Ordinal Score "(S,P)"systems

Suppose you allow six grades. Call them 0 (worst), 1, 2, 3, 4, and 5 (best). The candidates are Bush, Gore, and Nader. The Republicans bullet vote, because they do not like Nader any better than they like Gore. They think both of those are basically the same as Stalin. The DINOs bullet vote, because they are not used to the system, and cannot conceive of a better way to keep the hated Bush down than by throwing their full support behind Gore. They either have not heard that Nader is running, or wouldn't pay attention to such an announcement, on the grounds that fringe candidates cannot win.
Consider some different possible proportions of the electorate being DINOs and Republicans, and the remainder being Nader supporters. Let's say the Nader supporters honestly value Gore at 20% of the value of Nader, given a baseline of zero for Bush.
Are there some proportions of numerosities among these three factions such that in Score, the Nader supporters would see a better outcome if they voted Gore 4 than Gore 1? In your system, would they also see a better outcome if they voted Gore 4 than Gore 1, for some of the possible proportions? Are there some proportions for which the best outcome they could get in your system, using their optimal strategy for your system, is better than what they could get with Score using the same granularity in the range, and using their optimal strategy for that system?
I am working on a simulation framework to be based on a slider control that could be used to vary the proportions of the electorate belonging to factions. Coders could add any number of voting systems along with one or more alternative strategies for each system (the easiest to code, in each case, would probably be the naïve strategy, and I think it should be included with every system). Noncoder researchers could define factions, give the true sentiments for these factions, select strategies for them to use from among those provided by the coder contributors, and set their numerosities as a constant count of voters or a multiplier by the master slider control's leftward or rightward aspect. I'm thinking these aspects would evaluate to 1 when the slider is in the middle, and from 0 to 2 otherwise.

@cfrank From your video, I notice that the system would depend on past elections. I repeat that here for readers who don't hear out the video.

@JackWaugh Thank you for your thoughtful reply. Each candidate is given an “independent” ordinal score, so those kinds of results carry over from cardinal score. Altering the score given to Nader will only affect Nader’s score profile, it can’t cause Gore to beat out Bush or vice versa unless the voters change the scores they give to Gore and/or Bush. If a voter wants Nader to win, doesn’t mind Gore and wants Bush to lose, for example, then they should vote something like [B,N,G]=[0,5,3]. If they are really worried about Bush, then they should vote [0,5,5]. But this obviously increases the chance that Nader loses to Gore, so it’s up to the voter to manage their conflicting interests and indicate a ballot they think will serve them most effectively.
The system is very similar to score voting, the alterations I made were only to make the choice of cardinal scores less arbitrary.

@JackWaugh that can be true, but the system can also be made into a deterministic one by choosing a distribution or even by putting in place a deterministic method to select a distribution based on the present ballots.

@cfrank said in Ordinal Score "(S,P)"systems:
The system is very similar to score voting, the alterations I made were only to make the choice of cardinal scores less arbitrary.
Choice by the election designers, or choice by the voters?

@JackWaugh choice by the system designers. If anything I would actually prefer the scores to be more arbitrary for the voters, since this will prevent strategic behavior. That’s a part of why the system abstains from assigning the scores cardinal values. In the learning variant, the scores obtain their cardinal values via the manner in which they are used by the electorate, rather than the other way around.

@cfrank said in Ordinal Score "(S,P)"systems:
For an example of an (S,P)efficient election strategy, consider the voting system that chooses a candidate with a maximal area bounded in (S,P)space by their productionpossibilities frontier. This precludes (S,P)dominated candidates from winning the election.
Isn't this just score voting? Since for random variable X >= 0 with probability 1, EX = integral{0 to inf}[P(X>t)dt]
@cfrank said in Ordinal Score "(S,P)"systems:
Unfortunately, while the arbitrary nature of the scores has been eliminated, we have not been able to eliminate the arbitrary nature of the production of frontiers.
So the method you proposed could be considered part of a general class of methods of the following form:
Suppose X_1, X_2 ..., X_N are random variables between 0 and 1, where X_1 >= X_2 >= ... >= X_N with probability 1.
Then {(1,X_1), (2,X_2), ..., (N, X_N)} defines a random frontier.
For candidate C, let p(s) be the proportion of ballots such that candidate C is scored at least s.
Then the expected proportion of vertices on which p(s) >= X_s is (1/N)*sum{k=1 to N}[X_s <= p(s)]
(Note that if X_1 = ... = X_N ~ Unif(0,1), we get score voting.)Anyway, a criticism I have with the system that you have proposed in which the X_i are products of uniform distributions is that as i becomes larger, the distribution of the X_i become very leftskewed (edit: this should say rightskewed). For a 5 point scale, a candidate who receives a score of 5 on 1% of ballots, and 0 on all others will score about 34%.
Basically, it will be absolutely essential for a candidate to attain a (fairly small) critical mass of maxvalue scores, since that will gain them almost all of the points available on the highervalued scores. Among candidates who reach the critical mass, the winner will probably be whoever gets the most 1s (de facto turning into approval voting where anything greater than a 0 is full approval). 
@marylander Yes, with fixed uniform distributions SP voting does become score voting. But explicitly the system I am currently proposing (see the video explanation here, and don’t mind my brain fart at the beginning: https://app.vmaker.com/record/SGSydGYcwOW9Vf6d) uses distributions that are relevant models for the electoral data, not uniform. The toy model using products uniform distributions was a discarded idea.
I think from the SP framework one can make the argument that standard Score voting is very crude, since adopting the uniform distributions in an SP voting system seems fairly nonsensical compared with alternatives that reflect actual electoral behavior. One could use the order statistics of the uniform distribution, though, and that might not be terrible, but to me it seems like it would just be better to directly incorporate the actual data.
My guess is that if past electoral data were used to produce or to inform the distributions, they would become centered at lower fractions as the ordinal score is increased, and would actually skew to the right. I believe this eliminates the problem you suggested.
But I have no idea what the distributions would actually look like! They might not even be unimodal, and on the large scale they might not even stabilize. That’s partly why I also think it would be important for the distributions to be updated and for there to be a rigorous agreement about how and how often and when that should be done if the voting system were implemented for longrun use.

@cfrank said in Ordinal Score "(S,P)"systems:
My guess is that if past electoral data were used to produce or to inform the distributions, they would become centered at lower fractions as the ordinal score is increased, and would actually skew to the right. I believe this eliminates the problem you suggested.
I made a mistake, I should have said rightskewed.
In the past, various people have proposed using nonlinear levels for score voting. I have generally criticized this on the basis that it will seem arbitrary to voters. This method is not exactly one of those, but it is similar. Increasing a candidate's score from a 0 to a 1, a 1 to a 2, and so on, on a ballot, will all be worth different amounts (probably), but what amount it is worth depends on the distributions and the amount of support each candidate gets.
You have said that you are concerned that the values of score levels are arbitrary, so you might not be concerned that the amount that an additional point is worth under the proposed system is dependent on these factors; perhaps the approach you have proposed to assigning values to score levels gives them meaning. However, I am a bit skeptical of this.
First, the system might not pass Independence of Irrelevant Ballots. If several maxvalue ballots are added, then it will tend to reduce the value of highervalue scores relative to lowervalue ones on the rest of the ballots.
Second, I can think of cases where basing the distributions on past empirical data leads to points being devalued that clearly should not be devalued. Suppose that the first time a system like this is used, there are two polarizing candidates, and each candidate gets about half of the vote, with each ballot giving one candidate a 5 and the other a 0. The next election is similar, but this time, there is a compromise candidate who gets about a 3 from everybody. The compromise candidate will get a score of 60%, but if slightly less than half of voters gave this compromise candidate no points, then they will still score 60%, because no candidate in the first election got more than, say, 51% of scores above 0. While calibrating the distributions on the performances of two candidates is clearly insufficient and this example is probably unfair, I think it does highlight a general problem that a compromise candidate who gets an unprecedented amount of midrange support will stop receiving credit for additional midrange support beyond what has previously occurred.

@marylander Your criticisms are definitely fair and especially I hadn’t considered the midrange score devaluation problem, but I also agree that the calibration of the distributions is probably not an example that would reflect actual use. A winning candidate in this system probably needs to effectively pass thresholds of support of various kinds.
Also, any distribution or class of distributions could be selected as the prior, and more sophisticated methods could be implemented to update the distributions such as Bayesian statistics. It could be that a fixed distribution is chosen, I think the order statistics of uniform variables is not a terrible idea as the initial. But anyway it’s also all somewhat arbitrary, choosing a distribution is the same kind of problem as choosing cardinal score values. Furthermore with relevant updating the midrange support candidate will inform and reinforce the potential for compromise in future elections.
The irrelevant ballot problem can be fixed by simply ignoring or disallowing any ballots where the same score is given to all candidates, or more strictly any ballot that does not make use of both the minimal and maximal scores. I am not sure why rightskewed distributions would be problematic, in fact that is part of the functionality of the system.