What is Arrows Theorem?
Using the above definition of an Irrelevant Alternative, Arrow suggested the criterion that no Irrelevant Alternative should affect the result of an election. This is called the Independence from Irrelevant Alternatives Criterion. In short, Arrow proved that no method can satisfy all of a number of reasonable sounding criteria. In particular no method can: 1. Allow voters to rank candidates. 2. Always choose the majority winner in a two-candidate race. 3. Meet the Independence from Irrelevant Alternatives Criterion. This can be easily proven from the following example 40 A B C 35 B C A 25 C A B The pairwise results are: A over B, 65-35 B over C, 75-25 C over A, 60-40 It is clear that no matter which candidate we choose as winner, IIAC should allow us to hold a pairwise election between this and any other single candidate, and this candidate should still be the winner. Since each candidate is the majority loser of one of these pairwise contests, we cannot satisfy both 2 and 3.
