[EM] What are some simple methods that accomplish the following conditions?
Here is an interesting fact that might help inform the decision about that fraction:
If X beats Y pairwise then the implicit approval of X will be more than half of the implicit approval of Y.
In particular, the ballot CW (when there is one) will have more than half of the implicit approval of the implicit approval winner.
So perhaps in general we should require the method winner to have at least half of the approval of the approval winner (implicit or not).
But here is another thought: when trying to decide on some numerical parameter like the fraction in question, have each voter choose one of the possibilities on their ballot, and go with their median.
[When the median is to be chosen on a one dimensional spectrum, sincere voting is optimal strategy.]
So to be clear the possible "complaint" some voters might have (and you
think we should take seriously) is "We lied and the voting method
(instead of somehow reading our minds) believed us".
So therefore it is good to have a less expressive ballot because that
reduces the voter's opportunities to tell stupid lies and if the method
is simple enough then maybe also the temptation for them to do so.
To me that is absurd. If I agreed with that idea I would forget about
the Condorcet criterion and instead demand a method that meets
Later-no-Help,? such as? IRV or Bucklin or Approval.
But I've thought of a patch to address your issue.? We could have a rule
which says that if the winner's approval score is below some fixed
fraction of that of the most approved candidate, then a second-round
runoff is triggered between those two candidates. What do you
think of that?? What do you think that fraction should be?