When One Object Can Have Several Labels, Collective Classification Changes

An urban-greening proposal might support both biodiversity and climate adaptation, even if each resident must choose just one primary category. A new mathematical study asks how a collective classification can retain both memberships without inventing labels that nobody proposed—and what that choice implies for individual authority and equal treatment.

From one label to several

In the preprint Classification Aggregation with Multiple Memberships, Federico Fioravanti, Leo Kurata and Vasilis Stamatis study a collective decision problem: combining classifications when people disagree about where objects belong.

Each individual assigns exactly one category to every object and must use every category at least once. The collective outcome must also classify every object and keep every category represented, but it may give an object several categories. These are final memberships, not tentative options awaiting a later choice, and not fractional degrees of belonging.

The authors connect this change to an earlier impossibility result. In the standard finite-population setting with at least three categories, requiring a single collective label, respect for unanimous agreement and independence across objects forces the rule to copy one fixed person’s classification. Allowing multiple memberships opens a different set of possibilities.

Keeping accepted labels grounded

The paper maintains Independence: an object’s collective classification depends only on how individuals classify that object, not on their reports about other objects.

It also requires Supportiveness: every category assigned to an object must have at least one individual supporter for that object. This is stronger than merely respecting unanimous agreement.

The difference matters when people disagree. If some classify a proposal as Arts and others as Biodiversity, a rule could respect unanimity yet assign every category whenever disagreement occurs—including Climate, which nobody proposed. Supportiveness excludes that unsupported addition while permitting either supported category or both.

These conditions alone do not establish the paper’s main characterizations. The authors add separate requirements concerning individual authority or stability across changes in reports.

Who can secure a membership?

The first is Responsibility. Every accepted membership must have a supporter whose complete submitted classification keeps that membership accepted regardless of what everyone else reports. Merely helping a label reach a majority is not enough: the supporter’s report must suffice on its own.

The authors prove that Independence, Supportiveness and Responsibility characterize what they call junta rules. Here, “junta” means a designated coalition, with no restriction that it be small. For each object–category pair, the rule fixes a coalition and accepts that membership exactly when at least one coalition member proposes it.

Different people can therefore secure different labels for the same object. The coalitions may vary across objects and categories, but must satisfy common-member conditions that guarantee no object is unclassified and no category disappears. Treating category names symmetrically makes the coalition depend only on the object; treating object names symmetrically as well produces one coalition throughout.

Two routes through stability

For finite populations, Rejection Stability gives the same general junta characterization when added to Independence and Supportiveness. It requires a rejected membership to remain rejected when its supporters retain their own complete reports from either of two rejecting profiles. Combining separately insufficient supporting reports cannot make the membership succeed.

Split Stability instead concerns moving some support from one category to another that previously had no supporters at that object. Other memberships must remain unchanged, and at least one of the two categories must be accepted exactly when the original category was accepted before the split.

With at least three categories and more objects than categories, this condition characterizes the narrower class of object-dependent junta rules. Those size restrictions matter: when objects and categories are equally numerous, no admissible split exists.

The underlying winning coalitions form mathematical structures called grills. Their key property is that if a union of two coalitions wins, at least one component wins. The authors relate grills to the filters and ultrafilters used in social-choice theory.

Equal treatment has a price

Under Independence and Supportiveness, adding equal treatment of individuals—Anonymity—to Responsibility leaves only the rule that accepts every proposed membership. Within these assumptions, rejecting some supported labels while retaining Responsibility requires unequal individual authority.

Infinite populations separate the axioms. A rule accepting exactly those labels with infinitely many supporters satisfies the stability requirements but not Responsibility: no single supporter can secure acceptance. Responsibility continues to characterize junta rules even there.

What the results do not establish

This is an axiomatic study, not an evaluation of classification accuracy or a deployed budgeting system. Its application examples motivate the framework rather than demonstrate practical improvements. The authors do not recommend how to select a junta, and the requirement that every individual use every category may fit some applications better than others. Strategic incentives and models allowing individuals themselves to submit multiple labels remain questions for future research.