Leximin order
In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate, but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division. Definition A vector x = (x1, ..., xn) is leximin-larger than a vector y = (y1, ..., yn) if one of the following holds: The smallest element of x is larger than the smallest element of y; The smallest elements of both vectors are equal, and the second-smallest element of x is larger than the second-smallest element of y; ... The k smallest elements of both vectors are equal, and the (k+1)-smallest element of x is larger than the (k+1)-smallest element of y. Examples The vector (3,5,3) is leximin-larger than (4,2,4), since the smallest element in the former is 3 and in the latter is 2. The vector (4,2,4) is leximin-larger than (5,3,2), since the smallest elements in both are 2, but the second-smallest element in the former is 4 and in the latter is 3. Vectors with the same multiset of elements are equivalent w.r.t. the leximin preorder, since they have the same smallest element, the same second-smallest element, etc. For example, the vectors (4,2,4)