I thought up this voting method last week. It’s a hybrid between cardinal and ranking methods, but a sort of opposite of STAR. I think it strikes a uniquely good balance between simplicity for voters at the ballot box, expressiveness, resistance to basic strategies, accounting of relative preference strength, and (sort of) electing Condorcet winners. I would love to hear your thoughts!
Voters rate candidates on a 4-level scale, where the levels are labeled “Strong Approve”, “Approve”, “Oppose”, “Strong Oppose”. 
Choosing the Winner - Copeland + Approval
(Approval) Eliminate all candidates with approval (strong or mild) from fewer than 25% of voters.  
(Copeland) Match every candidate against every other candidate. Candidate “A” defeats candidate “B” if more voters rate A above B than rate B above A.  Candidates get one point for every other candidate they defeat. 
(Copeland) Rank the candidates by the number of points (i.e. pairwise wins). When one candidate defeats all other candidates (i.e. the Condorcet winner), this candidate will have the highest ranking.
(Approval) Resolve any ties by using overall approval rates. 
Elect the highest ranked candidate.
The exact number of levels in the scale is not critical. I chose 4, but I can see strong arguments for 5, 6, or 7 levels, particularly for elections with many competitive candidates. However, it should be clear which levels on the scale constitute “approval”.
The 25% approval threshold was chosen somewhat arbitrarily. (Also, this step may not be valuable.) 10% seems pointlessly low; 50% seems prohibitive. This threshold could alternatively be defined in terms of number of candidates (e.g. top 5 most approved candidates). The goals here are to minimize the impact of dark horses and losers on the final result and to incentivize all candidates to find at least a modicum of approval.
If no candidate meets the threshold, simply elect the candidate with the most approval votes.
In cases where voters omit ratings for candidates, approved candidates can be considered preferred over unrated candidates.
This method could count close pairwise matchups (e.g. margins below 1% of votes) as ties, granting both candidates half a point in the matchup. Explicitly allowing pairwise ties might reduce the probability of majority rule cycles and avoid costly recounts.
Ties in the Copeland score could also be broken by a different scoring metric. I favor approval here, as it keeps the incentive to give extreme ratings low.
Strengths of its cardinal properties
- The rating scheme is simple enough for any voter to understand, and it would fit easily enough on a ballot. It is certainly more approachable than a full ranking scheme (when there are 5+ candidates) or a scoring scheme with a large number of ratings.
- The rating scheme (a Likert scale) is pretty expressive and comports with how social scientists often measure human opinions.
- In a race with 5+ candidates, voters are forced to differentiate between stronger preferences (giving candidates different ratings) and weaker preferences (giving candidates the same rating). Therefore, only stronger preferences influence the results. This is, in a sense, the goal of score methods. Still, most pairwise preference relations are preserved.
- The ratings can be used to avoid the most pathological issues with ranking methods. For example, true dark horses can be eliminated by an initial approval threshold.
- Some omitted preferences can be inferred from the ratings (approval > unrated).
- Ties arising from majority rule cycles can be broken using voter ratings, which are more understandable than heuristics based on pairwise defeat strength. This also addresses the Copeland method’s problem with ties.
Strengths of its ordinal properties
- While this method is not a true Condorcet method, as it derives voters’ preferences from a cardinal scale with limited slots, it seems extremely likely that it would elect the Condorcet winner, should one exist. In absence of a Condorcet winner, Copeland winners are still in the Smith Set. It is also highly unlikely that it would elect a Condorcet loser.
- The method (sort of) inherits other attractive features of ranking systems, like the mutual majority criterion.
- Few assumptions are made about the relative distances between locations on the rating scale.
- The strategic incentives to exaggerate preferences are low (unlike vanilla score voting).