Auctions: Efficiency and Existence of Equilibrium

拍卖：效率和均衡的存在

• 批准号: 9905599
• 负责人: Philip Reny
• 金额: $ 12.65万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-09-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9905599&HistoricalAwards=false
• 关键词: Auctions; Efficiency; Existence; Equilibrium

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.

这个提案由三个项目组成。第一个（与希伯来大学的Motty Perry教授合作）旨在将Vickrey的多单位拍卖扩展到投标人不对称、价值相互依赖和商品异质的环境中，同时保持交易后的高效结果。维克里著名的拍卖能够有效地将许多同质商品分配给任意数量的（可能是不对称的）竞标者，只要他们每个人都有私人估值和向下倾斜的需求。我们的重点将放在更普遍的相互依存的价值观上。不幸的是，当价值相互依赖时，维克瑞的拍卖不能保证产生有效的结果。除了实现事后效率外，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，我们已经解决了待售商品均质且需求向下倾斜的特殊情况。在这种解决方案中，平衡是简单的，因为它对竞标者的要求并不比他们在单一单位的Vickrey拍卖中与其他单一竞标者的竞争更复杂。此外，与标准拍卖一样，我们的拍卖规则与环境细节无关。我们的目的是继续沿着这条研究路线，扩大这一结果，以便允许可能的异质和互补的商品。第二个项目（也是与Perry教授合作）将以第一个项目为基础。我们希望采用在上述第一个项目中开发的密封投标拍卖作为设计有效的上升多单元拍卖的指导。开发一种有效的升序拍卖非常重要，因为升序拍卖通常更容易操作，因此在实践中比密封拍卖更有可能表现出色。其中一个原因是，以同质商品为例，我们高效的密封竞价拍卖要求竞标者提交竞价函数，而我们寻求的上升竞价最多只要求竞标者跟踪有限数量的退出价格，并在价格时钟到达时退出各种单位。我们已经解决了单个单元的情况，这确实是相关的比较。在我们的单单位上升拍卖中，竞标者只需要跟踪他当前的退出价格，并在价格时钟达到该价格时退出。我们的单股上行拍卖和标准上行拍卖的区别在于，我们的竞标者在以较低价格“退出”后，可以以更高的价格活跃起来。我们希望我们可以将这次拍卖扩展到多单位的情况下。正如我们已经在单单元案例中所做的那样，我们期望我们的同步拍卖设计和分析将在多单元上升拍卖的设计以及其均衡描述中发挥指导作用。第三个项目（与希伯来大学的Shmuel Zamir合作）主要研究首价拍卖中非递减竞价函数中纯策略均衡的存在性。虽然人们对首价拍卖进行了广泛的研究，并得出了大量的存在结果，但在可能是最自然的环境中却没有这样的结果：N个可能不对称的、具有关联信号和相互依赖价值的竞标者。我们的印象是，这种环境之所以出现困难，只是因为标准的单交叉特性可能在其中失效。也就是说，即使其他竞标者采用非降标函数，也很可能出现这样的情况：当一个人的信号较低时，高标是比低标更好的答复，而当一个人的信号较高时，则是更差的答复。在这种情况下，大多数存在性证明都失效了。但似乎较弱的单交性足以推动证明通过，而且似乎在我们感兴趣的环境中，这种较弱的单交性也成立。虽然我们希望这将产生一个非常宽容的存在结果，但我们也对一个人能走多远感兴趣。从这些方面来看，竞标者信号的维度似乎是至关重要的。当信号是一维的时候，我们较弱的单交叉特性是成立的，但当信号是多维的时候，它很容易失效。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，非递减出价函数的纯策略均衡可能不存在。事实上，我们已经有了一个粗略的例子，我们希望改进它。如果幸运的话，我们新的一维信号的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首价拍卖理解。