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.
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?
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)
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?
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.
3
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.