[EM] Voting paradoxes article

[Markus Schulze]

Dear David,

I am not sure whether I have understood you correctly.
That's why I want to ask you to explain your thoughts
using the following two examples?

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Example 1:

There are 120 voters and 4 candidates for 2 seats.
8 voters vote A > C > B > D.
8 voters vote A > C > D > B.
8 voters vote B > C > A > D.
8 voters vote B > C > D > A.
8 voters vote C > A > B > D.
8 voters vote C > A > D > B.
8 voters vote D > A > B > C.
8 voters vote D > A > C > B.
7 voters vote A > D > B > C.
7 voters vote A > D > C > B.
7 voters vote B > D > A > C.
7 voters vote B > D > C > A.
7 voters vote C > B > A > D.
7 voters vote C > B > D > A.
7 voters vote D > B > A > C.
7 voters vote D > B > C > A.

If I understand you correctly, then either
(A and B) or (C and D) must be elected in
Example 1. Because of the Anonymity Criterion
and the Neutrality Criterion, you can presume
without loss of generality that C and D are
elected.

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Example 2:
Now, candidate E is added so that there are
120 voters and 5 candidates for 2 seats.
8 voters vote A > E > C > B > D.
8 voters vote A > E > C > D > B.
8 voters vote B > C > A > E > D.
8 voters vote B > E > C > D > A.
8 voters vote C > A > B > E > D.
8 voters vote C > A > D > B > E.
8 voters vote D > A > B > E > C.
8 voters vote D > A > C > B > E.
7 voters vote A > D > B > E > C.
7 voters vote A > E > D > C > B.
7 voters vote B > E > D > A > C.
7 voters vote B > E > D > C > A.
7 voters vote C > B > A > E > D.
7 voters vote C > B > D > A > E.
7 voters vote D > B > A > E > C.
7 voters vote D > B > C > A > E.

What does

> Consider an election of n seats. Are there any combinations
> of n "Condorcet winners" who win in any consideration of n+1
> candidates? If there are more than one combinations which do
> this it becomes more complex and involves more head-to-head
> stuff- but, for simplicity, ignore this. I would argue that
> this setup (with your choice of election rules for the n+1->n
> election and the head-to-head untying elections) is independent
> of the removal of irrelevant alternatives wherever an answer
> exists- just like Condorcet!

mean in this context?

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Markus Schulze