Auctions: Efficiency and Existence of Equilibrium

这一提案由三个项目组成。第一个（与教授莫蒂佩里，希伯来大学联合）的目的是扩展维克瑞的多单位拍卖的环境与不对称的投标人，相互依存的价值观，和异质商品，同时保持一个事后有效的结果。维克瑞著名的拍卖能够在任意数量的、可能不对称的投标人之间有效地分配许多同质商品，只要他们每个人都有私人估价和向下倾斜的需求。我们的重点将放在相互依存的价值观的更一般的情况下。不幸的是，当价值相互依赖时，Vickrey的拍卖并不能保证产生有效的结果。除了实现事后效率，本研究的一个重要方面是拍卖的简单性。例如，在最近的一篇论文中，我们已经解决了供销售的商品是同质的并且需求是向下倾斜的特殊情况。在这个解决方案中，均衡是简单的，因为它不需要投标人比他们在单一单位维克瑞拍卖中与其他单一投标人竞争更复杂。此外，与标准拍卖一样，我们的拍卖规则与环境的细节无关。我们的目的是继续沿着这条研究路线，扩展这一结果，以便考虑到可能的异质性和互补性商品。第二个项目（也是与佩里教授联合进行的）将建立在第一个项目的基础上。我们希望采用上述第一个项目中开发的密封投标拍卖作为设计有效的多单位递增拍卖的指导。开发一个有效的递增拍卖是很重要的，因为递增拍卖通常更容易玩，所以更有可能在实践中表现得比密封投标同行。其中一个原因是，在同质商品的情况下，我们的高效密封投标拍卖要求投标人提交投标函数，而我们寻求的递增拍卖最多只要求投标人跟踪多个退出价格，并在价格时钟到达时退出各种单位。 我们已经解决了单单位的情况，这确实是有关的比较。在我们的单单位递增拍卖中，投标人只需要跟踪他当前的退出价格，并在价格时钟到达时退出。我们的单单位递增拍卖和标准递增拍卖之间的区别在于，在我们的拍卖中，投标人可以在以较低价格“退出”后以较高价格活跃。我们希望我们可以将这种拍卖扩展到多单元的情况。正如我们已经做了单单位的情况下，我们希望我们的同步拍卖的设计和分析将发挥指导作用，在设计的多单位上升拍卖，以及在描述其equilibrium.The第三个项目（与Shmuel Zamir，希伯来大学联合）集中在非减投标函数在第一价格拍卖纯策略均衡的存在。虽然第一价格拍卖已被广泛研究和许多存在的结果，有没有这样的结果，在什么也许是最自然的环境：N可能不对称的投标人与附属的信号和相互依赖的价值。我们的印象是，这种环境之所以出现困难，仅仅是因为标准的单交叉属性在其中可能会失效，也就是说，即使其他投标人采用非递减投标函数，也很可能出现这样的情况：当一个人的信号较低时，高投标比低投标更好，但当一个人的信号较高时，高投标比低投标更差。大多数存在性证明在这种情况下就失效了。但似乎一个较弱的单交叉性质足以推动证明通过，而且似乎这个较弱的单交叉性质在感兴趣的环境中也成立。虽然我们希望这将产生一个非常宽容的存在结果，但我们也对人们能走多远感兴趣。沿着这些线，似乎投标人的信号的维度是至关重要的。虽然当信号是一维时，我们较弱的单交叉属性成立，但当信号是多维时，它很容易失败。事实上，在我们看来，在非常标准的假设下，当信号是多维的时，非减出价函数中的纯策略均衡可能不存在。事实上，我们已经有了一个粗糙的例子，我们希望改进。幸运的话，我们新的一维信号存在性结果以及多维信号的反例将提供比我们目前所享受的更完整的对一价拍卖的理解。

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.