The property that a group's choice between any a and b should be a function only of the choices of the individuals in the group between a and b. In particular, it should not change if some individuals in the group change their minds about the merits of c and/or d. To most, but not all, analysts of electoral systems, independence seems a highly desirable property, and the inconsistency of independence with other desirable properties which is proved by Arrow's impossibility theorem therefore seems disturbing. Others, who disagree with the claim that independence is desirable, are less worried by Arrow's theorem and happier with voting systems, such as the Borda count, which violate the independence of irrelevant alternatives.
To understand what is at stake, consider four skaters A, B, C, and D, and three judges X, Y, and Z. All four skaters are candidates for the open competition, and A, B, and D are also candidates for the under‐25 competition. Both competitions are judged at the same time. The judges rank the candidates on their performance. Their rankings, in descending order, are:
Judge X: ABCD
Judge Y: BCDA
Judge Z: CDAB
By the Borda rule, C wins the open competition, and B wins the under‐25 competition. Then an argument breaks out about the real quality of C's performance. The judges look again at the video replay, and change their minds in various ways, now reporting the following rankings:
Judge X: ACBD
Judge Y: BDAC
Judge Z: DACB
No judge has changed her mind about the relative performance of the three under‐25 contenders—their ranking remains unchanged. But the winner of the under‐25 competition is now A, and B comes in only at third place. Thus the Borda rule violates the independence of irrelevant alternatives. See also path dependence.