Finite-dimensional utilities

Sufficient axioms are identified for the existence of a finite- dimensional quasilinear utility function whose lexicographically ordered vectors preserve a decision maker's preference order on a mixture set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal M}$\end{document}. It is shown that those axioms are also necessary for the linear lexicographic representation when the underlying set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal M}$\end{document} is a mixture space.