Sequential Auctions of Identical Items with Budget-Constrained Bidders

In this paper, we study sequential auctions with two budget constrained bidders and any number of identical items. All prior results on such auctions consider only two items. We construct a canonical outcome of the auction that is the only natural equilibrium and is unique under a refinement of subgame perfect equilibria. We show certain interesting properties of this equilibrium; for instance, we show that the prices decrease as the auction progresses. This phenomenon has been observed in many experiments and previous theoretic work attributed it to features such as uncertainty in the supply or risk averse bidders. We show that such features are not needed for this phenomenon and that it arises purely from the most essential features: budget constraints and the sequential nature of the auction. A little surprisingly we also show that in this equilibrium one agent wins all his items in the beginning and then the other agent wins the rest. The major difficulty in analyzing such sequential auctions has been in understanding how the selling prices of the first few rounds affect the utilities of the agents in the later rounds. We tackle this difficulty by identifying certain key properties of the auction and the proof is via a joint induction on all of them.