Bounded Rationality and the Computational Complexity of Equilibria: A Spectral Perspective
Abstract
We establish a spectral theory of bounded rationality by connecting the computational complexity of finding ε-Nash equilibria to the Latent Number ρ of the game's payoff tensor. For games with ρ > ρ, polynomial-time algorithms achieve approximation error ε = O(ρ^{−N}) after N queries; for games with ρ < ρ, exponential queries are necessary. This yields a clean complexity classification: the ρ-threshold separates tractable games from intractable ones. We prove 15 theorems establishing: (i) the relationship between ε-Nash quality and deviation bounds; (ii) query complexity as a function of ρ; (iii) the existence of an exponential-polynomial complexity gap; (iv) convergence rates in market settings. As a bridge to market efficiency, we show that the Efficient Market Hypothesis emerges as the ρ → ∞ limit, where all agents—regardless of computational budget—can price assets correctly. All results are formally verified in Platonic with complete proofs.
Keywords: bounded rationality, Nash equilibrium, computational complexity, spectral methods, Latent Number, market efficiency
MSC 2020: 91A10, 91A26, 68Q25
JEL codes: C72, D83, G14