If there are three or more candidates, there is no voting system (other than a dictatorship) for which both the Condorcet winner criterion and the independence of irrelevant alternatives hold.

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This is known as Arrow's impossibility theorem, which states that if there are three or more candidates, no voting system can guarantee to always produce a fair result by satisfying all of the following criteria: (1) the Pareto criterion (if every voter prefers A to B, then the group prefers A to B), (2) the independence of irrelevant alternatives (adding or removing a candidate should not change the winner), and (3) the non-dictatorship criterion (there is no individual who always determines the group outcome). The Condorcet winner criterion is a form of the Pareto criterion, and it is impossible to satisfy all three criteria simultaneously.

If There Are Three or More Candidates, There Is No