The Impossibility of Consistent Scalar Aggregation: Why Multidimensional Poverty Indices Must Fail Distributional Sensitivity

Multidimensional ordinal data are frequently summarized by a single scalar index. Two
desiderata are commonly demanded: robust monotonicity under joint dominance improvements,
and distributional sensitivity among the deprived. A canonical robust dominance order on a finite
ordinal product grid is strict lower orthant dominance. A canonical distributional sensitivity
principle is a spread axiom that penalizes concentration in the distribution of deprivation severity
among the poor.
A known defect of global spread axioms on ordinal scales is Pareto perversity: they can
penalize perfect concentration at the best end of the ordinal support. A standard repair in poverty
measurement is censoring: apply spread sensitivity only within an identified poor subpopulation.
This paper formalizes that joint-censoring repair and proves that the scalar aggregation deadlock
persists. Fix any nontrivial poverty identification rule and consider the conditional distribution
among the identified poor. Fix any finite severity scale on the poor set with at least two levels,
and fix any order-sensitive severity concentration preorder that treats sufficiently endpoint-peaked
severity distributions as strictly maximally concentrated. If the best corner outcome is classified
as non-poor, then for every strictly positive distribution P there exists a strictly positive Q such
that

P ≺lower orthant Q ≺joint censored P.

Consequently, no scalar index can be strictly increasing under both relations.

The proof isolates a structural orthogonality: mixing a distribution with sufficiently large
mass at a single non-poor dominance anchor improves all dominance thresholds while leaving the
conditional distribution among the poor invariant. This orthogonality yields mixed cycles and a
general impossibility theorem for scalar indices under joint-censored distributional sensitivity.

This version adds five further results in full detail. First, the mixed-cycle impossibility
extends beyond strict positivity: the set of distributions that lie on strict mixed cycles contains
an open and dense subset of the full probability simplex, and every boundary distribution
admits an ε-mixed-cycle arbitrarily nearby. Second, the results are stated as a sharp design
frontier: strict anchored dominance and within-poor focus cannot be jointly satisfied by any
scalar index whenever strict anchored dominance is nontrivial. Third, a substantive logical defect
is repaired: weak within-poor monotonicity is not automatically satisfied by dominance-based
scalars and remains inconsistent with strict dominance under mixed cycles. Fourth, the argument
is generalized beyond the best corner of a product grid: the mixed-cycle mechanism holds on
any finite partially ordered outcome space equipped with any dominance order generated by
down-sets that share a common non-poor dominance anchor, and it admits a measurable-space
formulation for anchored dominance families under a uniform slack condition.

Fifth, and strategically, the paper is extended from a pure no-go statement to a complete
classification theorem and a representation-dimension theory. Under anchored robust dominance,
strict scalarizability together with conditional-invariant within-poor sensitivity holds if and
only if the within-poor strict relation is empty (no nontrivial strict within-poor sensitivity).
When scalarizability fails, the minimal dimension required for a coordinatewise multi-index
representation is characterized and, in the canonical two-axiom case, equals 2.