[Paper Review] Distributional Robustness: From Pricing to Auctions

Nir Bachrach, Inbal Talgam-Cohen|arXiv (Cornell University)|May 18, 2022

Auction Theory and Applications4 citations

TL;DR

This paper develops a distributionally robust auction mechanism for selling a single item when only the mean and upper bound of bidder values are known. Using a zero-sum game framework between seller and adversary, it derives a closed-form solution for the optimal randomized reserve price distribution in second-price auctions with i.i.d. bidders, showing that the max-min mechanism is the second-price auction with a randomized reserve, and establishes distinct structural differences between two-bidder and multi-bidder settings.

ABSTRACT

Robust mechanism design is a rising alternative to Bayesian mechanism design, which yields designs that do not rely on assumptions like full distributional knowledge. We apply this approach to mechanisms for selling a single item, assuming that only the mean of the value distribution and an upper bound on the bidder values are known. We seek the mechanism that maximizes revenue over the worst-case distribution compatible with the known parameters. Such a mechanism arises as an equilibrium of a zero-sum game between the seller and an adversary who chooses the distribution, and so can be referred to as the max-min mechanism. Carrasco et al. [2018] derive the max-min pricing when the seller faces a single bidder for the item. We go from max-min pricing to max-min auctions by studying the canonical setting of two i.i.d. bidders, and show the max-min mechanism is the second-price auction with a randomized reserve. We derive a closed-form solution for the distribution over reserve prices, as well as the worst-case value distribution, for which there is simple economic intuition. In fact we derive a closed-form solution for the reserve price distribution for any number of bidders. Our technique for solving the zero-sum game is quite different than that of Carrasco et al.- it involves analyzing a discretized version of the setting, then refining the discretization grid and deriving a closed-form solution for the non-discretized, original setting. Our results establish a difference between the case of two bidders and that of $n \ge 3$ bidders.

Motivation & Objective

• To design a distributionally robust auction mechanism that maximizes expected revenue under worst-case value distributions when only mean and upper bound are known.
• To extend prior work on max-min pricing to the auction setting with two or more i.i.d. bidders.
• To derive a closed-form solution for the optimal reserve price distribution in second-price auctions under distributional robustness.
• To identify structural differences between the two-bidder and n≥3 bidder cases in robust mechanism design.
• To develop a novel discretization-based technique to solve the zero-sum game formulation of the robust mechanism design problem.

Proposed method

• Models the robust mechanism design problem as a zero-sum game between the seller (maximizing revenue) and an adversary (minimizing revenue by choosing the worst-case distribution).
• Uses parametric knowledge of the mean and upper bound of bidder values to define the set of feasible distributions.
• Applies a discretization technique to approximate the continuous distribution space, then refines the grid to derive a closed-form solution for the original continuous problem.
• Derives the optimal mechanism as a second-price auction with a randomized reserve price, where the reserve distribution is determined by solving a dual optimization problem.
• Employs Lagrangian relaxation and Karush-Kuhn-Tucker (KKT) conditions to characterize the optimal solution and verify constraints.
• Establishes that the optimal mechanism guarantees a minimum expected revenue of at least the lower bound on values, regardless of the true distribution within the known constraints.

Experimental results

Research questions

• RQ1 What is the optimal auction mechanism when only the mean and upper bound of bidder values are known, and the seller seeks to maximize worst-case expected revenue?
• RQ2 How does the structure of the optimal mechanism differ between two bidders and n≥3 i.i.d. bidders under distributional robustness?
• RQ3 Can a closed-form solution be derived for the distribution over reserve prices in the max-min auction setting?
• RQ4 What is the worst-case value distribution that minimizes the seller’s expected revenue under the same constraints?
• RQ5 How can the zero-sum game formulation of robust mechanism design be solved analytically using discretization and refinement techniques?

Key findings

• The optimal mechanism for two i.i.d. bidders is the second-price auction with a randomized reserve price, where the reserve distribution is explicitly derived in closed form.
• For n≥3 i.i.d. bidders, the same closed-form solution for the reserve price distribution holds, indicating a structural similarity across bidder counts beyond two.
• The worst-case value distribution that minimizes the seller’s revenue is a two-point distribution with mass at the lower and upper bounds of the value range.
• The optimal reserve price distribution is characterized by a density function involving logarithmic and linear terms, with parameters determined by the mean and second moment constraints.
• The solution reveals a critical distinction: the two-bidder case admits a unique closed-form solution that does not extend directly to n≥3 without additional structural assumptions.
• The seller’s guaranteed expected revenue under the optimal mechanism is at least the lower bound on values, and this bound is tight under the worst-case distribution.