Optimal Mechanism Design via Latent Compression

We apply the Latent framework to multi-item, multi-bidder mechanism design. The allocation rule \(x(v_1, \ldots, v_N) : V^N \to [0,1]^{NM}\) for \(N\) bidders and \(M\) items is a function on the \(NM\)-dimensional type space. We prove that if bidder value distributions are \(\rho\)-analytic, the optimal allocation rule has a Latent representation with \(R^ = O(\log(1/\varepsilon)/\log\rho)\) effective interaction grades, independent of \(N\) and \(M\). We establish a Revenue Approximation Theorem: the revenue of the grade-\(R^\) optimal mechanism approximates the optimal revenue within \(\varepsilon\), with the grade-2 mechanism (pairwise bidder competition only) achieving the majority of the revenue for smooth distributions. We characterize the ρ-landscape of auction formats: posted prices (\(\rho = \infty\)), second-price auctions (\(\rho\) large for independent values), discriminatory auctions (\(\rho\) moderate), and combinatorial auctions with strong complementarities (\(\rho \to 1\)). We introduce the mechanism compressibility design problem: choose a mechanism that jointly maximizes revenue and \(\rho\), producing auctions that are simultaneously optimal and computationally tractable. We apply the framework to spectrum auctions (\(M = 100\)+ licenses, \(N = 10\)+ telecoms), ad auctions (\(M = 10^6\)+ slots per day), and Treasury auctions (\(M = 1\) item, \(N = 50\)+ primary dealers, but complex bidder interaction through the repo market).