Compendium of Advances in Game Theory: Classical, Differential, Algorithmic, Non-Archimedean and Quantum Game
Bourama Toni
TL;DR
This compendium surveys advances across classical, differential, algorithmic, intelligent, quantum, non-Archimedean, and p-adic game theory, linking foundational equilibrium concepts to modern computational and informational extensions. It highlights how Nash equilibrium, mean-field, and evolutionary dynamics provide scalable analyses for large populations, while Algorithmic Game Theory and Security Games address computational challenges and real-world defense applications. The work introduces quantum and p-adic generalizations of games, including quantum strategies with entanglement and p-adic probability structures, and demonstrates potential improvements in coordination and efficiency through these non-classical frameworks. By integrating socio-cultural modeling with advanced mathematical tools, the compendium aims to guide researchers toward robust, multi-scale analyses of strategic interactions in an era of artificial intelligence and post-human mathematical creativity.
Abstract
This compendium features advances in Game Theory, to include: Classical Game Theory: Cooperative and non-cooperative. Zero-sum and non-zero sum games. Potential and Congestion games. Mean Field games. Nash Equilibrium, Correlated Nash Equilibrium and Approximate Nash Equilibrium. Evolutionary Game Theory. Intelligent Game: Differential Game Theory. Algorithm Game Theory and Security Games. Quantum Games and Quantumization of classical games such as the Battle of the Sexes. Non-Archimedean and p-adic game theory and its growing relevancy as the domains of game-theoretic application expand. p-adic quantum game to leverage and combine the distinguishing features of non-Archimedean analysis and quantum information theory. This is a novel game-theoretic approach with great potential of application. In times of exponential growth of artificial intelligence and machine learning and the dawn of post-human mathematical creativity, this compendium is meant to be a reference of choice for all game theory researchers.