  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.
