Prefer Accept Reject voting
Prefer Accept Reject (PAR) voting works as follows:
 Voters Prefer, Accept, or Reject each candidate. On ballots which don't explicitly use "Reject", or for candidates with less than 25% "Prefer", blanks count as "Reject"; otherwise, blanks count as "Accept".
 Tally 1 point for each "Prefer" for each candidate.
 Out of the candidates (if any) with no more than 50% "Reject", find the one with the most points. For every ballot which doesn't "Prefer" this frontrunner, add 1 point for each "Accept".
 If the frontrunner still has the most points, they win. Otherwise, the winner is the candidate with fewest "Reject" ratings.
To express it in a single sentence: if the mostpreferred nonmajorityrejected candidate, X, has more nonreject votes than any other candidate has nonreject votes that aren't below X, then X wins; otherwise, the leastrejected candidate wins.
Note that the procedure above will always elect a candidate with no more than 50% "Reject", if any exist. This is because, if any exist, one of them will be the frontrunner, and they will thus score points equal to at least 50% of the voters.
Each candidate's score at the end can be seen as an approval total, and is thus suitable for combining with approval totals from other jurisdictions in a system like the National Popular Vote Interstate Compact.
A related system which passes FBC is FBPPAR. This has the same steps, except that voters can choose to mark any of their "preferred" candidates as "stand aside". "Stand aside" preferences are counted as rejections when finding the leader, but as preference when assigning points.
Relationship to NOTA
If all the candidates in the first round got a majority of reject, then the voters have sent a message that none of the candidates are good, akin to a result of "none of the above" (NOTA). PAR still gives a winner, but it might be good to have a rule that such a winner could only serve one term, or perhaps a softer rule that if they run for the same office again, the information of what percent of voters rejected should be next to their name on the ballot.
Criteria compliance
PAR voting passes the majority criterion, the mutual majority criterion, the majority loser criterion, Local independence of irrelevant alternatives (under the assumption of fixed "honest" ratings for each voter for each candidate), Independence of clone alternatives, Monotonicity, polytime, resolvability, and O(N²) summability. (It is also possible to run it in no more than 3 counting rounds, each of which is O(N) summable.)
There are a few criteria for which it does not pass as such, but where it passes related but weaker criteria. These include:
 It fails Independence of irrelevant alternatives, but passes Local independence of irrelevant alternatives.
 It fails the Condorcet criterion, but if there is a voted majority Condorcet winner X, then any faction (defined as the set of voters who prefer X>Y,Z, or alternately as the set who prefer Y>X>Z, for some Y and Z) can ensure that X beats Z, using semihonest ballots (X>>Z or YX>>Z, respectively).
 It fails the participation criterion but passes the semihonest participation criterion.
 It fails the laternohelp criterion, but passes if there is at least one candidate rejected by under 50%.
It fails the consistency criterion, the Condorcet loser criterion, reversibility, the Strategyfree criterion, and the laternoharm criterion.
Favorite betrayal?
PAR voting fails the favorite betrayal criterion (FBC). For instance, consider the following "nondisqualifying centersqueeze" scenario: (
 30: AX>B (That is, on 35 ballots, A and X are preferred, B is accepted, and C is rejected)
 5: AX>C
 15: B>A
 10: B>AC
 40: C>B
None are majorityrejected, and C is the frontrunner. Points are: A, 60; B, 55; C, 55; X, 35. A wins. However, if 11 of the last group of voters strategically betrayed their true favorite C, the situation would be as follows:
 30: AX>B (That is, on 35 ballots, A and X are preferred, B is accepted, and C is rejected)
 5: AX>C
 15: B>A
 10: B>AC
 29: C>B
 11: B
Now, C is not viable with 51% rejection; so B is the leader. Since C is no longer the leader, B gets the 34 points from C voters, and wins. The strategy succeeded; the strategic voters are better off.
However, there are several ways to "rescue" FBClike behavior for this system.
For one, we could add a "stand aside" option to the ballot, as described in FBPPAR.
For another, the B>AC voters could simply reject C, the strongest rival of their favorite, and B would win with no need for favorite betrayal.
And finally, note that in any scenario where it fails that for some small group, there is a rational strategy for some superset of that group which does not involve betrayal. For instance, in first scenario above, if 16 of the C>B voters switch to CB, then B is the leader and wins without them having to rate C below their true feelings.
An example
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
 Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
 Nashville, with 26% of the voters, near the center of Tennessee
 Knoxville, with 17% of the voters
 Chattanooga, with 15% of the voters
The preferences of the voters would be divided like this:
42% of voters (close to Memphis) 
26% of voters (close to Nashville) 
15% of voters (close to Chattanooga) 
17% of voters (close to Knoxville) 





Assume voters in each city preferred their own city; rejected any city that is over 200 miles away or is the farthest city; explicitly accepted the closest other city; and left the rest blank. In the following table, the quantities in parentheses are explicit votes, while the implicit totals (including blank votes) are unparenthesized.
City  P  (A)  A  (R)  R  tally 

Memphis  42  (0)  0  (58)  58  42 
Nashville  26  (42)  74  (0)  0  100 
Chattanooga  15  (43)  43  (0)  42  58 
Knoxville  17  (15)  15  (0)  68  32 
Memphis is rejected by a majority, and is nonviable. Chattanooga and Knoxville both get less than 25% preference, so they are also nonviable. Nashville is the leader, and wins with a tally of 100%. This is a strong equilibrium; no rational strategy from any faction or combination thereof would change the winner. Knoxville and/or Chattanooga could each prevent the other from being disqualified and even get the other to temporarily be the frontrunner, but Nashville would still win with a tally of at least 68 (the ballots of Nashville and Memphis).
(If Memphis voters rejected Nashville, then Chattanooga or Knoxville could win by conspiring to reject Nashville and accept Memphis. However, Nashville could stop this by rejecting them. Thus this strategy would not work without extreme foolishness from both Memphis and Nashville voters, and extreme amounts of strategy from the others.)
Discussion
Logic for 25%preferred threshold (step 2)
The 25%preferred threshold in step 1 is not purely arbitrary; it is exactly enough so that, in a 3candidate election where all voters give all three grades, there will always be at least 1 candidate who passes the thresholds to not be disqualified. In other words: if a minority supports a rejected candidate, while a majority divides preferences between two candidates while accepting the other, then at least one of those two will not be disqualified. This does not hold for an election with 4 or more candidates, because the majority could split its preferences more than two ways; but even in those cases, it is usually reasonable to hope that the top 3 candidates combined will get enough preferences to ensure that at least one of them is above the 25% threshold.