# If the real preferences of a population include a condorcet winner, does that mean that any condorcet method is invulnerable to manipulation?

This is probably true, and easy to prove, but I am not coming up with a proof...

Say we have a population, with each individual casting a ballot of preferences Individual1 might say, eg, A > B > C, meaning that individual1 prefers candidate A to B, but B to C.

A condorcet method aways picks a condorcet winner if the given ballots that have such a candidate. We say that an set of individuals can "manipulate" the result if they can cast ballots of preferences that do not correspond to their true preferences, and with that cause the winner of the election to change to a more preferred candidate.

Is it the case that:

1. If the true preferences have a condorcet winner
2. The election method is a condorcet method

Then it is impossible for any set of individuals to manipulate the result?

No, they're subject to manipulation just like everything else. Warren D. Smith has an example on his website:

#voters Their vote
6       A>C>B>D
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


Here we have the true preferences, which have a Condorcet winner C. Under any Condorcet method, C will win, but A's supporters can manipulate the outcome to their advantage by burying C on their ballots, tactically changing their votes from A>C>B>D to A>B>D>C:

#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


Now A wins under Condorcet methods like Tideman ranked pairs, Simpson-Kramer min-max, and Schulze beatpaths.

Note that this effect was achieved by creating a condorcet cycle (see Warren D. Smith's site for a table detailing the results of two contender elections)