Identifying Unmet Demand under Participation Restrictions in Public Libraries

A public library can restrict an activity and subsequently possess few records of participation in it. Interpreting those records requires a distinction between current use, responses to a specified alternative offer, interests formed through experience, and the public reasons for supporting an opportunity. This paper develops an account of that distinction for shared library spaces and group activities. It reconstructs a binary governance gate under explicit assumptions, derives partial-identification bounds for population coverage and recording error, and examines how capacity and coordination alter the relation between demand and attendance. A history-based counterexample distinguishes an unobserved prior disposition from demand indexed to an actual offer that may itself change interest. A proposed design randomizes formative exposure before a common subsequent offer and identifies the resulting prospective response contrast under stated assignment and measurement conditions. A separate branch-level construction accommodates within-branch interference and distinguishes population weighting from equal branch weighting. Further analysis addresses feedback between observed use and continued access, changing outcome categories, and the preservation of records needed for later comparison. Documented public-library policies and existing professional evaluation practices motivate a prospective study; the numerical examples remain constructed. The philosophical discussion examines preference formation, practical agency, competing uses, refusal, and authority over public priorities. The contribution is an explicit connection among an inferential target, the policy conditions that generate its evidence, the assumptions needed to interpret that evidence, and the decisions left open by it. Empirical implementation and a broader originality review remain research obligations.