IMPLEMENTATION OF REDUCED FORM AUCTIONS: A GEOMETRIC APPROACH
IMPLEMENTATION OF REDUCED FORM AUCTIONS: A GEOMETRIC APPROACH
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简化形式拍卖的实施：几何方法
DOI：
10.2307/2938181
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发表时间：
1991
期刊：
影响因子：
6.1
通讯作者：
Kim C. Border
中科院分区：
文献类型：
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作者：
Kim C. Border
拍卖是将单个不可分割的对象分配给几个竞争竞标者之一的机制。获胜者是被授予对象的投标人。拍卖规则指定了两个功能。首先是竞标者获胜的概率，这是每个人的出价。第二个是每个投标人向卖方付款，作为所有投标以及他是否获胜的付款。例如，首价拍卖会授予最高出价者的对象，概率为（只要没有领先投标），获胜者就会支付他的出价，而失败者则无需支付任何费用。拍卖中的竞标者差异很大。这些差异是由投标人类型捕获的。一种类型可能是投标人对出售对象的个人估值，他的风险规避程度，也许是他对物体的信息。 （Maskin and Riley（1984）讨论了出价者类型的许多不同经济有意义的例子。）从卖方和其他投标人的角度来看，每个投标人的类型都是随机变量。在此分析中，我们将注意力仅限于拍卖，其中类型是根据已知概率分布独立和相同分布的。启示原则断言，每次拍卖在战略上都等同于拍卖，在该拍卖中，竞标者通过宣布其类型而竞标，没有投标人有任何撒谎的动机。这样的拍卖称为激励兼容直接拍卖。我们将注意力仅限于直接拍卖的概率功能，并让激励兼容条件限制付款功能。每个投标人可以通过平均其他投标人的类型来计算他赢得的概率，以自己的类型为条件。将投标人类型与获胜的可能性相关的功能是拍卖的减少形式。关于“最佳”拍卖的文献通常解决了最大化卖方预期收入的问题。为此，有关拍卖概率函数的所有相关信息均以其简化形式包含。这是减少的表格，决定了每个投标人的行为，因此是卖方的预期收入。在对称拍卖中，每个投标人的降低形式都是相同的，因此预期收入是在还原形式上定义的功能，这是一个变量的函数，即类型。这使得卖方的问题有些解决。要设计拍卖，卖方必须能够识别降低的形式并恢复基础拍卖。还原形式满足直观的可行性条件。给定一组类型，
AN AUCTION IS A MECHANISM for allocating a single indivisible object to one of several competing bidders. The winner is the bidder who is awarded the object. The rules of the auction specify two functions. The first is the probability with which a bidder wins, as a function of everyone's bids. The second is the payment each bidder makes to the seller, as a function of all the bids and whether or not he wins. For instance, a first-price auction awards the object to the highest bidder with probability one (providing there are no tie bids), the winner pays his bid, and the losers pay nothing. The bidders in an auction differ significantly. These differences are captured by the bidder's type. A type may be the bidder's personal valuation of the object for sale, his degree of risk aversion, or perhaps his information about the object. (Maskin and Riley (1984) discuss a number of different economically meaningful examples of bidder types.) From the viewpoint of the seller and the other bidders, each bidder's type is a random variable. In this analysis we confine attention to auctions in which the types are independently and identically distributed according to a known probability distribution. The Revelation Principle asserts that every auction is strategically equivalent to an auction in which bidders bid by announcing their type and no bidder has any incentive to lie. Such an auction is called an incentive compatible direct auction. We will confine our attention to the probability functions for direct auctions, and let the incentive compatibility conditions restrict the payment functions. Each bidder can compute the probability that he wins, conditional on his own type, by averaging over the types of the other bidders. The function relating a bidder's type to his probability of winning is the reduced form of the auction. The literature on "optimal" auctions usually addresses the problem of maximizing expected revenue for the seller. For this purpose, all the relevant information about the probability function of an auction is contained in its reduced form. It is the reduced form that determines each bidder's behavior and hence the seller's expected revenue. In a symmetric auction each bidder's reduced form is identical, so that expected revenue is a functional defined on reduced forms, which are functions of one variable, namely, types. This makes the seller's problem somewhat tractable. To design an auction, a seller must be able to recognize a reduced form and recover the underlying auction. Reduced forms satisfy an intuitive feasibility condition. Given a set of types, the