our assignments so far are consistent, we must show that all points that receive a total value of 3 are indifference points. Here, again, the preferential independence assumption is used, since the relation between and is independent of z (and similarly for the other pairs); hence and similarly for the other pairs. Hence, the 3 point is defined consistently. We can continue like this by induction and define the per-commodity functions in all integer points, then use continuity to define it in all real points. An implicit assumption in point 1 of the above proof is that all three commodities are essential or preference relevant. This means that there exists a bundle such that, if the amount of a certain commodity is increased, the new bundle is strictly better. The proof for more than 3 commodities is similar. In fact, we do not have to check that all subsets of points are preferentially independent; it is sufficient to check a linear number of pairs of commodities. E.g., if there are different commodities, , then it is sufficient to check that for all , the two commodities are preferentially independent of the other commodities. Uniqueness of additive representation An additive preference relation can be represented by many different additive utility functions. However, all these functions are similar: they are not only increasing monotone transformations of each other (as are all utility functions representing the same relation); they are increasing linear transformations of each other. In short, An additive ordinal utility function is unique up to increasing linear transformation. Constructing additive and quadratic utility functions from ordinal data The mathematical foundations of most common types of utility functions — quadratic and additive — laid down by Gérard Debreu enabled Andranik Tangian to develop methods for their construction from purely ordinal data. In particular, additive and quadratic utility functions in variables can be constructed from interviews of decision makers, where questions are aimed at tracing totally 2D-indifference curves in coordinate planes without referring to cardinal utility estimates. See also Preference (economics) Multi-attribute utility Consumer theory Marginal utility Lattice theory Convex preferences References External links Lexicographic preference relation cannot be represented by a utility function. In Economics.SE Recognizing linear orders embeddable in R2 ordered lexicographically. In Math.SE. Murray N. Rothbard, "Towards a Reconstruction of Utility and Welfare Economics" Utility