Exact Minimum-Change Pure Nash Equilibrium Repair in Binary Polymatrix Cactus Games: Source-Complete Separators, Committed Reanchoring, and a Treewidth-Two Complexity Boundary
Year: 2026
Type: preprint
Abstract: This preprint develops a structural and algorithmic framework for exact minimum-change pure Nash equilibrium repair in binary polymatrix games on cactus graphs. The problem is to determine whether a pure Nash equilibrium exists within a prescribed Hamming distance of a reference action profile, and to maintain exact repair solutions under local payoff modifications and committed reference changes. The central construction uses source-complete separator interfaces that distinguish joint multi-boundary payoff contributions from one-boundary maximum-support envelopes. Together with a balanced source-token hierarchy and bounded quotient composition rules, these interfaces yield an exact dynamic maintenance algorithm with explicit preprocessing, update, witness extraction, and abstract-space bounds. The paper establishes a worst-case successful-operation bound under a specified abstract arithmetic, indexing, and reference-counted allocation model. It also develops epoch-safe optimization and commit operations, distinguishes equilibrium nonexistence from budget infeasibility, and proves weak NP-hardness of one-edit equilibrium repair on a restricted noncactus family of treewidth-two book graphs. Version 1.7 incorporates corrections arising from adversarial mathematical and source-level audits, including surviving-boundary aggregation, zero-budget witness accounting, deterministic preprocessing, rational bit-space accounting, and history-independent retained-state ownership. A separate reproducibility archive provides executable research implementations, deterministic hierarchy construction, exact-payoff verification, intermediate separator-state comparisons, transactional regression tests, and allocator and memory-reachability audits. The computational evidence supports the mathematical construction but does not constitute universal certification of worst-case physical runtime or garbage-collector behavior. The algorithmic framework also applies to corresponding generic additive-threshold constraint systems; no Nash-specific asymptotic advantage is claimed. Keywords: Binary polymatrix games; pure Nash equilibrium; cactus graphs; dynamic algorithms; equilibrium repair; graph separators; minimum Hamming distance; parameterized algorithms; computational complexity; dynamic programming.
Author: Kelly Rhys
Language: English
Topic: Game Theory and Applications
Field: Decision Sciences
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7225589457