A Sheaf Framework for Strategic Multi-Agent Systems: From Consensus to Nash Equilibria
For researchers in multi-agent systems, this provides a rigorous mathematical foundation for combining geometric, logical, and strategic reasoning, but the contribution is primarily theoretical with no empirical validation.
The paper introduces a unified categorical framework integrating event calculus, ensemble formation, and game-theoretic rewards into a Grothendieck topos, proving that Nash equilibria correspond to global sections of a best-response sheaf and demonstrating expressiveness via an immunological case study.
The coordination of heterogeneous autonomous agents in dynamic, adversarial environments requires simultaneous satisfaction of geometric constraints, logical consistency, temporal reasoning, and strategic optimization. Existing sheaf- and topos-theoretic frameworks provide powerful tools for geometric consensus, knowledge alignment, and causal planning, but lack explicit models for value, reward, and strategic choice. This report presents a unified categorical framework that integrates event calculus, SCEL-like ensemble formation, and game-theoretic reward structures into a single Grothendieck topos of time-space histories. We introduce the notion of a \emph{game sheaf} whose stalks contain utility functions and policy distributions, and restriction maps encode both parallel transport and best-response dynamics. We prove that Nash equilibria correspond to global sections of a derived best-response correspondence sheaf, while cohomological obstructions classify failures of strategic consistency. A detailed case study of an immunological ``bastion defense'' scenario -- heterogeneous agents forming attack/defense ensembles under resource constraints -- demonstrates the framework's expressiveness. This synthesis provides a rigorous foundation for verifiable, autonomic, and economically rational multi-agent systems.