An outcome is Pareto efficient when no feasible alternative weakly improves every agent's utility and strictly improves at least one. For strict preference orders over distinct alternatives, an outcome unanimously ranked below another is inefficient. Onto strategyproof social choice functions on an unrestricted domain are Pareto efficient: rank-raising monotonicity and unanimity rule out selecting a unanimously dominated alternative.
A feasible change weakly increasing every participant's utility and strictly increasing at least one. An assignment is Pareto optimal exactly when no Pareto improvement exists. This comparison permits coordinated changes by several participants; it is different from a profitable unilateral deviation used to test a Nash equilibrium.
Two strictly positive nonproportional utility gradients admit a transfer direction with and . Choose with ; strict Cauchy-Schwarz inequality makes this interval nonempty. Small opposite transfers in this direction strictly improve both differentiable utilities. Absence of such trades therefore forces positive proportionality of all gradients in an unconstrained allocation space.
The Pareto frontier consists of feasible payoff vectors that no other feasible vector weakly improves in every coordinate and strictly improves in at least one. A Nash bargaining solution with positive gains lies on this frontier, because increasing either gain without decreasing the other raises its Nash product.
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Pareto efficiency, also known as Pareto optimality, is an economic concept that describes a situation in which resources are allocated in a way that no reallocation can make one individual better off without making at least one other individual worse off. In simpler terms, an allocation is Pareto efficient if there are no possible changes that could improve someone's situation without harming someone else's situation.