Optimal Mechanisms for Value Maximizers with Budget Constraints via Target Clipping

Abstract

We study the design of revenue-maximizing mechanisms for value-maximizing agents with budget constraints. Agents have return-on-spend constraints requiring a minimum amount of value per unit of payment made and budget constraints limiting their total payments. The agents' only private information are the minimum admissible ratios on the return-on-spend constraint, referred to as the target ratios. Our work is motivated by internet advertising platforms, where advertisers are increasingly adopting automated bidders to purchase advertising opportunities on their behalf. Instead of specifying bids for each keyword, advertisers set high-level goals, such as maximizing clicks, and targets on cost-per-clicks or return-on-spend. The platform then automatically purchases opportunities by bidding in different auctions. We present a model that abstracts away the complexities of the auto-bidding procurement process that is general enough to accommodate many allocation mechanisms such as auctions, matchings, etc. We reduce the mechanism design problem when agents have private target ratios to a challenging non-linear optimization problem with monotonicity constraints. We provide a novel decomposition approach to tackle this problem that yields insights into the structure of optimal mechanisms and show that surprising features stem from the interaction between budget and return-on-spend constraints. Our optimal mechanism, which we dub the target-clipping mechanism, has an appealing structure: it sets a threshold on the target ratio of each agent, targets above the threshold are allocated efficiently, and targets below are clipped to the threshold.