score interval: score with additional protection against the chicken dilemma

[edit] This initial post about "score+" is obsolete, see the second post for a description of "score interval voting".[/edit]
Short summary:
 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 laternoharm, but I think the real issue is the chicken dilemma. LNharm 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.
Edit:
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 05. 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 55, B 03, C 00.
Likewise the second group votes A 03, B 55, C 00. 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 MCAP and MCAAP?). 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.
It passes:
clone independence, favorite betrayal, monotonicity, summability, LNHelp, cancellation, reversal symmetry, consistency, IIA (like score, assuming voters don't normalize)
It partly satisfies:
chicken dilemma (see above),
LNHarm (for the first round only)
It fails:
CW, CL, Smith, majority, pluralityNote 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.

There is a simpler variant that uses plain score and normalizes the scores in the second round.
 Voters score candidates on a scale 05.
 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.
 Normalize the scores for the remaining candidates.
 The remaining candidate with the highest score is declared winner.
Or even simpler (but less clean):
 Voters score candidates on a scale 05.
 The two highest scoring candidates and everyone with equal or over 50% of possible score enter the runoff.
 Normalize the scores for the remaining candidates.
 The remaining candidate with the highest score is declared winner.
This second variant behaves like STAR in the case when there is at most two candidates with 50%+ scores. This means, it's a simple Chicken Dilemma improvement to STAR.

Isn't the chicken dilemma kind of a myth? It contradicts the theorem that approval and score voting elect the Condorcet winner under 100% tactical voting.

Most likely, yes. The ideas here are a theoretical exploration.

Is it additive? Is it balanced?

@isocratia said in score interval: score with additional protection against the chicken dilemma:
Isn't the chicken dilemma kind of a myth? It contradicts the theorem that approval and score voting elect the Condorcet winner under 100% tactical voting.
Kind of. The issue is we don't know for sure what model of tactical voting is the most accurate.
Many voters don't have the capacity to strategize, so they rely on instructions from people they trust. That can be a friend who's smarter than them (basically the same as the individual strategy model, which gives the result you ask for), but could also be a political party, or a favorite candidate.
The last one is the most concerning possibility, but my guess is it gets basically canceled out by the second. The party and candidate have opposite incentives here (the party says to vote for both copartisans, while the candidates want you to vote only for them).
To slightly reduce the chicken incentive, I've suggested replacing appointments or byelections with countbacks.

@jackwaugh What do you mean by "additiv" and "balanced"?

@casimir if interested you can check here and read the ”Motivation” section: https://github.com/cfrankston728/symmetric_quantilenormalized_score
I’m not selling the method, but this point has been discussed between myself and @JackWaugh before. In rough terms, a certain “cancellation property” (aka “balance”) of interest seems necessary, and an “Abelian group” property (aka “additivity”) seems sufficient, for certain desirable properties of a voting system. But it isn’t clear where necessary and sufficient combine into a universal criterion—probably because we don’t know exactly what we mean by “desirable properties.”
In @JackWaugh’s language, additivity is a strictly stronger condition than balance (which in the strictest sense is a very weak condition—see the “password attacking” argument). Additivity confers nice properties, but is restrictive in terms of the kinds of systems we can try to consider or develop while satisfying it.

@casimir, By "additive", I mean that the outcome only depends on a sum of the votes, where "adding" votes is commutative and associative.
By "balanced", I mean that for every vote that's allowed, another vote that's allowed will cancel it if both are submitted in the same election. That's Frohnmayer balance.

As far as I can tell it's not additive, because it depends on how candidates compare on each ballot. Balance may be possible, but I have to check in detail. Casting a complementary ballot in the first round (which is possible) may inhibit the ability to cast a complementary ballot in the second round.

@casimir said in score interval: score with additional protection against the chicken dilemma:
depends on how candidates compare on each ballot
Recall that preferences can be coded in a matrix, and matrices add.

@isocratia said in score interval: score with additional protection against the chicken dilemma:
Isn't the chicken dilemma kind of a myth? It contradicts the theorem that approval and score voting elect the Condorcet winner under 100% tactical voting.
Perfectlyinformed, perfectlyrational tactical voting generate a Condorcet winner. This is true of all (relevant) voting systems (including plurality), and is the inherent nature of a Condorcet winner.
The problem is that a very wide variety of factors stand in the way of "perfect." There is decent hindsight evidence suggesting that Gary Johnson would have been the Condorcet winner of the 2016 US Presidential election. Consider the informational, institutional, and political barriers preventing the Democrats from nominating Gary Johnson, advancing a Johnson strategy instead of a Clinton strategy. We'd sooner have held the election on Jupiter.
In prisoner's dilemma, it is just as rational for two criminals who have telepathy to have each other's backs as it is for two criminals who are isolated to sell each other out. Neither outcome should be surprising.
Chicken dilemma is similar. If Sanders voters have perfectly singlepeaked preferences and only care about maximizing their preferences in this one election, then they will line their perfectlyrational and wellbehaved butts up behind Clinton to beat Trump.
But man, political hostagetaking is so in right now. "No, screw that, you elect me if you don't want the other side to win!" This approach might even be mathematically rational if your previous assumptions about this person's utilities were wrong. Maybe Sanders and his voters care more about influencing future elections than winning just this one. Maybe Matt Gaetz just want to run for Florida Governor. Maybe someone is a fullon accelerationist, who believes the best way to address problems is to first make them worse.
I can't think of a point in American history where intragroup picking of "BETRAY" has ever been so prolific as it is right now.

It's not exactly true of plurality voting. In plurality, there can be multiple MyersonWeber equilibria (fixed points where the voters vote tactically based on beliefs about the likely winners that are later matched by the actual result). The Condorcet winner wins in some but not all.
To use a contrived example:
~50% of voters: A > B > C
~50% of voters: A > C > BIf the voters believe that A is one of the two likely frontrunners, then A will win with 100% of the vote. That is one MyersonWeber equilibrium.
But if the voters believe that B and C are the likely frontrunners, then A will lose with 0% of the vote. That is the other MyersonWeber equilibrium. And even though they would all be better off if they all voted for A, any individual voter unilaterally switching their vote to A will only make the outcome worse from their own perspective.
Approval voting eliminates this kind of absurd equilibrium simply by allowing voters to vote for multiple candidates instead of "switching" their vote from one candidate to another. In the example above, if voters believe that B and C are the frontrunners, then they all approve A anyway, and A wins with 100% approval. So the second, suboptimal equilibrium is eliminated.

@chocopi said in score interval: score with additional protection against the chicken dilemma:
Perfectlyinformed, perfectlyrational tactical voting generate a Condorcet winner. This is true of all (relevant) voting systems (including plurality), and is the inherent nature of a Condorcet winner.
I think this is missing the most important caveat, which is mentioned above. The problem isn't information, it's that any such move has to be perfectly coordinated. Voters need to take actions that are individually irrational, despite there being no way to enforce these strategies. The reason Johnson lost is because it would be individually irrational for a voter to switch from backing Trump to backing Johnson, unless they expected everyone else to do the same. (Whereas anyone who preferred Johnson to Clinton and Trump should've approved Johnson.)
IIRC Vox did a poll with approval in 2016, and found Johnson neckandneck with Clinton (which lines up with polls they'd be competitive in a oneonone).

@chocopi said in score interval: score with additional protection against the chicken dilemma:
In prisoner's dilemma, it is just as rational for two criminals who have telepathy to have each other's backs as it is for two criminals who are isolated to sell each other out. Neither outcome should be surprising.
Chicken dilemma is similar. If Sanders voters have perfectly singlepeaked preferences and only care about maximizing their preferences in this one election, then they will line their perfectlyrational and wellbehaved butts up behind Clinton to beat Trump.Wait, but it's not similar. The prisoner's dilemma is taught in game theory classes as a way to teach students a simple game with a dominant (alwaysbest) strategy. The chicken dilemma is a way to teach students about equilibriumrefinement.
Basically, there are some games with multiple Nash equilibria, i.e. there are many "potentially rational" strategies you could use. To predict which one someone will actually use, we need to put reasonable constraints on what kinds of beliefs we'd consider rational. There's two pure Nash equilibria for chicken: If you're 100% certain your opponent will swerve, you should plow ahead. If you're 100% certain they won't, you should swerve.
But what if you're not 100% certain what your opponent will do? Then you have to look for the tremblinghandproper equilibrium. Say I know my opponent isn't perfect: they can make mistakes or slipups in playing a strategy, or maybe some of their voters won't get the message about the strategy I'm telling them to use. Basically, I'm going to rule out any situations where 100% of my opponent's voters use a certain strategy. Maybe we can say that for each voter, there's a 99% chance they'll use the correct strategy, but a 1% chance they'll mess up their ballot.
This is a very powerful refinement, and you can use it to prove that both bulletvoting and friendlyvoting are irrational in the chicken model. 100% bulletvoting risks throwing the election to the Republicans. 100% friendlyvoting means you can't the election to the other subfaction with 100% probability. The rational strategy is a mixed strategy, i.e. in equilibrium some Sanders voters will support Clinton and others won't.
For example, say Clinton's faction is polling 35% and Sanders' faction is polling 25%, with Trump at 40%. Then, the optimal strategy is for >20% of Sanders' voters to approve Clinton (or equivalently, for all of them to give a 20/100 on a score ballot). This lets Clinton win with more than 1/5 * .25 + .35 = 40% of the vote, while keeping the margin between Clinton and Sanders as tight as possible in case there's a polling error.
Where exactly the scores end up depends on details like exact utilities and uncertainty. You'll get more cooperation with better polling; that lets you rule the smallest faction out of contention with 100% certainty. You also get more cooperation with more polarization (i.e. if voters care more about blocking the other party than about supporting their favorite).
The main point, though, is that bulletvoting is irrational in this chicken dilemma. Clinton is still (by far) the mostlikely winner in this situation, assuming she's the Condorcet winner. We can also work out where, exactly Clinton will end up with perfect strategy, and the answer is "just above 40%, barely edging out Trump and Sanders". In approval, the optimal strategy is to randomize whether you approve Clinton. (In score, I expect a fair chunk of voters to use the intermediate scores instead because that's simpler, but it gives the same result.)

@casimir said in score interval: score with additional protection against the chicken dilemma:
Or even simpler (but less clean):
Voters score candidates on a scale 05.
The two highest scoring candidates and everyone with equal or over 50% of possible score enter the runoff.
Normalize the scores for the remaining candidates.
The remaining candidate with the highest score is declared winner.This second variant behaves like STAR in the case when there is at most two candidates with 50%+ scores. This means, it's a simple Chicken Dilemma improvement to STAR.
This is getting close to practical.
It might even comply with state constitutions that require a winning candidate to receive the “the highest”, “the greatest” or “the largest” number of votes, or “a plurality of votes”. (need good lawyers.)
Could call it STIR, Score Than Instant Runoff.

Normalize the scores for the remaining candidates.
I like normalization well enough, but are we going to be able to explain and justify it to John Q. Public?

@casimir said in score interval: score with additional protection against the chicken dilemma:
There is a simpler variant that uses plain score and normalizes the scores in the second round.
 Voters score candidates on a scale 05.
 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.
 Normalize the scores for the remaining candidates.
 The remaining candidate with the highest score is declared winner.
Or even simpler (but less clean):
 Voters score candidates on a scale 05.
 The two highest scoring candidates and everyone with equal or over 50% of possible score enter the runoff.
 Normalize the scores for the remaining candidates.
 The remaining candidate with the highest score is declared winner.
This second variant behaves like STAR in the case when there is at most two candidates with 50%+ scores. This means, it's a simple Chicken Dilemma improvement to STAR.
Here's a particularly simple and attractive method:
 Eliminate all candidates scored below 50%.
 Use quadratic voting to pick the best remaining candidate. (Rebrand it as equalweight voting, by framing it as taking each ballot and dividing by its "weight"—i.e. sum of squares.)
Why the first elimination step? Well in score, approval, etc. with optimal strategy and perfect information, only one candidate should get over 50% of the vote. This candidate should be the Condorcet winner. That means that for candidates scoring over 50%, voters don't have enough information to know which (if any) is the Condorcet winner.