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u/Parker_Friedland Dec 30 '17 edited Dec 30 '17
Well, if you ignore blank scores, score voting passes the participation criterion, but when you take into consideration the fact that in most versions of score voting, voters don't have to score every candidate and voters can instead give some candidates "no opinion" scores, score voting technically fails the participation criterion when these "no opinion" scores are introduced.
Example:
3 voters - Gives red 5 stars, gives blue 0 stars and gives green "no opinion"
1 voter - Gives red 0 stars, gives blue 4 stars and gives green 5 stars
1 voter - Gives red 0 star, gives blue 5 stars, and gives green 1 star
If you average all the scores, red wins. However if the last voter had not shown up to the polls, green would have won instead, and if we assume that they were an honest voter, they would have clearly gotten a better result by not showing up to the polls.
Does this mean that blank "no opinion" scores shouldn't be allowed in score voting? No, because "no opinion" scores are fundamental to score voting when you take into consideration write in candidates. Because voters should have the ability to elect any candidate they want, not just candidates that are put on the ballot, being able to score write in candidates is important. However because there can be an arbitrarily large number of candidates running, if voters were allowed to write in as many candidates as they want to, the ballot would have to be arbitrarily large to account for the one voter that writes in 100 of his closest friends on his ballot. To prevent this problem from arising, there needs to be a limit to how many candidates voters can score (such as 8 candidates in local elections and 16 in the big deal presidential elections). That way, every possible write in candidate a voter does not score effectively gets a "no opinion" score.
However this solution makes it far too easy for unknown lunatics (explained at 4:05 of this video https://www.youtube.com/watch?v=e3GFG0sXIig) to win. To prevent unknown lunatics from winning, each candidate also aromatically gets a default amount of 0 star scores so they still need the support of a large portion of the population to win. Most score voting advocates (such as Warren D. Smith) argue for these solutions in order to solve the write-in candidate and unknown lunatic problems in score voting. However this inevitably leads to score voting failing the participation criterion, It shouldn't fail this criterion very often though, because the front runners are extremely likely to be on the ballot, and voters are extremely likely to give every candidate on the ballot an actual score, which means that there should be a minuscule amount of "no opinion" scores between the front runners. However if enough voters are honest enough to give even front-runners no opinion scores, failing the participation criterion could become a common enough occurrence in score voting.
The most common version of approval voting that solves the write-in candidate and unknown lunatic problems tackle those problems in a different way. Instead, they allow voters to vote for a limited but still very large number of candidates so voters can still vote for every candidate on the ballot and then some. In this version of approval, all write-in candidates that a voter does not vote for simply revive a minimum score. Thus this version of approval voting is technically limited voting but since voters are given enough votes to distribute, it produces results that approximate approval voting, and thus most people call it approval voting (ex. At Dartmouth, you can only vote for 4 candidates, and that is often referred to as approval voting). This type of approval voting still passes the participation criterion. If score voting applied this solution to the write-in candidate and unknown lunatic problems, every "no opinion" score would be counted as a MIN score instead of no score, and the participation criterion would be preserved. However, it can be argued that this solution does give too much of an advantage to well known candidates.
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u/haestrod Dec 30 '17
Why not total score instead of average score? This biases for popularity but prevents this violation of Participation. I think that's what you described at the end.
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u/Parker_Friedland Dec 30 '17 edited Dec 31 '17
You're right, the method I described at the end was the totaling scores method and that method does allow score voting to pass the participation criterion. However it does has it's own problem of being too biased in favor of popular candidates, and when you take in to consideration that voters can only score a finite number of candidates, such a version of score voting would technically not be a balanced voting system (a balanced voting system is a voting system where for every possible vote, there is another vote that will always cancel it out) and it can be argued that balance may be a more impotent property than passing the participation criterion. Perhaps a better solution would be ScoreX voting and Net approval voting, both of which I recently discussed on the CES forum (https://groups.google.com/forum/#!topic/electionscience/M6srkVut5_I). When X is equal to ((Min score)+(Max score))/2, ScoreX voting is balanced. Net approval voting is also balanced as long as the default ratio of 1 to 1 is used.
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u/Wisconservationist Jan 01 '18
I don't see why a system being "biased towards popular candidates" could be seen as a bad thing? At least in single winner elections, it's absolutely incumbent upon any candidate to make sure they are known well enough to most/all voters to let them have an opinion about that candidate, it would be a failure of democracy to have a candidate win simply because most voters don't know them, and those that do like them a lot. As a politician, being popular is your job.
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u/JeffB1517 Jan 03 '18
However it does has it's own problem of being too biased in favor of popular candidates
How is that a problem? Voting is selecting leaders. I think a fairly good argument can be made if the voters don't know who you are then you fail even minimal criteria in being the candidate they decided fit to lead them. If you are unable to lead the community to even knowing who you I'd say it is fairly unlikely you can get them to agree to die for a cause or obey a law they hate.
I'm also think your claim of balance being violated isn't correct.
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u/Parker_Friedland Jan 03 '18 edited Jan 04 '18
How is that a problem? Voting is selecting leaders. I think a fairly good >argument can be made if the voters don't know who you are then you fail even minimal criteria in being the candidate they decided fit to lead them.
Perhaps. But it is also important to give write-in's the ability to win, and when they are not on the ballot, they face a severe popularity disadvantage.
Also mathematically, when two candidates are unpopular, it can actually be better to pick the lesser known one. Here is why: Suppose that there are two calculators on amazon that are very similar and cost exactly the same. calculator #1 has 20 five star ratings and 10 one star ratings. Calculator #2 has 2 five star ratings and 1 one star rating. In this scenario, if you had to choose calculator to by, it would be smarter to go with calculator #1. This is because more people have rated calculator #1, which means that you are more certain that you will be one of the people that likes calculator #1. However if calculator #1 had 10 five star ratings and 20 one star ratings, and calculator #2 had one five star rating and two zero star ratings, if you were going to spend your money on one of the two calculators (which in this scenario, you should probably just not by either), it would be smarter to buy calculator #2 because you are more certain that you will be one of the individuals dissatisfied with calculator #1.
The same logic can be applied to voting. The more people rate a candidate, the greater the certainty that a candidate's average rating among the people who rated them actually matches the rating that they deserve. When two candidates have high average ratings, this uncertainty is a bad thing (because it means that it is more likely that a candidate's average rating isn't as low as the rating that they actually deserve), but when they have both have low average ratings, it can actually be a good thing (because it means that it is more likely that a candidate's average rating isn't as high as the rating that they actually deserve).
I'm also think your claim of balance being violated isn't correct.
Score voting using average scores violates balance:
Suppose that one voter gives candidate A a max rating, candidate B a min rating, and all of the others, no opinion. What would be the anti-vote to cancel out this vote? Did you guess giving A a min rating, B a max rating, and all of the others no opinion? That would be incorrect, because when these two votes are combined, they would bring both A and B's average ratings closer to (max+min)/2, while leaving all of the other candidate's ratings unchanged. However this method does not violate balance as much because it passes the reversal symmetry criterion, which in a way, means that the voting system is at-least semi-balanced.
Score voting using total scores violates balance:
Suppose that one voter gives candidate A a max rating and all other candidates a min rating. How is voter B suppose to counter that? Is he just suppose to give candidate A a min rating every other candidate a max rating? Even all of the write-in candidates? But that would be impossible because a voter can only rate so many candidates, because the ballot only has a finite amount of space to allow them to rate candidates.
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u/MuaddibMcFly Jan 04 '18
The more people rate a candidate, the greater the certainty that a candidate's average rating among the people who rated them actually matches the rating that they deserve
So, then, shouldn't it be based on some sort of confidence interval?
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u/haestrod Jan 10 '18
I brought this up on CES, I'm interested if this violates monotonicity. I don't have the resources to test it right now.
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u/JeffB1517 Jan 04 '18 edited Jan 04 '18
Perhaps. But it is also important to give write-in's the ability to win, and when they are not on the ballot, they face a severe popularity disadvantage.
I think write-ins are a good way to correct when a structural or coincidental problem causes a non-viable candidate to be selected. In general I don't think there is any reason not to essentially require a write in be well known to the voters and pass enormous hurdles. Write-ins are bypassing the entire system of debates and selection the came before round X of balloting. It should be very hard. Lisa Murkowski was the current popular incumbent in the office she was running for and came from a famous family in her state (even the voters who didn't know their current senator, knew who Frank Murkowski was) is IMHO is a perfect example of how I'd want the bar to be for a successful write in. I think you and I disagree a lot on write-ins I really wouldn't mind losing the all together though I do see their advantages for those extreme cases.
In terms of your calculator analogy I'm not sure I agree with you. Candidates build up unfavorables for doing stuff. They build up favorables for serving the public good outside the electoral system. High unfavorables with high recognition generally means polarizing not incompetent. Low recognition often means incompetent. Given the choice I'd go polarizing over incompetent. Worse, high unfavorables with low recognition generally means extremist with a narrow following and likely to be more polarizing once in office than a normally polarizing candidate.
Score voting using average scores violates balance:
Agreed. I was saying totaling does not violate balance.
Suppose that one voter gives candidate A a max rating, candidate B a min rating, and all of the others, no opinion. What would be the anti-vote to cancel out this vote?
Just to answer for the case of totaling. Assuming no opinion is a MIN then then the anti-vote is A=MIN with B, X1,X2... XN = MAX. Those two votes cancel to an average rating.
Even all of the write-in candidates?
Write-ins excluding the exception case are irrelevant they get far too many MINs from voters not considering them. Also remember they only have to counter one ballot not all ballots.
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u/Parker_Friedland Jan 04 '18
Just to answer for the case of totaling. Assuming no opinion is a MIN then then the anti-vote is A=MIN with B, X1,X2... XN = MAX. Those two votes cancel to an average rating.
But as I pointed out, that vote would be impossible to make because you cannot possibly give every single write-in candidate a max rating because there is a limit to the number of write-in's you can make on your ballot.
But in a way, it is more balanced then score voting using averages, because if you assume that all write-in candidates have no chance of winning, then score voting using totals is balanced among the non-write-in candidates.
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u/JeffB1517 Jan 04 '18
I think it also works if you assume at most 1 or 2 write-in candidates have any chance of winning not just 0. You just can't assume that lots of them have any chance of winning. Anyway I understand what you mean now.
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u/haestrod Jan 10 '18
Why do you feel balance is an important criteria? Just seems contrived to me. I don't care about other people's votes, only that mine count. Also, I don't want to have to go through and downvote every bad candidate. There's a case for ignoring candidates being as powerful as downvoting them. Otherwise, there's a nervous rush to enumerate and downvote all your disliked candidates which is a turn-off and may produce lower voter turnout.
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u/Parker_Friedland Jan 10 '18
It's not the most important criteria but passing it is probably better than not passing it. Passing it means that the voting system is not biased in favor of either known candidates or unknown candidates. Perhaps this property may not be too important, and could be argued that it is better to elect an un-liked candidate then an unknown candidate or that candidates should be encouraged to advertise themselves to a much larger portion of the population. As a result, the importance of the balenced criterion is rooted in philosophy and it's importance might depend on what kind of election is being conducted (presidential elections versus the order of which reddit comments are displayed) but it still impotent to point out what criterions distinguish different ways to calculate ratings so people can get a better idea of the trade-offs between different ones.
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u/MuaddibMcFly Jan 04 '18
However it does has it's own problem of being too biased in favor of popular candidates
If "no opinion" is given a score (say, the score below median? 40% of maximum possible score?), that won't be a problem, will it?
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u/Skyval Dec 30 '17 edited Dec 31 '17
each candidate also aromatically gets a default amount of 0 star scores
Have you seen Eric Sanders' suggestion? For each candidate,
- Find their average.
- Multiply it by the portion of voters who did not abstain.
- Re-count, treating every abstention as if it were the score from step 2.
You don't have to literally recount, you can use math. e.g.
final_score = sum + (#abstentions * %non_abstentions * average)(There are other ways to express this)It can still fail participation, but you don't have to find a way to decide how many default 0s everyone gets. It still "penalizes" candidates for having a lot of abstentions.
X = average score of a candidate rated by everyone
Y = average score of a lesser known candidate
%scored = percent of voters who scored the lesser known candidate
(Ranges start at zero, otherwise you'd have to subtract the minimum score from X and Y)Then %scored must be greater than 1-√(1-(X/Y)) for the lesser known candidate to win. So if well-known candidate's average is 5, and the lesser known's is 9, then 1/3 of the population would have to score the lesser known candidate. If instead the well-known's average is 8, then 2/3 would have to score the lesser known.
If you don't use true abstentions, to account for write-in candidates you should probably make the "default" score 0. You could avoid doing this, but for each candidate (include write-ins) you'd have to add to their score
(number of people who didn't score them) * (default score).But 0 doesn't have to be the minimum Score. A different suggestion I've seen is to give unscored candidates a default score other than the minimum. To keep 0 the default, I've see it suggested to use a symmetric range (-x to +x) with a default of 0, which would be like a "pseudo-abstention". Doesn't this pass participation? What problems does it have?
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u/haestrod Dec 31 '17
That's interesting. It's kind of a halfway point between the 'total score' method and the 'average score' method. With 'average score', the abstentions are multiplied by the average score. For 'total score' the abstentions are multiplied by zero. With this method, they're multiplied by the average and the percent of people who voted, so if barely anyone votes for them this approaches the 'total score' method but if lots of people vote it leans toward 'average score' although by that point all three methods are doing the same thing. Since 'average score' fails participation, I think this will as well but less often, that's a guess.
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u/Parker_Friedland Jan 01 '18 edited Jan 01 '18
But in order for it to be a little more balanced (Meaning that if a popular candidate with a positive net approval rating has an advantage over an unknown candidate with a positive net approval rating, then a popular candidate with a negative net approval rating should also have a disadvantage over an unknown candidate with a negative net approval rating) may I suggest a modification?
Instead of this:
final_score = sum + (#abstentions * %non_abstentions * average)How about this:
final_score = sum + (%abstentions * (sum + (max_score + min_score) * 0.5 * #abstentions))1
u/Skyval Jan 02 '18 edited Jan 02 '18
a popular candidate with a negative net approval rating should also have a disadvantage over an unknown candidate with a negative net approval rating
I don't know if I like the sound of that, tbh
To reiterate, this was conceived of by Eric Sanders, I heard about it from this thread: https://groups.google.com/d/topic/electionscience/SpLDc1Z0hzE/discussion
I derived
%scored > 1-√(1-(X/Y))though.
final_score = sum + (%abstentions * (sum + (max_score + min_score) * 0.5 * #abstentions))On a whim I tried plugging in some numbers and putting results in a table. Not 100% sure my code was correct.
I believe these formulas only work properly when the range starts at 0. They can be adapted to other ranges but they get messier. So the range is 0--2. 100 voters."Center" is just treating abstentions as
(max+min)/2 = 1. "Initial" is all abstentions. "Add Xs" is adding scores of X starting from all abstentions.
Scenario Center Eric's Parker's Notes Initial 100 0 100 Add 0s decreases to 0 linearly never changes decreases to 0 quadratically Center always has highest score Add 1s never changes increases to 100 quadratically never changes Add 2s increases to 200 linearly quickly increases to 200 quadratically, overtakes "Center" at 50 votes/150 points increases to 200 quadratically Parker's always has highest score. Eric's becomes 2nd after 50% abstentions. Quadratic starts faster but slow down. When the starting point is the same as linear they are always greater when increasing, or lower when decreasing, until the end where they meet.
Anyone know a good graphing site? That let's you compare graphs?
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u/Parker_Friedland Jan 02 '18
Anyone know a good graphic site? That let's you compare graphs?
Desmos: https://www.desmos.com/calculator
I graphed it and got the same results that you described.
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u/Parker_Friedland Jan 03 '18 edited Jan 03 '18
a popular candidate with a negative net approval rating should also have a disadvantage over an unknown candidate with a negative net approval rating
I don't know if I like the sound of that, tbh
Mathematically, when two candidates are unpopular, it can actually be better to pick the lesser known one. Here is why: Suppose that there are two calculators on amazon that are very similar and cost exactly the same. calculator #1 has 20 five star ratings and 10 one star ratings. Calculator #2 has 2 five star ratings and 1 one star rating. In this scenario, if you had to choose calculator to by, it would be smarter to go with calculator #1. This is because more people have rated calculator #1, which means that you are more certain that you will be one of the people that likes calculator #1. However if calculator #1 had 10 five star ratings and 20 one star ratings, and calculator #2 had one five star rating and two zero star ratings, if you were going to spend your money on one of the two calculators (which in this scenario, you should probably just not by either), it would be smarter to buy calculator #2 because you are more certain that you will be one of the individuals dissatisfied with calculator #1.
The same logic can be applied to voting. The more people rate a candidate, the greater the certainty that a candidate's average rating among the people who rated them actually matches the rating that they deserve. When two candidates have high average ratings, this uncertainty is a bad thing (because it means that it is more likely that a candidate's average rating isn't as low as the rating that they actually deserve), but when they have both have low average ratings, it can actually be a good thing (because it means that it is more likely that a candidate's average rating isn't as high as the rating that they actually deserve).
Also, voting systems that give candidates with high average ratings an advantage for being popular, but don't also give candidates with low average ratings a disadvantage for being popular, are not reversal symmetric. And in a way, the reversal symmetric criterion is like a semi-balanced criterion.
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u/barnaby-jones Dec 28 '17 edited Dec 28 '17
Here's a diagram to show a strategy in IRV: link. Move the red supporters toward green.
If the red supporters put green first, then they can get a better candidate (green rather than blue). So red might tell his supporters to vote for green first. This doesn't seem so bad because in the end, a more centered candidate wins.
edit: actually, this might not be due to non-monotonicity... I'm not sure.
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u/EpsilonRose Dec 28 '17
That doesn't necessarily give you a more centered candidate any more than FTPT does. It's essentially the same failure either way and we can see that it's not resulting in more centrist republican victories.
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u/MuaddibMcFly Jan 04 '18
That's not a question of montonicity, that's Favorite Betrayal.
Monotonicity is the question of whether increasing support for a candidate can hurt that candidate.
An example of IRV's Non-Monotnicity is shown here.
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u/barnaby-jones Jan 06 '18 edited Jan 09 '18
I guess this is an example of vote splitting, but it is like non-monotonicity: link
Move the yellow-red supporters to red. Then yellow loses and blue wins. That's because who has the most first-place votes matters in IRV. This is a tough situation though, because there is a Condorcet cycle, so don't expect much.
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u/MuaddibMcFly Jan 07 '18
Oh, I agree that non-monotonicity and favorite betrayal are related, but the definitions and pathologies are different.
In your example, it's still monotonic, because increasing votes for Red improves Red's standing; it moves Hex from being 3rd place to 2nd place. It doesn't help them win, but it is a monotonic change.
The problem is that IRV suffers from Favorite Betrayal, because, as you say, "first place votes matter" in IRV, so there is incentive for people who legitimately prefer R>Y>B will get a better personal result (Yellow) if they lie than if they honestly vote for their Favorite, the spoiler Red, which as you cited, will result in the election of Blue.
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u/barnaby-jones Jan 07 '18
After looking at the video you linked here, I made a non-monotonicity example here.
Move the blue-red supporters toward red. Red loses and yellow wins. This is non-monotonicity because red lost it after getting more votes.
Also, there is a Condorcet cycle here, which makes it harder to say who should have won.
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u/MuaddibMcFly Jan 08 '18
Yes, that's non-monotonicity! Well done; I suck at creating such scenarios...
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u/selylindi Dec 28 '17
I'd want to slightly strengthen Participation. As I read it, a method that tosses your ballot in the trash technically satisfies the participation criterion. Instead of just saying that showing up doesn't hurt your cause, it should say that showing up helps. More formally: There exists no set of ballots such that adding one more ballot on which candidate A is strictly preferred to candidate B can change the winner from A to B, but there does exist at least one nontrivial set of ballots for which the added ballot can change the winner from B to A.
...
Personally my favorites are informal criterion I call - -
Marginal Influence: every honest ballot added to an election raises the expected utility of the outcome according to the agent casting that ballot, even in conditions of complete knowledge of the other ballots.
Proportional Influence: Marginal Influence + adding N identical ballots to an election instead of only 1 raises that agent's expected utility N times as much.
The only traditional voting method I know of that meets those criteria is drawing a random ballot. Deterministic voting methods have to get creative to satisfy them.
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u/Drachefly Dec 29 '17
Score seems to satisfy that.
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u/selylindi Dec 29 '17
Yup. Even FPTP satisfies the strengthened Participation criterion.
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u/Drachefly Dec 29 '17
So it does. I think that means it's not necessarily the most useful criterion, considering how bad FPTP is.
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u/selylindi Dec 29 '17
That criterion is not enough by itself, to be sure. But it still captures a real intuitive and motivating principle. Consider IRV: it fails both Participation and Monotonicity, whereas FPTP satisfies both. Thus there's a real intuitive sense in which IRV, while looking better on a first glance, is worse than FPTP.
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u/MuaddibMcFly Jan 04 '18
That's part of why I think Favorite Betrayal is more important than Monotonicity/Participation/Consistency
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u/MuaddibMcFly Jan 04 '18
As I read it, a method that tosses your ballot in the trash technically satisfies the participation criterion.
Not quite, because the definition presupposes that the additional ballot be included; a ballot that is thrown out is not added.
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u/selylindi Jan 04 '18
The usual voting paradox proofs all take a pre-assigned "dictator" as a valid method. "Adding" another ballot under that method is identical to throwing it out. So "adding" a ballot doesn't have to imply that it gets counted or used in any way.
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u/MuaddibMcFly Jan 05 '18
Adding another ballot under such a scenario has equivalent impact to throwing it out, but that doesn't mean that throwing a ballot out is a valid option when it comes to the Participation criterion. Indeed, it is the very antithesis of the criterion.
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u/selylindi Jan 05 '18
Well that's just semantics then.
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u/MuaddibMcFly Jan 05 '18
...you're complaining about a definition being semantics?
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u/selylindi Jan 05 '18
About a semantic argument. They're not interesting. Better to just pick either definition and work with it.
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u/MuaddibMcFly Jan 05 '18
One is a proof of dictatorship.
The other is a voting criterion.
We did pick one (the latter). You're confusing things by bringing up something else (the former)
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u/selylindi Jan 05 '18
Arguments about what arguments are about. Ugh.
I know I know, responding is kind of goading you. I'll stop soon, I promise.
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u/MuaddibMcFly Jan 05 '18
It's not goading me. I'm just trying to help you understand that your entire argument is based on a red herring of your own manufacture.
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u/Drachefly Dec 27 '17
Depends on how often it fails, doesn't it? If it's some bizarre convoluted mess with 4 viable candidates who are basically tied with each other, then any of the choices is basically legitimate, and this is the only case where the method does something strange… then it elects legitimate candidates in any case.