Auctions: Efficiency and Existence of Equilibrium

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There are three projects making up this proposal. The first (joint with Professor Motty Perry, Hebrew University) is aimed at extending Vickrey's multi-unit auction to environments with asymmetric bidders, interdependent values, and heterogeneous goods, while maintaining an ex-post efficient outcome. Vickrey's celebrated auction is able to efficiently allocate many homogeneous goods among any number of, possibly asymmetric, bidders so long as they each have private valuations and downward-sloping demands. Our focus will be on the more general case of interdependent values. Unfortunately, Vickrey's auction is not guaranteed to yield an efficient outcome when values are interdependent. In addition to achieving ex-post efficiency, an important aspect of the present research is simplicity of the auction. For example, in a recent paper we have already solved the special case in which the goods for sale are homogeneous and demand is downward-sloping. In this solution, the equilibrium is simple in the sense that it requires no more sophistication from the bidders than they would need to play well in a single-unit Vickrey auction against a single other bidder. Moreover, as in standard auctions, the rules of our auction are independent of the details of the environment. Our intent is to continue along this line of research by extending this result so as to allow for possibly heterogeneous and complementary goods.The second project (also joint with Professor Perry) will build upon the first. We hope to employ the sealed-bid auction developed in the first project above as a guide toward the design of an efficient ascending multi-unit auction. Developing an efficient ascending auction is important since ascending auctions are typically much simpler to play and so are more likely to perform well in practice than their sealed-bid counterparts. One reason for this is that in the homogeneous goods case for example, our efficient sealed-bid auction requires bidders to submit bid functions, while the ascending auction we seek would at most require bidders to keep track of finitely many drop-out prices and drop-out on various units when the price clock reaches them. We have already solved the single unit case and this is indeed the relevant comparison. In our single-unit ascending auction a bidder need only keep track of his current drop-out price and drop out when the price clock reaches it. The difference between our single-unit ascending auction and a standard ascending auction is that in ours, bidders are allowed to become active at a higher price after having "dropped out" at a lower price. We are hopeful that we can extend this auction to the multi-unit case. As we have already done with the single-unit case, we expect that our simultaneous auction design and analysis will play a guiding role in both the design of a multi-unit ascending auction as well as in the description of its equilibrium.The third project (joint with Shmuel Zamir, Hebrew University) centers on the existence of pure strategy equilibria in nondecreasing bidding functions in first-price auctions. While first-price auctions have been extensively studied and numerous existence results are available, there is no such result in what is perhaps the most natural environment: N possibly asymmetric bidders with affiliated signals and interdependent values. It is our impression that this environment has presented difficulties only because the standard single-crossing property can fail within it. That is, it is quite possible that even if other bidders employ nondecreasing bid functions it might be the case that a high bid is a better reply than a low bid when one's signal is low, but a worse reply when one's signal is high. Most existence proofs break down when this is the case. But it appears that a weaker single-crossing property suffices to push a proof through and it also appears that this weaker single-crossing property holds in the environment of interest. While we are hopeful that this will produce a very permissive existence result, we are also interested in how much further one can go. Along these lines it appears that the dimensionality of the bidders' signals is critical. While our weaker single-crossing property holds when signals are one-dimensional, it can easily fail when they are multi-dimensional. Indeed, it appears to us that under very standard assumptions a pure strategy equilibrium in nondecreasing bid functions may fail to exist when signals are multi-dimensional. In fact, we already have a crude example of this nature that we hope to refine. With any luck, our new existence result with one-dimensional signals together with a counterexample for multi-dimensional signals will provide an even more complete understanding of first-price auctions than we currently enjoy.