At first, it seems "obvious" that it is desirable for a voting system to obey the "Condorcet property" that, whenever any candidate X exists who would be preferred over any opponent by more voters in a pairwise election (deducing the X versus Y preferences from the ballots), then X must always win.

But Peter Fishburn [*Paradoxes of Voting*, Amer. Political Science Review 68 (1974) 537-546]
pointed out the following. He constructed an election with 9 candidates and
101 voters (each providing a full
rank order ballot) in which the candidates named "X" and "Y" received the following numbers of
kth-rank votes:

k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | total |
---|---|---|---|---|---|---|---|---|---|---|

X | 0 | 30 | 0 | 21 | 0 | 31 | 0 | 0 | 19 | 101 |

Y | 50 | 0 | 30 | 0 | 21 | 0 | 0 | 0 | 0 | 101 |

In this election, it seems "obvious" that Y should be preferred over X. After all, no matter where you draw the line, Y gets more votes than X above that line. (For example: draw the line at the top rank? Y gets 50 top-rank votes, versus X gets 0. Draw the line at the third rank? Y gets 50+30=80 top-thru-third-rank votes, versus X gets 30.) And not just "more," but indeed "a lot more" – always at least 23% more, which is generally regarded as a strong and convincing victory. (And indeed, pretty much every non-Condorcet voting method ever invented, would elect Y here.)

*However,* in Fishburn's election X was the beats-all winner.
Therefore, your two "obvious" perceptions *conflict* and hence at least one
of them must be wrong.
Fishburn concluded that the Condorcet property is *not* an always-desirable one.
Therefore,
"non-Condorcet" voting systems
that do not always elect beats-all winners,
should not be downgraded or dismissed.

Fishburn did not actually write down the ballots in his election, so we shall. Here's an election that does the job:

#voters | their vote |
---|---|

19 | Y>A>B>C>D>E>F>G>X |

31 | Y>A>B>C>D>X>F>G>E |

10 | E>X>Y>G>F>D>C>B>A |

10 | F>X>Y>G>E>D>C>B>A |

10 | G>X>Y>E>F>D>C>B>A |

21 | G>F>E>X>Y>D>C>B>A |

Its defeats matrix is:

Canddt | A | B | C | D | E | F | G | X | Y |

A | * | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 0 |

B | 51 | * | 50 | 50 | 50 | 50 | 50 | 50 | 0 |

C | 51 | 51 | * | 50 | 50 | 50 | 50 | 50 | 0 |

D | 51 | 51 | 51 | * | 50 | 50 | 50 | 50 | 0 |

E | 51 | 51 | 51 | 51 | * | 39 | 29 | 50 | 31 |

F | 51 | 51 | 51 | 51 | 62 | * | 60 | 50 | 31 |

G | 51 | 51 | 51 | 51 | 72 | 41 | * | 50 | 31 |

X | 51 | 51 | 51 | 51 | 51 | 51 | 51 | * | 51 |

Y | 101 | 101 | 101 | 101 | 70 | 70 | 70 | 50 | * |

X beats every opponent pairwise by a 51-to-50 margin.
Should this really force X's victory and should it really outweigh the huge
apparent preference for Y over X?
Keep in mind, the sole reason X beats Y pairwise is due to a *single* voter.
Meanwhile, Y beats every opponent besides X by a huge landslide margin
(70-to-51 or 101-to-30) and only loses to X by a tiny 51-to-50 margin in
votes, and Y has a huge advantage over X in *strength* of preference –
as indicated both by the table at the very top, and also by the fact that
the 51 voters who prefer X over Y all do so by the tiniest amount possible (adjacent rankings),
while the 50 who prefer Y over X all do so by large amounts.
If you agree, even in this election *alone*,
that X is not the best winner, then you have *admitted* that the
Condorcet criterion is undesirable. That is because the Condorcet criterion states
that in *every* election where a beats-all winner exists, it must win.
Condorcet brooks *no* exceptions. (Any voting method that fails to elect
a Condorcet winner, in even a single case, is not a "Condorcet method.")
So if you admit even a *single*
exception, then you have admitted Condorcet's criterion is not right.
And I think we *do* have to admit that this election *is* an exception.

Condorcet's error was in assuming that something that usually sounds
good, must *always* be good. Actually, something that is usually good, is exactly that
– usually good – and therefore we should want, or at least not exclude,
voting systems like range voting that usually,
but not always, yield
beats-all winners when they exist.
(Range Voting
with strategic voters yields Condorcet winners
under reasonable assumptions, but that is a different issue; this page has only been about
honest voters.)

Condorcet self-contradiction example #1 and #2.