A Constant Factor Prophet Inequality for Online Combinatorial Auctions

A Google TechTalk, presented by Andrés Cristi, 2023-06-27 A Google Algorithms Seminar ABSTRACT: In online combinatorial auctions m indivisible items are to be allocated to n agents who arrive online. Agents have random valuations for the different subsets of items and the goal is to allocate the items on the fly so as to maximize the total value of the assignment. A prophet inequality in this setting refers to the existence of an online algorithm guaranteed to obtain, in expectation, a certain fraction of the expected value obtained by an optimal solution in hindsight. The study of prophet inequalities for online combinatorial auctions has been an intensive area of research in recent years, and constant factor prophet inequalities are known when the agents’ valuation functions are submodular or fractionally subadditive. Despite many efforts, for the more general case of subadditive valuations, the best-known prophet inequality has an approximation guarantee of O(log log m). We prove the existence of a constant factor prophet inequality for the subadditive case, resolving a central open problem in the area. Our prophet inequality is achieved by a novel, but elementary, sampling idea which we call the Mirror Lemma. This lemma is essentially concerned with understanding algorithms for which the set of items that are allocated and those that are not, distribute equally. The other main ingredient is a nonstandard application of Kakutani’s fixed point theorem. Bio: Andrés is currently a postdoctoral researcher at the Center for Mathematical Modeling at Universidad de Chile, and in 2024 will join EPFL as an assistant professor. He received his PhD from Universidad de Chile and was advised by José Correa and Paul Dütting. His research is focused on the interplay between optimization and incentives, this is, situations where the outcome depends on the actions of strategic agents. He is particularly interested in allocation problems with a dynamic aspect, where decisions are made on the fly. He also often considers data-driven approaches, in which decisions are directly made using observations rather than standard distributional assumptions. Modern platforms like routing apps, online advertisers, and online marketplaces face these challenges on a daily basis, and his work is centered on understanding the fundamental aspects that drive decision-making in these settings.