An Optimal Bidimensional Multi-Armed Bandit Auction for Multi-unit Procurement

We study the problem of a buyer (aka auctioneer) who gains stochastic rewards by procuring multiple units of a service or item from a pool of heterogeneous strategic agents. The reward obtained for a single unit from an allocated agent depends on the inherent quality of the agent; the agent's quality is fixed but unknown. Each agent can only supply a limited number of units (capacity of the agent). The costs incurred per unit and capacities are private information of the agents. The auctioneer is required to elicit costs as well as capacities (making the mechanism design bidimensional) and further, learn the qualities of the agents as well, with a view to maximize her utility. Motivated by this, we design a bidimensional multi-armed bandit procurement auction that seeks to maximize the expected utility of the auctioneer subject to incentive compatibility and individual rationality while simultaneously learning the unknown qualities of the agents. We first assume that the qualities are known and propose an optimal, truthful mechanism 2D-OPT for the auctioneer to elicit costs and capacities. Next, in order to learn the qualities of the agents in addition, we provide sufficient conditions for a learning algorithm to be Bayesian incentive compatible and individually rational. We finally design a novel learning mechanism, 2D-UCB that is stochastic Bayesian incentive compatible and individually rational.