On bidding for a fixed number of items in a sequence of auctions

We consider the problem of a firm (“the buyer”) that must acquire a fixed number (L) of items. The buyer can acquire these items either at a fixed buy-it-now price in the open market or by participating in a sequence of N>L auctions. The objective of the buyer is to minimize his expected total cost for acquiring all L items. We model this problem as a Markov Decision Process and establish monotonicity properties for the optimal value function and the optimal bidding strategies.