Revenue Approximation in Mechanism Design via Grade Truncation

We establish formally verified foundations for revenue approximation in mechanism design through grade truncation. The core framework shows that mechanism revenue decomposes additively across grade levels, with each successive grade contributing geometrically decaying revenue increments bounded by C/ρ^k for constants C > 0 and ρ > 1. This yields explicit approximation guarantees: a grade-k mechanism captures optimal revenue up to an O(ρ^{-k}) error. We prove that competition effects dominate pure optimization in multi-agent settings, quantified through gap monotonicity in squared differences. The framework is formalized in the Platonic proof system with 27 verified theorems and zero domain-specific axioms, relying only on bootstrap arithmetic over ℝ.

Keywords: mechanism design, revenue approximation, auction theory, grade truncation, formal verification