A Reachable-State Operator Formulation of Deferred Acceptance: Progress Invariants and Structural Diagnostics
Abstract
We give a reachable-state operator formulation of one-to-one deferred acceptance with strict, possibly incomplete preference lists. A local update operator is defined for each active proposer, while a scheduler selects which local update is applied. On the states reachable from the canonical empty initial state, three invariants are immediate: the set of proposed edges grows strictly, every proposer visits each acceptable receiver at most once, and every receiver holds its most-preferred proposal received so far. These invariants yield termination after at most |E| proposals and stability of every terminal reachable state. The classical rejection lemma then gives proposer optimality and schedule independence. We separate these trajectory statements from two logically different results: the distributive-lattice structure of the full stable-matching set and one-sided strategy-proofness. A diagnostic table records which proof obligation is lost when a model changes the bipartition, ordinal comparisons, proposal irreversibility, or receiver choice rule. The paper makes no new complexity claim; its purpose is a precise operator-level account of the classical proof architecture and of the limits of that architecture.
Related
- Structural Realizations and Invariants of Universal Obstruction Cohomology: Goodwillie Towers, Chromatic Filtrations, and Quantum Contextuality
- Exact Avoid-or-Compress Extrema for Dense Reachable Images: Defects Four and Five
- Counting Reachable Positions in a Simplified Model of Connect Four
- Reachable sets under Kolmogorov dynamics in the probability simplex
- Computing Invariant Reachable Sets through Sum-of-Squares Programming
BibTeX
@misc{ishida2026reachablestate,
title = {{A Reachable-State Operator Formulation of Deferred Acceptance: Progress Invariants and Structural Diagnostics}},
author = {Yoshiteru Ishida},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19237},
eprint = {2609.19237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19237},
}
RIS
TY - UNPB
TI - A Reachable-State Operator Formulation of Deferred Acceptance: Progress Invariants and Structural Diagnostics
AU - Yoshiteru Ishida
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19237
UR - https://arxiv.org/abs/2609.19237
AB - We give a reachable-state operator formulation of one-to-one deferred acceptance with strict, possibly incomplete preference lists. A local update operator is defined for each active proposer, while a scheduler selects which local update is applied. On the states reachable from the canonical empty initial state, three invariants are immediate: the set of proposed edges grows strictly, every proposer visits each acceptable receiver at most once, and every receiver holds its most-preferred proposal received so far. These invariants yield termination after at most |E| proposals and stability of every terminal reachable state. The classical rejection lemma then gives proposer optimality and schedule independence. We separate these trajectory statements from two logically different results: the distributive-lattice structure of the full stable-matching set and one-sided strategy-proofness. A diagnostic table records which proof obligation is lost when a model changes the bipartition, ordinal comparisons, proposal irreversibility, or receiver choice rule. The paper makes no new complexity claim; its purpose is a precise operator-level account of the classical proof architecture and of the limits of that architecture.
ER -