Arrow's Impossibility Theorem
I seem to be posting a lot about paradoxes recently. I'll probably take a little break from paradoxes after this one.
Summary #
In 1951, Kenneth Arrow demonstrated that it is not possible to have a "fair" voting system that satisfied the following three criteria (imagine the group is voting on which fruit to eat: apples or pears):

If every voter prefers apples to pears, then the group prefers apples to pears. (Sound familiar? It's called Pareto efficiency.)

If every voter prefers apples to pears, then even if bananas are added to the set of options, the group will still prefer apples to pears.

There is no dictator.
This is known as Arrow's Impossibility Theorem.
Commentary #
The actual details of the theorem are interesting, and I refer you to Wikipedia (for those who are interested). There are situations, however where item 2 (where we added bananas) doesn't hold: imagine the game rockspaperscissors. In such a case, adding an alternative transforms the straightforward choice into a cyclic choice. I sometimes see this scenario when people compare different aspects of multiple candidates' platforms (or when they're choosing which car / laptop / soap / pants to purchase).
Sometimes, the tradeoffs are hard; but sometimes they're impossible.