Efficient Dynamic Allocation with Uncertain Valuations

In this paper we consider the problem of efficiently allocating a given resource or object repeatedly over time. The agents, who may temporarily receive access to the resource, learn more about its value through its use. When the agents' beliefs about their valuations at any given time are public information, this problem reduces to the classic multi-armed bandit problem, the solution to which is obtained by determining a Gittins index for every agent. In the setting we study, agents observe their valuations privately, and the efficient dynamic resource allocation problem under asymmetric information becomes a problem of truthfully eliciting every agent's Gittins index. We introduce two bounding mechanisms, under which agents announce types corresponding to Gittins indices either at least as high or at most as high as their true Gittins indices. Using an announcement-contingent affine combination of the bounding mechanisms it is possible to implement the efficient dynamic allocation policy. We provide necessary and sufficient conditions for global Bayesian incentive compatibility, guaranteeing a truthful efficient allocation of the resource. Using essentially the same method it is possible to approximately implement truthful mechanisms corresponding to a large variety of surplus distribution objectives the principal might have, for instance a dynamic second-price Gittins index auction, which maximizes the principal's revenue subject to implementing an efficient allocation policy