en.wikipedia.org/wiki/Pareto_front
2 corrections found
a Pareto front represents the set of solutions where no solution outperforms any other solution in the set at every objective, and every solution not in the set is outperformed by at least one solution in the Pareto front in every objective.
This wording uses too-strong dominance criteria. A Pareto-optimal point is one that is not worse in all objectives and strictly better in at least one objective than another point—not necessarily better in every objective.
Full reasoning
The standard definition of Pareto dominance is weaker than the article's wording.
For a minimization problem, solution (x_a) Pareto-dominates (x_b) when:
- (x_a) is no worse in every objective, and
- (x_a) is strictly better in at least one objective.
That means a non-Pareto point can be dominated even if the dominating point is equal on some objectives. Likewise, points on the Pareto front are those not dominated by any feasible point under that standard rule—not points that cannot be beaten only when another solution is better in every objective.
A simple counterexample shows why the article's phrasing is too strong: in a two-objective minimization problem, ((1,2)) dominates ((1,3)) because it is equal on the first objective and better on the second. The article's "in every objective" wording would incorrectly exclude that standard domination case.
So the sentence overstates the requirement for Pareto dominance and gives a misleading definition of what makes a point belong to the Pareto front.
2 sources
- Pareto optimization with small data by learning across common objective spaces - PMC
Definition 1 (Pareto Dominance) A solution xa is said to Pareto dominate solution xb if ∀i ∈ {1,2,...,m}: fi(xa) ≤ fi(xb) and ∃j ∈ {1,2,...,m} such that fj(xa) < fj(xb).
- Comparing Solutions under Uncertainty in Multiobjective Optimization
Definition 1 (Pareto dominance). The objective vector z dominates the objective vector w, z≺w, if and only if zj ≤ wj for all j ∈ {1, …, m} and zk < wk for at least one k ∈ {1, …, m}.
at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers.
This is only generally true for interior allocations under standard smoothness assumptions. Pareto-efficient boundary or corner allocations can be efficient even when consumers’ marginal rates of substitution are unequal or undefined.
Full reasoning
The sentence is stated too broadly.
In standard exchange-economy analysis, equal marginal rates of substitution (MRS) characterize interior Pareto-efficient allocations under smooth, well-behaved preferences. But that is not true for all Pareto-efficient allocations.
A direct counterexample is given in an open-access economics paper: if one agent's bundle lies on the boundary of the consumption set, the allocation can still be Pareto efficient despite an inequality of marginal rates of substitution. In corner cases, one consumer's MRS may even be undefined.
So the unqualified claim "at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers" is incorrect as a general statement. It needs a qualification such as "for interior Pareto-efficient allocations" (with the usual differentiability/convexity assumptions), or it should explicitly note the boundary/corner exceptions.
2 sources
- Locally efficient and strategy-proof allocation mechanisms in exchange economies
In an economy with two agents and two goods, consider allocation where one agent's consumption is on the boundary of the agent's consumption set. For many reasonable preferences, the allocation will be Pareto efficient despite an inequality of marginal rates of substitution.
- Extremes of the Edgeworth Box
Thus, interior efficiency equalizes marginal rates of substitution; at a binding boundary, a non-negative shadow value means a small feasible transfer would harm the bound-side agent at least as much as it helps the other, so no Pareto improvement exists.