Optimal Auctions with Convex Perceived Payments

TL;DR

This paper extends optimal auction theory to settings with convex perceived payments, deriving bounds and heuristics for revenue maximization, and demonstrating practical near-optimal solutions through experiments.

Contribution

It generalizes Myerson's auction model to convex perceived payments, providing bounds and heuristics for both Bayesian and robust auction design.

Findings

Derived upper and heuristic lower bounds on revenue

Bound-based heuristics support monotonic allocation rules

Experimental results show near-optimal auction solutions

Abstract

Myerson derived a simple and elegant solution to the single-parameter revenue-maximization problem in his seminal work on optimal auction design assuming the usual model of quasi-linear utilities. In this paper, we consider a slight generalization of this usual model---from linear to convex "perceived" payments. This more general problem does not appear to admit a solution as simple and elegant as Myerson's. While some of Myerson's results extend to our setting, like his payment formula (suitably adjusted), others do not. For example, we observe that the solutions to the Bayesian and the robust (i.e., non-Bayesian) optimal auction design problems in the convex perceived payment setting do not coincide like they do in the case of linear payments. We therefore study the two problems in turn. We derive an upper and a heuristic lower bound on expected revenue in our setting. These bounds are easily computed pointwise, and yield monotonic allocation rules, so can be supported by Myerson payments (suitably adjusted). In this way, our bounds yield heuristics that approximate the optimal robust auction, assuming convex perceived payments. We close with experiments, the final set of which massages the output of one of the closed-form heuristics for the robust problem into an extremely fast, near-optimal heuristic solution to the Bayesian optimal auction design problem.