Strategic Scaling: A Game-Theoretic Framework for the Principle of Scale in Complex Systems

This paper introduces a novel game-theoretic framework for understanding how strategic behavior evolves under transformations of scale. Traditional game theory assumes fixed structures of interaction, but real-world systems—particularly those involving AI governance, institutional design, and collective action—change dynamically in size, complexity, and payoff logic. These changes often invalidate previously optimal strategies and lead to systemic breakdowns. We formalize this insight through the concept of a Scale Operator S_k, which transforms the components of a game as it scales, and we introduce the Double Bind Index (DBI) to quantify cross-scale strategic contradictions. We present a taxonomy of scale-sensitive games—including linear-scaling, threshold-tipping, scale-reversing, recursive, and Batesonian double bind games—and use case studies in AI policy to demonstrate how equilibrium behaviors collapse, invert, or generate emergent risks under structural expansion. This framework expands traditional game theory into a new domain: one in which scale is not a neutral multiplier, but a force that reconfigures the very logic of rational interaction.