Online Combinatorial Allocation with Interdependent Values

TL;DR

This paper extends online secretary algorithms to combinatorial settings with interdependent valuations, achieving competitive ratios for various valuation classes and designing truthful mechanisms for strategic environments.

Contribution

It introduces $2e$-competitive algorithms for combinatorial secretary problems with interdependent values, matching single-choice guarantees and extending to strategic settings with truthful mechanisms.

Findings

Achieved $2e$-competitiveness for submodular and XOS valuations.

Extended results to strategic settings with a $4e$-competitive truthful mechanism.

Matched approximation guarantees of non-interdependent secretary problems under interdependence.

Abstract

We study online combinatorial allocation problems in the secretary setting, under interdependent values. In the interdependent model, introduced by Milgrom and Weber (1982), each agent possesses a private signal that captures her information about an item for sale, and the value of every agent depends on the signals held by all agents. Mauras, Mohan, and Reiffenh\"auser (2024) were the first to study interdependent values in online settings, providing constant-approximation guarantees for secretary settings, where agents arrive online along with their signals and values, and the goal is to select the agent with the highest value. In this work, we extend this framework to {\em combinatorial} secretary problems, where agents have interdependent valuations over {\em bundles} of items, introducing additional challenges due to both combinatorial structure and interdependence. We provide $2e$-competitive algorithms for a broad class of valuation functions, including submodular and XOS functions, matching the approximation guarantees in the single-choice secretary setting. Furthermore, our results cover the same range of valuation classes for which constant-factor algorithms exist in classical (non-interdependent) secretary settings, while incurring only an additional factor of $2$ due to interdependence. Finally, we extend our study to strategic settings, and provide a $4e$-competitive truthful mechanism for online bipartite matching with interdependent valuations, again meeting the frontier of what is known, even without interdependence.