Linear Logical Mechanism Design: Proof-Theoretic Foundations of Auction Strategy-Proofness and VCG Incentive Compatibility

Description (Abstract / Summary) Title: Linear Logical Mechanism Design: Proof-Theoretic Foundations of Auction Strategy-Proofness and VCG Incentive Compatibility Abstract: This paper extends the framework of "Linear Logical Economics" to mechanism design theory, providing a proof-theoretic foundation for auction strategy-proofness and the incentive compatibility of the Vickrey-Clarke-Groves (VCG) mechanism. Traditional mechanism design verifies incentive compatibility through utility inequalities; in contrast, this study reformulates it as a structural property of proof figures within Jean-Yves Girard's Linear Logic. We model an auction mechanism as an axiomatic system where bidders' private valuations are formalized as linear logic contexts and bidding strategies as proof search processes. Within this framework, we derive three key results: 1. Strategy-Proofness as the Normal Form Theorem: We prove that "truthful reporting" corresponds to a cut-free proof figure (normal form). Any misreporting introduces a "Cut" (logical detour) into the proof. Under the VCG payment rule, the cut-elimination process of this detour necessarily consumes more resources (utility) than the direct path, thereby establishing strategy-proofness as a topological consequence of the Strong Normalization Theorem. 2. Logical Revelation Principle: We demonstrate that the Revelation Principle is isomorphic to the Cut-Elimination Theorem. The fact that any indirect mechanism can be replaced by a direct mechanism corresponds to the logical fact that any proof with cuts can be transformed into a cut-free normal form. 3. Gibbard-Satterthwaite Impossibility via Resource Conservation: We reinterpret the impossibility of non-dictatorial strategy-proof mechanisms in voting (without money) as the failure of cut-elimination in a system lacking a resource-absorbing "payment" formula. This approach offers a novel perspective where economic incentives are not external constraints but internal syntactic requirements of the logical system itself.