Bossiness and Accessibility in Constrained Allocation: A Menu Normal Form for Efficient and Individually Strategy-proof Mechanisms

Abstract

We study deterministic allocation with an outside option, arbitrary finite feasibility constraints, and the full domain of strict preferences. Our main economic result answers negatively Imamura and Kawase's (2025, p. 503) question of whether accessibility is necessary without endowments: in a two-school market with one unitcapacity school, a computer-assisted four-student example is inaccessible yet admits a Pareto-efficient, individually rational, and individually strategy-proof mechanism. The mechanism is bossy, highlighting the gap between individual and group strategyproofness. More generally, we combine the fixed-opponents menu principle with an individual-rationality word reduction and prove that implementation exists if and only if a finite labelled menu relation admits a legal colouring. This yields a constructive normal form for arbitrary finite joint constraints. It also gives finite recognition and exact mechanism reconstruction. Its two-school specialisation gives restriction heredity, product-cylinder closure, and a general bossiness implication for inaccessible constraints. A natural structural characterisation of implementable constraints remains open.