A mathematical framework for comparing utility across people starts with a demanding question: can people's preferences, recorded only as rankings of alternatives, support a common unit of measurement? An arXiv preprint addresses that question with a formal theory built around a virtual numeraire, an abstract commodity used as a shared utility unit. A profile is interpersonally comparable under the paper's definition when the same virtual commodity serves as a utility unit for every person.
The work is a modeling analysis of abstract societies and allocations, not a study of recruited participants or observed outcomes. Its social-state space is a convex, relatively open Euclidean set with at least two dimensions, and individuals are indexed by S. No empirical dataset is described.
A shared scale with strict rules
The proposed unit is tested through two requirements: common-transfer invariance and monotonicity. In plain language, the virtual commodity must behave consistently when it is transferred within the model, while the preference ordering changes in the required direction. The framework uses these conditions to test for a common unit based on ordinal preference information.
Theorem 1 links the virtual commodity to ordinary utility representations. Every virtual numeraire corresponds to a utility unit for some representation, and every representation has such a unit. When a unit is shared, the associated utility representations are fixed up to an additive constant: their zero points may differ, while the common scale remains the same.
A single virtual commodity must be a common utility unit for every member of the profile. The definition identifies a shared measurement structure; it does not assert that every profile has one.
The mathematical boundary of comparability
For profiles with at least two individuals, Proposition 2 describes exactly which changes in utility labels preserve comparability across all comparable profiles. Every person must use the same positive rescaling of the utility scale, although each person may have a different additive shift. Different zero points are allowed, but person-by-person rescaling is not.
Theorem 3 offers a geometric test for profiles that satisfy a constant-rank condition, meaning the relevant gradient structure keeps the same rank across the model. Utility gradients describe the local direction of utility change. The theorem says a profile is comparable if and only if the affine hull of those gradients never contains the zero vector. In ordinary terms, the set formed by the relevant affine combinations must not include zero.
The paper also examines whether common units exist locally, meaning in a limited neighborhood of the model. When a preference subprofile has a locally separable representation, Theorem 4 restricts every common local virtual numeraire to a simple parametric family. In the nonseparable case, Theorem 6 gives a necessary-and-sufficient compatibility condition under three technical conditions labeled CR.1, CR.2 and CR.3, together with an index that does not vanish locally. It smoothly parameterizes the solution set when that condition holds.
From measurement to welfare rules
Once a profile meets the paper's comparability definition, Theorem 2 addresses how a social planner may combine utilities. On comparable profiles, an Arrovian planner, meaning one operating under the paper's stated social-choice axioms, and satisfying cardinal unit-comparability is characterized as utilitarian. Here, utilitarian aggregation means summing utilities measured in the common unit to form a social objective. The result is conditional on the comparable domain and those axioms.
For expected-utility profiles, the framework connects choices about utility translations with weighted sums of expected utilities. A particular translation direction makes the comparable-utilitarian objective affine-equivalent, or equivalent after a linear rescaling and added constant, to relative utilitarianism. These are analytical results for expected-utility preference models, rather than findings from observed decisions.
The analysis also considers dilation and mixed units. Those constructions correspond to normalized-log and other parametric aggregation forms, with expected-utility summation and Nash social welfare as the two endpoint cases. The links apply within the expected-utility profile and common-unit assumptions used by the model.
An illustrative tax choice
The paper applies the framework to an illustrative tax-reform example. Under the common-unit utilitarian criterion, the example selects policy A. Under the money-metric sum, it selects policy B; the paper describes that sum as incoherent because its units cannot be rationalized in a common unit. The example therefore presents different policy rankings under different welfare criteria.
That contrast is a model result, not an observed policy test. The example is an illustrative calculation within the model rather than evidence from an observed policy outcome.
The limits of the framework
The existence results are not unconditional. The definition of comparability does not assert that a common unit exists for every profile. In the nonseparable case, Theorem 6 makes local existence depend on a necessary-and-sufficient compatibility condition and an index that is non-zero locally, then gives a smooth description of the solutions when the condition holds.
The framework treats a common unit as a mathematical structure to be checked inside the model. When the shared unit is available, Theorem 2's axiomatic result identifies utilitarian aggregation on comparable profiles.
The document is identified as an arXiv preprint, version 1, dated 25 August 2026, with a separate August 27, 2026 date shown in the front matter. Several colleagues are acknowledged for helpful discussions, but no funding source is reported in the supplied material.
Paper data and sources
Original title: Interpersonally Comparable Utility
Authors: Peter Caradonna, Zachary Raines
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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