Mode Connectivity in Auction Design
Christoph Hertrich, Yixin Tao, László A. Végh
TL;DR
This work provides a theoretical foundation for the success of differentiable economics in auction design by establishing mode connectivity for RochetNet and Affine Maximizer Auctions. It proves that, under conditions like $\varepsilon$-reducibility or sufficiently large menu sizes, locally optimal auction menus can be connected via simple, piecewise-linear paths with near-optimal revenue along the path. The results hinge on discretization techniques and introduce concrete piecewise-linear constructions (three- to five-piece paths) that preserve revenue up to $\varepsilon$. This work thus explains why gradient-based training often finds high-quality auctions and offers practical guidance on menu design and network size; it also sets the stage for extensions to more complex architectures and weighted settings.
Abstract
Optimal auction design is a fundamental problem in algorithmic game theory. This problem is notoriously difficult already in very simple settings. Recent work in differentiable economics showed that neural networks can efficiently learn known optimal auction mechanisms and discover interesting new ones. In an attempt to theoretically justify their empirical success, we focus on one of the first such networks, RochetNet, and a generalized version for affine maximizer auctions. We prove that they satisfy mode connectivity, i.e., locally optimal solutions are connected by a simple, piecewise linear path such that every solution on the path is almost as good as one of the two local optima. Mode connectivity has been recently investigated as an intriguing empirical and theoretically justifiable property of neural networks used for prediction problems. Our results give the first such analysis in the context of differentiable economics, where neural networks are used directly for solving non-convex optimization problems.