Formalizing Cooperation and Competition: An AI-Assisted Mathematical Addendum to a 2018 Planning Thesis

Haitham Masaud Amar

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arXiv · 2610.29637

Formalizing Cooperation and Competition: An AI-Assisted Mathematical Addendum to a 2018 Planning Thesis

Haitham Masaud Amar

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Abstract

We present an AI-assisted mathematical addendum to Strategic Trip Planning: Striking a Balance Between Competition and Cooperation (Amar, 2018). We reconstruct selected relationships between strategic choice, cooperative stability, participant approval and enforcement. The thesis's Nash-to-core proposition is under-specified until outsider opportunities are stated. We verify two-person implications under fixed outsiders and explicit efficiency conditions. Under isolation, the original example has an empty TU core; its funded counterpart uses an explicit external subsidy. For a monotone resource model we characterize the exact financing gap. A finite negotiation procedure implemented in Lean is proved to terminate with participant approval, joint-cost non-worsening and local optimality against permitted approved moves. Supporting formalizations address bargaining geometry and regret-based efficiency. A resource-game counterinstance shows why monotonicity alone provides no uniform price-of-anarchy bound. Reproducible computations compare a documented heuristic reconstruction with the verified procedure on 44 source-based instances and 96 starting cases from four additional synthetic network families. Acceptance-matched heuristic and randomized pairwise-search controls contextualize the optimization results. The mathematical ingredients are established results; the contribution is a traceable reconstruction linking source statements, explicit assumptions, checked proofs, executable code and retained counterinstances. The study illustrates how AI-assisted investigation can turn engineering intuitions from legacy research into precise mathematical claims whose conditions and computational consequences can be inspected and reproduced.

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Haitham Masaud Amar. 2026-10-21. Formalizing Cooperation and Competition: An AI-Assisted Mathematical Addendum to a 2018 Planning Thesis. https://arxiv.org/abs/2610.29637

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