Interaction Decay in ρ-Analytic Games: Grade Structure and Truncation Bounds

We establish rigorous bounds on the decay of interaction terms in ρ-analytic N-player games. For games whose payoff functions admit a convergent power series representation with analyticity radius ρ > 1, we prove that the grade-r interaction component decays as ρ^{-r}. This exponential decay enables finite-grade approximations with controlled truncation error. We derive explicit bounds for equilibrium perturbation under truncation, prove grade-2 dominance for pairwise interactions, characterize the phase transition at ρ = 1, and establish Shapley value grade bounds. As applications, we show that linear Cournot oligopoly is exactly grade-2, and prove revenue loss bounds for mechanism design under truncation. All 50 theorems are formally verified in the Platonic proof system with zero axioms beyond real arithmetic.

Keywords: game theory, analyticity radius, grade decomposition, truncation error, equilibrium perturbation, Shapley value

MSC 2020: 91A10, 91A06, 26E05