Mechanism Design via the Interim Relaxation
Kshipra Bhawalkar, Marios Mertzanidis, Divyarthi Mohan, Alexandros Psomas
TL;DR
The paper develops a general framework for revenue-maximizing mechanisms under downward-closed constraints with additive agents by combining interim relaxation techniques with two-level online contention resolution schemes (tOCRS). It proves that when provided with interim rules feasible in expectation and a (b,c)-selectable tCRS/tOCRS, one can obtain Bayesian Incentive Compatible and Bayesian Individually Rational mechanisms with approximation factor roughly α/(b c), including sequential implementations. The framework yields strong, concrete results: a sequential mechanism achieving a $ rac{2e}{e-1} ext{(≈3.16)}$-approximation for matroid feasibility, and efficient procurement extensions via an OCRS for Stochastic Knapsack that achieve a $(3+e^{-2})$-approximation to the optimal procurement value. It also provides explicit two-level constructions for Knapsack and Multi-Choice Knapsack, along with efficient, Bernoulli-factory-based implementations, and introduces a new OCRS for Stochastic Knapsack that improves prior guarantees in certain regimes. The work thereby unifies and extends prior Bayesian mechanism design with online rounding, enabling practical, end-to-end mechanisms across single- and multi-parameter settings, including procurement, under broad feasibility constraints.
Abstract
We study revenue maximization for agents with additive preferences, subject to downward-closed constraints on the set of feasible allocations. In seminal work, Alaei~\cite{alaei2014bayesian} introduced a powerful multi-to-single agent reduction based on an ex-ante relaxation of the multi-agent problem. This reduction employs a rounding procedure which is an online contention resolution scheme (OCRS) in disguise, a now widely-used method for rounding fractional solutions in online Bayesian and stochastic optimization problems. In this paper, we leverage our vantage point, 10 years after the work of Alaei, with a rich OCRS toolkit and modern approaches to analyzing multi-agent mechanisms; we introduce a general framework for designing non-sequential and sequential multi-agent, revenue-maximizing mechanisms, capturing a wide variety of problems Alaei's framework could not address. Our framework uses an \emph{interim} relaxation, that is rounded to a feasible mechanism using what we call a two-level OCRS, which allows for some structured dependence between the activation of its input elements. For a wide family of constraints, we can construct such schemes using existing OCRSs as a black box; for other constraints, such as knapsack, we construct such schemes from scratch. We demonstrate numerous applications of our framework, including a sequential mechanism that guarantees a $\frac{2e}{e-1} \approx 3.16$ approximation to the optimal revenue for the case of additive agents subject to matroid feasibility constraints. We also show how our framework can be easily extended to multi-parameter procurement auctions, where we provide an OCRS for Stochastic Knapsack that might be of independent interest.