Characterization of Pareto dominance

The purpose of this note– inspired by and using a number of properties from Barbara and Jackson [2] – is to characterize the Pareto dominance relation. The set-up and interpretation is completely analogous to [2]. In particular, situations are considered where different actions are to be compared on the basis of their consequences, which are described by vectors of real numbers. Ranking actions then becomes equivalent with ranking vectors and a decision criterion can be seen as a binary relation on the set of real vectors (see [2, p. 35]). If the coordinates of a vector x ∈ R measure positive attributes, like a firm’s profit, utility of a decision maker, or the quantity of a certain good to a nonsatiable consumer, it Pareto dominates a vector y ∈ R if xi yi for all coordinates i, with strict inequality for at least one coordinate. Conversely, if coordinates measure negative attributes (loss, disutility, quantities of ‘bads’, etc.), x Pareto dominates y if xi yi for all coordinates i, with strict inequality for at least one coordinate. If an alternative x is not Pareto dominated in a given set of alternatives, it is Pareto optimal.