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.