[Autocratic]

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In our two party system I don't think this would make much of a difference.

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That's not true, it would actually force several third parties into the spotlight if they were allowed on ballots.

Also, the Condorcet winner is not a voting method - the method is the Copeland method, the Condorcet winner is a candidate who wins all of its match ups under the Copeland method, and the Condorcet criterion for voting. I didn't look at your links so perhaps that was covered.

The major problem with the Copeland method is that it violates the Pareto condition, which states that if everyone prefers candidate X to Y, Y cannot win. Observe:

3 candidates, A, B, and C. Here is a chart of numbered preferences for several groups of people:

9 voters vote: ABC (A is the first choice, C the third).
10 voters vote: BAC
11 voters vote: CAB

Now, counting just first place votes (meaning by plurality), C wins, with 11 first place votes. Here is when you test the Condorcet criterion:

A vs B: A wins with 20 voters to 10 preferring it.
A vs C: A wins with 19 votes to 11.
B vs C: B wins with 19 to 11.

Here, A is the winner with the Copeland method, not C. Though it's accepted that no voting system is perfect.