Condorcet-cardinal hybrid methods
These are Condorcet methods that use cardinal methods a.k.a graded voting (Approval voting, Score voting, etc.) to resolve cycles.
One of the strongest reasons to prefer rated Condorcet methods to other Condorcet methods is that they can do better than other Condorcet methods in situations where cycles are strategically induced because they allow voters to prioritize which pairwise matchups they want to win. In the below example, the top line of voters honestly voted A>C>B>D, making C the Condorcet winner. But...:
Now A's 6 supporters naturally are not happy about that. What can they do to make their man win? There is one, and basically only one, change to their "A>C>B>D" vote they can make which causes A to win. That is to "bury" C with "A>B>D>C." (Well, the alternate burials A>D>B>C and A>D>C>B also work in this example, but they both are at-least-as-dishonest votes and anyway still count as burying C.) If they do this, we get
#voters Their vote 6 A>B>D>C 2 B>C>A>D 3 B>A>C>D 2 C>D>A>B 2 C>B>A>D 5 D>C>A>B 1 D>A>C>B
whereupon A wins using every one of these Condorcet methods: Tideman ranked pairs, Basic Condorcet, Simpson-Kramer min-max, and Schulze beatpaths. (Success!)[1]
This problem is averted with Smith//Score or Smith//Approval if the C>A voters (voters who prefer C to A) move their approval threshold between C and A, because they can make C have 11 approvals to A's 10. Essentially, they can re-simulate the pairwise matchup between C and A (where C has 11 votes to A's 10) using min-max strategy to fix the result. This isn't as easy with Condorcet-IRV hybrid methods or defeat-dropping Condorcet methods; for the most part (ignoring things like the Tied at the top rule, etc.), the only way for C>A voters to fix the result is to Favorite Betray. In some sense, this all takes advantage of how rated methods have Nash Equilibriums on the Condorcet winner.