Approximately Optimal Mechanism Design for Competing Sellers

TL;DR

The paper addresses revenue design when two identical sellers compete for a single buyer who values a copy of the good according to a known distribution $D$. It shows that in a Stackelberg setting, where one seller commits to a mechanism and the other best-responds, the leader can guarantee at least $\text{Rev}(D)/4$ revenue (for regular or DMR $D$) by using a simple single-lottery mechanism, with the follower’s best response being a fixed-price mechanism. A key technical lemma shows that the follower’s best response to any single lottery is indeed a fixed price, enabling a reduction to a monopolist-like problem through an auxiliary distribution $D_s$ and virtual-value analysis; the result highlights a fundamental trade-off between commitment and randomness in oligopolistic mechanism design. The paper also proves a lower bound of $\text{Rev}(D)/e$ for the leader in some instances and shows that Nash equilibria can yield zero revenue even when a monopolist could profit, underscoring the value of commitment and randomization in this setting. Collectively, these findings illuminate how simple, structured mechanisms can closely approximate monopoly revenue in competitive, sequential sale environments, while identifying natural gaps and directions for extending the model to richer markets.

Abstract

Two sellers compete to sell identical products to a single buyer. Each seller chooses an arbitrary mechanism, possibly involving lotteries, to sell their product. The utility-maximizing buyer can choose to participate in one or both mechanisms, resolving them in either order. Given a common prior over buyer values, how should the sellers design their mechanisms to maximize their respective revenues? We first consider a Stackelberg setting where one seller (Alice) commits to her mechanism and the other seller (Bob) best-responds. We show how to construct a simple and approximately-optimal single-lottery mechanism for Alice that guarantees her a quarter of the optimal monopolist's revenue, for any regular distribution. Along the way we prove a structural result: for any single-lottery mechanism of Alice, there will always be a best response mechanism for Bob consisting of a single take-it-or-leave-it price. We also show that no mechanism (single-lottery or otherwise) can guarantee Alice more than a 1/e fraction of the monopolist revenue. Finally, we show that our approximation result does not extend to Nash equilibrium: there exist instances in which a monopolist could extract full surplus, but neither competing seller obtains positive revenue at any equilibrium choice of mechanisms.

Figures (2)

• Figure 1: Constructing the distribution $D_s$ (red) from the distribution $D$ (black). The density below $p$ is unchanged, the density in the interval $[*]{ p, s }$ is squished into a smaller interval $[*]{ p, a + \lparen*\rparen{ 1 - z } \cdot s }$, and the density above $s$ is scaled down by a factor of $1 - z$ with the remaining probability moved to an atom at $0$.
• Figure 2: Allocation rule of a general mechanism (black) and the corresponding bottom proper mechanism constructed in \ref{['lemma:fixedprice-reduction']} (red). The allocation below $s$ is changed to a step function with the same area under the curve and the same probability of allocation at $s$.

Theorems & Definitions (70)

• Theorem 1.1
• Lemma 1.2: Key Lemma
• Theorem 1.3
• Theorem 1.4
• Definition 2.2
• Proposition 2.3: Revenue equals expected virtual welfare Mye81
• Definition 2.4
• Theorem 3.1: Formal version of \ref{['thm:oneoptionalice-informal']}
• Lemma 3.2: Formal version of \ref{['lemma:fixedprice-informal']}
• proof : Proof of \ref{['thm:oneoptionalice-formal']}