Characterizing lexicographic preferences

We characterize lexicographic preferences on product sets of finitely many coordinates. The main new axiom is a robustness property. It roughly requires this: Suppose x is preferred to y; many of its coordinates indicate that the former is better and only a few indicate the opposite. Then the decision maker is allowed a change of mind turning one coordinate in favor of x to an indifference: even if one less argument supports the preference, the fact that we started with many arguments in favor of x suggests that such a small change is not enough to give rise to the opposite preference.