Bounds on the Inefficiency of Sequential Auctions

We study the sequential second price auction of multiple units of a homogeneous commodity. It is well known that such auctions can have inefficient equilibria. For the case of two bidders, we show that the value of the allocation obtained in a the unique subgame perfect equilibrium is at least $1-e^{-1}$ of the value of the efficient allocation. Furthermore, we show that this bound is asymptotically tight.