The method I have in mind is reminiscent of random ballot Banks.
Let's say that a "needle" is a chain of candidates such that no candidate is beaten either pairwise or approvalwise by any candidate below it in the chain, but every candidate in the chain beats every candidate below it either pairwise or approvalwise, if not both.
A "needle point" is a candidate at the top of a maximal needle.
Note that Approval Winners and Condorcet Winners are always among the needle points, and that when these two coalesce into one, there are no other needle points.
Here's the procedure that I have in mind for picking a needle point at random. To show that the method can be explained to a beginner I will not mention needles or needle points.
1. List the candidates in order of approval.
2. For each candidate X find a candidate Y=f(X) as follows:
2a. Initialize a list L with X.
2b. Starting at the bottom of the approval list and working your
way up, integrate as many candidates into the list as possible
consistent with the requirement that no member of the list can beat
either pairwise or approvalwise any member above it, but each
member must beat either pairwise or approvalwise every member below
it.2c. The top member of the list is Y.
3. Let S = { Y | Y=f(X) for some candidate X }, i.e. S is the image of the function f.
4. Select a ballot B at random.
5. Transform B into B' by striking from B those candidates that are not members of the set S.
6. The highest ranked candidate on B' is the winner.
[end of method description]
Remark. It worries me that there may be some needle points that are not in the set S, i.e. that our method of constructing needles is not general enough to generate all needle points.
Any ideas?
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