Simple Mechanisms for Profit Maximization in Multi-item Auctions

We study a classical Bayesian mechanism design problem where a seller is selling multiple items to a buyer. 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 1/11 the optimal profit, when the buyer has a constraint-additive valuation over independent items. The approximation factor becomes 6 when the buyer is additive. Our result holds even when the seller's costs are correlated across items. We introduce a new class of mechanisms called permit-selling mechanisms. These mechanisms have two stages. For each item i, we create a separate permit that allows the buyer to purchase the item at its cost. 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 i by only paying the cost $c_i$ if the buyer has purchased the permit for item i in the first stage. We show that the best permit-selling mechanism or the best posted price mechanism is already a constant factor approximation to the optimal profit (6 for additive, and 11 for constrained additive). Indeed, we do not require the optimal permit-selling mechanism, only selling the permits separately or as a grand bundle suffices to achieve the above approximation ratio. Our proof is enabled by constructing a benchmark for the optimal profit via a novel dual solution and a new connection to revenue maximization in multi-item auctions with a subadditive bidder.