In evaluating closed-bid competitive procurement auctions, the most crucial issue is to determine the probability of placing a winning bid for a given markup level. There has long been disagreement on how this should be done due to the absence of a mathematical derivation of one of the main evaluation techniques¡ªGates' method.Gates' method is shown in this paper to be valid if, and only if, bids can be described using the proportional hazards family of statistical distributions. When markup values are included in Gates' method, it is seen that the underlying statistical distribution required for the method to work is closely related to the Weibull distribution. Likelihood based methods are suggested for parameter estimation and an illustrative example is provided by analysis of Shaffer and Micheau's 1971 construction contract bidding data
As Crowley 2000 has observed, "there has been a lingering,unresolved debate concerning two contrasting bid models intro-duced in articles by Friedman 1956 and Gates 1967 each model provides an approach to evaluating closed-bid competitive procurement auctions amongst known competitors the distinc-tion between the models being in the determination of the prob-ability of placing a winning bid." The main reason for the disagreement has been the lack of a convincing proof of the Gates' formula, described by Gates as being based on a "balls in the urn" model. Immediately upon publication of the Gates paper,Stark 1968 questioned the validity of his model while Benjamin 1969 , recognizing that Gates provides no mathematical proof,attempted to rectify the situation but was unable to do so. Rosen-shine's 1972 later attempt met with equal lack of success as did Gates himself Gates 1976 , leading Englebrecht-Wiggans 1980 to re ect that Gates' model appears to be an empirical fit formula that is still without "mathematical justification."
For a worked example, we refer to the data published in a previ-ous paper by Shaffer and Micheau 1971 . This comprises all identified bids for a series of 50 sealed bid building contract auc-tions over the period 1965¨C1969. For the purposes of the ex-ample, it is assumed that the bids for the first 49 contracts are known but only the identity of the bidders are known for the 50th contract¡ªOurs, Dick, Leon, Les, Lyle, and Rob. The task, there-fore, is to estimate the theoretical probabilities associated with each of these being the lowest bidder for the 50th contract. From the above, there are three main contending models, comprising:1 the exponential; 2 the Weibull; and 3 the general PH family.

在评价公开招标拍卖的竞争性采购，最关键的问题是，以确定的概率放置一个中标的某一标记的水平。人们长期以来一直分歧就如何这样做，因为没有一个数学推导的一个主要评价技术¡ ª盖茨method.Gates '的方法显示在此文件，如果是有效的，且仅当，出价可被称为使用比例风险家庭的统计分布。当标记值包括在盖茨的方法，它是看到了潜在的统计分布所需的方法来工作是密切相关的Weibull分布。然的方法，提出了参数估计和说明问题的例子是由分析谢弗和Micheau 1971年施工合同招标数据
作为2000年克劳利说： “有了挥之不去的，悬而未决的辩论有两种截然不同的收购模式介绍的文章，弗里德曼1956年和1967年盖茨每个模型提供了一个办法，评价公开招标拍卖的竞争性采购除已知的竞争对手的区别，法之间的模式是在确定的概率放置中标。 “主要原因分歧一直缺乏令人信服的证据，盖茨的公式，盖茨所描述的是基于一种“球在瓮”的模式。出版后立即在盖茨文件， 1968年斯塔克的有效性提出质疑他的模型，而本杰明1969年，认识到盖茨没有提供数学证明，试图纠正这种情况，但未能这样做。罗森，光泽的1972年后试图会见了平等缺乏成功也盖茨本人盖茨1976年，领导Englebrecht - Wiggans 1980年重新等，盖茨模型似乎是一个合适的经验公式，仍然是“数学的理由。 ”
对于工作的例子中，我们指的是数据公布在previ -皮文件谢弗和Micheau 1971 。这包括所有投标确定的一系列密封投标50个建设合同拍卖筹措期间的1965年¨ C1969 。为施行前充裕，这是假定，出价为49个合同第一次被称为但只有人的身份投标人是众所周知的第50合同¡ ª我们，迪克，里昂，法国，莱尔和罗伯。的任务，还有，前面，是估计的理论概率与这些被报价最低者为50的合同。从以上的，有三个主要的对立模式，包括： 1的指数; 2威布尔;和3 PH值一般家庭。