elect the CW?
>>
> Indeed, that would be the case. But I think voters may not realize the
> ramifications of their selections in early use of any new method, so an
> option to reconsider might be beneficial.

I dooooooon't like that.  I think elections should be decisive.

I ain't UK, but I hate Brexit almost as much as I hate Trump and Trumpism, and I think that the original Brexit question in 2016 was malformed.  It should have, from the very beginning, have been understood to be a two-referendum deal.  The first (in 2016) to get the ball rolling (or to end the discussion if the question fails), then Parliament and the Government gets to negotiate the best deal they can with the EU, and **then** that actual deal is put to referendum for a final yes/no vote.  If it passes both times, it's hard to declare it's not the will of the people in a democracy.  But this is essentially a "constitutional" level political question.  In the States, that would need 2/3 of Congress and 3/4 of the states to change.  Changing something "constitutionally" should not be easy.  It should never have been decided by a thin, simple majority in a single election.  (Fuck, what are the next generation of Brits gonna think about their stupid parents generation and elders fucking up their future?)

But, in general, if people get to weigh in on something, or on a candidate, and the people speak and then later you give them the "option to reconsider" and in between all sorts of money and election shenanigans get dumped in there, and then the "voice" of the people changes to a different collective opinion.  Especially if the margin is thin, if we're the voters that originally prevailed and then they tell us we didn't in the "reconsideration", then it's time for pitchforks and torches and trouble in the streets.  This is why, when there is a FPTP election with no majority, and the law calls for a (delayed) runoff, and only half of the electorate that voted on the original election day show up for the runoff, and the original plurality winner is defeated in the runoff with half of the usual voters voting, there is trouble and the legitimacy of the upset winner is questioned.

Delayed runoffs suck.  They suck green donkey dick.  Almost as bad as FPTP sucks.  This is why we want Ranked-Choice Voting even in cases where majorities are required.

Still a partisan for Condorcet with the simplest meaningful rules for dealing with possible cycles.

Someday, when I have time to form some questions, I'll ask Chris B for detail (maybe examples) why he doesn't like RP with Margins.  Seems simple to me.  And meaningful.

L8r,
 

--

r b-j                         [hidden email]

"Imagination is more important than knowledge."
 

 

 

 


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Re: [EM] (3) Best Single-Winner Method

C.Benham
In reply to this post by Ted Stern

I was going to type simply "Good question."

One of the major justifications MJ has is that it  (like Relevant Ratings and IBIFA and Bucklin and Approval)
meets the Favorite Betrayal Criterion (aka FBC) which isn't compatible with Condorcet.

In other words, MJ's  (main) excuse for failing Condorcet is that it meets FBC.

But when you add a 2-candidate runoff you always have a possible Push-over incentive and so no
compliance with FBC.

And so the whole thing becomes a weird method that meets Condorcet but probably fails Smith.

Chris Benham

On 18/06/2019 2:54 pm, Ted Stern wrote:


On Mon, Jun 17, 2019, 18:51 robert bristow-johnson <[hidden email]> wrote:



---------------------------- Original Message ----------------------------
Subject: Re: [EM] (3) Best Single-Winner Method
From: "Ted Stern" <[hidden email]>
Date: Mon, June 17, 2019 5:24 pm
To: "steve bosworth" <[hidden email]>
Cc: "[hidden email]" <[hidden email]>
--------------------------------------------------------------------------

> I would certainly favor Majority Judgment over many other options, if there
> were no other choice, though I'd prefer if it included a runoff against the
> Condorcet Winner if one exists.
 

I know that i am just stuck in Condorcet land and I haven't completely groked MJ (because i don't like the ballot), but can you guys help me understand how **any** candidate beats the Condorcet Winner in a runoff?


Voters are not rational. Also, in a runoff, they are not necessarily the same voters. 

If you have an election method and subject that winner of that method to the CW (whom must have been chosen from ranked-ballot results) in a runoff, does not always the CW win?  If so, then why not just elect the CW?

Indeed, that would be the case. But I think voters may not realize the ramifications of their selections in early use of any new method, so an option to reconsider might be beneficial. 


--

r b-j                         [hidden email]

"Imagination is more important than knowledge."
 

 

 

 


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Re: [EM] (3) Best Single-Winner Method

Ted Stern
Chris, your point about pushover strategy is a good one.  I was making the suggestion in an effort to persuade Steve.

Another example which demonstrates problems with MJ is the following:

101: A > B > C > D > E > F > G
101: B > A = C > D > E > F > G
101: C > B = D > A = E > F > G
050: D > C = E > B = F > A = G
099: E > D = F > C = G > B > A
099: F > E = G > D > C > B > A
099: G > F > E > D > C > B > A

which is taken from https://en.wikipedia.org/wiki/Majority_judgment#Outcome_in_political_environments .  Majority Judgment selects B while Condorcet (or MCA/Bucklin) selects C, who seems to be minimizing variance.

Smith//Relevant-Ratings is an interesting method (essentially what Chris was suggesting earlier).  It seems to be able to handle several burial attack strategies successfully.

On Tue, Jun 18, 2019 at 8:26 AM C.Benham <[hidden email]> wrote:

I was going to type simply "Good question."

One of the major justifications MJ has is that it  (like Relevant Ratings and IBIFA and Bucklin and Approval)
meets the Favorite Betrayal Criterion (aka FBC) which isn't compatible with Condorcet.

In other words, MJ's  (main) excuse for failing Condorcet is that it meets FBC.

But when you add a 2-candidate runoff you always have a possible Push-over incentive and so no
compliance with FBC.

And so the whole thing becomes a weird method that meets Condorcet but probably fails Smith.

Chris Benham

On 18/06/2019 2:54 pm, Ted Stern wrote:


On Mon, Jun 17, 2019, 18:51 robert bristow-johnson <[hidden email]> wrote:



---------------------------- Original Message ----------------------------
Subject: Re: [EM] (3) Best Single-Winner Method
From: "Ted Stern" <[hidden email]>
Date: Mon, June 17, 2019 5:24 pm
To: "steve bosworth" <[hidden email]>
Cc: "[hidden email]" <[hidden email]>
--------------------------------------------------------------------------

> I would certainly favor Majority Judgment over many other options, if there
> were no other choice, though I'd prefer if it included a runoff against the
> Condorcet Winner if one exists.
 

I know that i am just stuck in Condorcet land and I haven't completely groked MJ (because i don't like the ballot), but can you guys help me understand how **any** candidate beats the Condorcet Winner in a runoff?


Voters are not rational. Also, in a runoff, they are not necessarily the same voters. 

If you have an election method and subject that winner of that method to the CW (whom must have been chosen from ranked-ballot results) in a runoff, does not always the CW win?  If so, then why not just elect the CW?

Indeed, that would be the case. But I think voters may not realize the ramifications of their selections in early use of any new method, so an option to reconsider might be beneficial. 


--

r b-j                         [hidden email]

"Imagination is more important than knowledge."
 

 

 

 


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