score interval: score with additional protection against the chicken dilemma
 This initial post about "score+" is obsolete, see the second post for a description of "score interval voting".[/edit]
- Have a score vote 0 to 5
- Match all candidates by score
- Regard every defeat as a tie when it can be reversed by adding all equal scores (divided by 5) to the loser's score.
- Eliminate all defeated candidates.
- Elect among the remaining candidates the one with most top scores.
With score and approval a common objection is that giving your second favorite any non zero score can cause them to beat your favorite. This is usual attributed to later-no-harm, but I think the real issue is the chicken dilemma. LN-harm is a very strict criteria and also excludes finding a compromise.
Say you have an election with three fractions.
voters A B C 35 5 3 0 25 3 5 0 40 0 0 5 score 250 230 200
Supporters of A and B cooperate, but under standard score voting B voters have an incentive to defect and give a worse rating to A in order to win.
voters A B C 35 5 3 0 25 2 5 0 40 0 0 5 score 225 230 200
Likewise A voter now have an incentive to defect, creating a spiral that causes both to lose and C to win.
voters A B C 35 5 0 0 25 0 5 0 40 0 0 5 score 175 125 200
This is a variation on the prisoner dilemma. Both cooperate, both win, one defects, one wins, both defect, both lose. To address this problem score+ alters the equation so that defecting still hurts your partner but doesn't give you any benefit. It doesn't fully avoid the CD as with a sufficient number of voters defecting it will still work, but the incentive for the individual voter is near zero.
To do this score+ counts votes who give either side some support as being able to shift the support in either direction. When A can beat B by 20 points, but enough B voters could withdraw their support to reverse the defeat, then the pair A versus B is counted as a tie. We then remove all candidates that have a defeat and find the winner among the remaining by some other metric than plain score.
Since the score winner can not be defeated in the first round, the set of remaining candidates will never be empty.
In this case 35 votes vote A 5, B 3 and 25 vote A 3, B 5, so they have 35x3+25x3 = 180 scores in common. We divide that by the max score 5 to get 36 points. While A beats B with 250 to 230 points, this can be reversed by adding the 36 points to B. Therefor this match is counted as a tie. C however is defeated by both A and B and therefor eliminated.
(Side note: To divide by 5 weakens the protective effect, but if one would use the full score, then almost no candidate would ever get eliminated and the whole processes would be useless.)
In the second round we only compare A and B by maximum score which A wins. Note that, if C where still present, C would win this round, because the first preference is split between A and B.
B voters also have little incentive to defect against A, like shown above. If they rate A 2, then the defeat B over A by 230 to 225 points can be reverse with the 31 points they still have in common. Only C is eliminated and in the second round A still wins.
There are two obvious limits to this approach.
It is still possible that all voters of one side defect and win, while the other side still cooperates. But such a coordinated effort would certainly become known to voters of the deceived side and they would retaliate with mutual destruction.
Also, since top scores are counted in the second round, the chicken dilemma is still present in top ratings. There are several alternative ways to resolve the second round, but I think this is the most consistent and it creates no new problems.
I'm not certain about the naming. Previously I thought about adding an additional top rating called "+" which would only be counted in the second round - therefor "score+". But that would be confusing and introduce some strange effects.
Here is the payoff matrix for the above example. When we assume that voters don't know who will win, then A and B voters can expect that either A or B wins when they cooperate.
With regular score A or B voters can increase the chance of their candidate winning when they defect.
A/B B cooperate B defect A cooperate 4/4 3/5 A defect 5/3 0/0
With score+, up to a certain point, defecting won't change the winner. Therefor cooperate/defect has the same payoff as cooperate/cooperate.
A/B B cooperate B defect A cooperate 4/4 4/4 A defect 4/4 0/0
Here is a variation on this idea which might be more elegant.
In the above description of score+, scores are used for the first round, but only approval for the second. It thereby loses part of the accuracy of score. The solution is to do two rounds of score.
Two round score works as follows:
- Voters mark up to two scores per candidate on a scale 0-5. When only one is marked it counts as two times the same score. No mark is zero.
- In the first round the higher scores are counted.
- Regard any defeat as a tie when it can be reversed by subtracting all common scores between those candidates of the winner's score.
- Eliminate all candidates with at least one defeat.
- In the second round the lower scores are counted.
- The remaining candidate with the highest score is declared winner.
For the voter it might feel like they give a benevolent and an egoistic score. Or - reminiscent of confidence intervals - an upper and a lower bound on what they are willing to score each candidate. It could be called "score interval voting" (SIV pronounced like sieve).
Let's go through the above example again.
The first group of voters love A, like B and hate C, but they also have an incentive to defect and give B a low score. So they vote A 5-5, B 0-3, C 0-0.
Likewise the second group votes A 0-3, B 5-5, C 0-0. The third group just votes for C.
By counting the higher score we get:
voters A B C 35 5 3 0 25 3 5 0 40 0 0 5 score 250 230 200
Again A and B share a lot of scores and therefor tie, while C gets eliminated.
Now we count the lower scores whereby A wins.
voters A B 35 5 0 25 0 5 40 0 0 score 175 125
Other than before, voters can choose not to defect all the way to zero, but pick some intermediary value.
There is an issue with this approach. The best scored candidate in the second round will always have a lower score than the best scored candidate in the first round. Say in the first round A has 300 points, but B wins the second with 250. Then A top voters might be upset. It would be possible to introduce some additional mechanism to address this, but I think it is justified to say that this method only works because the second round picks the winner. Also, the second round already is some form of automatic runoff.
By using upper and lower ratings, this method fails the equal vote criterion (two ballots can cancel each other) and reversal symmetry. This can be fixed when we allow voters to mark their ratings in reverse order, say using a "1" for the lower and a "2" for the upper rating for the respective rounds.
However I don't care for reversal symmetry and it only fails equal vote for both rounds combined, but passes for each round individually. Say a ballot votes one candidate 5/3, then one could vote 0/0 to balance the first round or 2/2 to balance the second round. To vote 0/2 they would need the above rule. If one ballot would cause a candidate to pass the first round and win the second, another ballot could prevent that candidate from winning in either round.
As far as I can tell, the "score interval voting" described in the second post avoids 1. clone spoilers/teams 2. favorite betrayal 3. chicken dilemma (under strategy) and is the only method I know of which does (besides maybe MCA-P and MCA-AP?). For it to pass CD I assume that voters will fully support their favorite and fully betray their friend in the second round, then they can fully support both in the first round.
This doesn't formally satisfy the chicken dilemma criterion, as the formal definition requires that the defecting party does not win. However there is almost no incentive to defect and I think that in practice that would be sufficient to prevent the CD. The CD criterion requires a strong Nash equilibrium, but here we have a weak Nash equilibrium (see end of first post).
Why should this be important?
Spoilers reduce the number of candidates running. In the extreme you end up with two candidates. FB paired with an environment which makes it seem like there are only two viable candidates will also lead to two party domination. In a chicken dilemma voters individually have the incentive to defect which may lead to the majority to fraction and lose. When the chicken dilemma is not addressed, then it could devolve to a tactic where one fraction of the majority defects and the other fraction supports them to avoid the greater evil. Which again resembles the lesser of two evil problem.
Like plain score, interval voting is very well behaved regarding many criteria. I didn't check that thoroughly but think that the list below is 95% correct.
clone independence, favorite betrayal, monotonicity, summability, LN-Help, cancellation, reversal symmetry, consistency, IIA (like score, assuming voters don't normalize)
It partly satisfies:
chicken dilemma (see above),
LN-Harm (for the first round only)
CW, CL, Smith, majority, plurality
Note that the failed criteria are restrictions on who the winner should be, while most of the passed criteria are about strategic voting and strategic nomination. I would argue (which may be another post at some point) that the winner definitions shouldn't be viewed as strict criteria to pass 100%, but goals to approach. One could also define that the utility winner should always be elected, which could only be satisfied by some pure cardinal methods, but instead we look at VSE and see that Ranked Pairs performs well on this metric. So instead of the Condorcet criterion I would rather look at Condorcet efficiency.