People carve the world into different categories. One person's “warm” spans 65-80°F; another's spans 70-85°F. One person distinguishes five shades of blue; another uses two words for the same range. When a group needs a shared vocabulary — a common set of categories — how should individual vocabularies be combined?
The paper (arXiv:2603.12958, March 2026) models vocabularies as partitions of a linearly ordered set, where each word corresponds to an interval. Individual vocabularies can differ in the number of words and the width of each interval. The question is whether an aggregation rule can be strategy-proof — meaning no agent benefits from misrepresenting their vocabulary.
The result: under specific preference constraints, strategy-proof aggregation exists, but it operates by merging intervals rather than averaging boundaries. The collective vocabulary is not a compromise between individual category systems — it is a structural combination that respects the interval property while preventing manipulation.
The structural lesson: aggregating categories is harder than aggregating values. When you average numbers, the result is a number. When you average partitions, the result may not be a valid partition — overlapping intervals, gaps, violated ordering. The constraint that the output must be a well-formed vocabulary (non-overlapping, complete coverage) restricts the space of possible aggregation rules far more tightly than numerical averaging. The strategy-proofness comes not from cleverness in the rule design but from the structural rigidity of the partition requirement — there are so few valid aggregation rules that manipulation has nowhere to go.