Simple Mechanisms for Profit Maximization in Multi-item Auctions

Abstract

We study a classical Bayesian mechanism design problem where a seller is selling multiple items to multiple buyers. We consider the case where the seller has costs to produce the items, and these costs are private information to the seller. How can the seller design a mechanism to maximize her profit? Two well-studied problems, revenue maximization in multi-item auctions and signaling in ad auctions, are special cases of our problem. We show that there exists a simple mechanism whose profit is at least 1 44 of the optimal profit for multiple buyers with matroid-rank valuation functions. When there is a single buyer, the approximation factor is 11 for general constraint-additive valuations and 6 for additive valuations. Our result holds even when the seller's costs are correlated across items.We introduce a new class of mechanisms called permit-selling mechanisms. For single buyer case these mechanisms are quite simple: there are two stages. For each item j, we create a separate permit that allows the buyer to purchase the item in the second stage. In the first stage, we sell the permits without revealing any information about the costs. In the second stage, the seller reveals all the costs, and the buyer can buy item j by paying the item price if the buyer has purchased the permit for item j in the first stage. We show that either selling the permits separately or as a grand bundle suffices to achieve the constant factor approximation to the optimal profit (6 for additive, and 11 for constrained additive). For multiple buyers, we sell the permits sequentially and obtain the constant factor approximation. Our proof is enabled by constructing a benchmark for the optimal profit by combining a novel dual solution with the existing ex-ante relaxation technique.

On Multi-Dimensional Gains from Trade Maximization

“…It is well known that in multi-item auctions, the revenue of selling the items separately is an O(log n)-approximation to the optimal revenue when there is a single additive buyer [25]. Cai and Zhao [11] provide an extension of this O(log n)-approximation to profit maximization. We build on this in Section 4.4 to upper bound the OPT-S(L, F L ) term, where with |L| items, we get a log(|L|) factor (Lemma 4.10).…”

“…In this section, we provide an upper bound for the optimal super seller profit from items in L. It is well known that in multi-item auctions the revenue of selling the items separately is a O(log n)-approximation to the optimal revenue when there is a single additive buyer [25]. Cai and Zhao [11] provide a extension of this O(log n)approximation to profit maximization. Combining this approximation with some basic observations based on the Cai-Devanur-Weinberg duality framework [8], we derive the following upper bound of OPT-S(L, F L ).…”

“…Myerson's ironed virtual value function 14 for the buyer's distribution for item i, D B i . To bound OPT-S(L, F L ), we need the following result from [11]. It provides a benchmark of the optimal profit using the Cai-Devanur-Weinberg duality framework [8]: The profit of any BIC, IR mechanism is upper bounded by the buyer's virtual welfare with respect to some virtual value function, minus the sellers' total cost for the same allocation.…”

“…Cai and Zhao [11] also give a logarithmic upper bound of the optimal profit for a single additive buyer, using the sum of optimal profit for each individual item.…”

Non-Adaptive Matroid Prophet Inequalities

“…In a specific case of their analysis that analyzes the core of the core (a double core-tail analysis follow the original of [Li and Yao, 2013]), they use [Feldman et al, 2015]. Work by Cai and Zhao [2019] approximates the optimal profit-seller revenue minus cost-for constrained-additive buyers. Like [Chawla and Miller, 2016], they also construct their benchmark using the ex-ante relaxation and use OCRS to bound a term here as well.…”