ParetoOrdnung

Paretoordnung, also called Pareto order, is a mathematical concept used to compare outcomes in multiple criteria by a componentwise relation. It is widely applied in economics, decision theory and multi-criteria optimization to formalize when one outcome is at least as good as another across all criteria.

Formally, let x and y be outcome vectors in a feasible set X ⊆ R^n, where each component i represents a criterion. We say x ≽ y (x Pareto-dominates y) if x_i ≥ y_i for all i. If x ≽ y and there exists at least one index i with x_i > y_i, then x strictly Pareto-dominates y. The relation x ≽ y is a partial order on X. An outcome x is Pareto-optimal (or Pareto-efficient) if there is no y ∈ X with y ≽ x and y ≠ x.

In economics, Pareto efficiency conveys that no feasible reallocation can make someone better off without making someone else worse off. The Pareto frontier (or Pareto set) consists of all Pareto-optimal outcomes and represents the trade-offs among criteria that cannot be improved without harming others. In decision analysis, the Pareto frontier helps illustrate competing criteria without imposing a subjective cardinal ranking.

Limitations include that the Pareto order does not specify a unique best outcome when several are non-dominated, and it does not resolve trade-offs between criteria without additional preferences or monetization. It is also defined relative to a chosen feasible set and may depend on measurement scales and direction of improvement.