拍卖设计的数值模拟研究
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摘要
拍卖是一种历史悠久的交易方式,也是国外产权交易的最重要的一种方式。目前我国国有产权的交易越来越频繁,成交的金额越来越大。涉及的领域包括林业资源产权交易、矿产资源产权交易、土地产权交易、无线频谱牌照交易,排污权交易等等。而调查显示这些国有产权的交易以协商交易居多,很少使用拍卖方式交易。使得这些交易还带有浓厚的计划经济色彩,政府指导的随意性很强,缺乏科学的理论依据。这样就会造成一些负面的后果,例如,国内的产权交易之后遗留的权益矛盾较多,这是不合理的交易方式和政府指导定价行为导致的。所以未来拍卖在国有产权交易方面的应用急需得到推广,从而建立起规范的交易制度与交易规则,减少社会矛盾,促进构建和谐社会。这使得加强对拍卖的研究变得非常必要。另外,在我国企业开始走向国际,进行国际产权收购时,屡屡付出高昂的学费,支付过高的购买成本,例如中国平安收购海富通,中铝投资力拓。加强对拍卖行为的研究还可以为海外产权的收购报价提供科学的参考基准,避免付出过高的代价。
本文的主要研究了如下问题。
①对称私有价值条件下,考虑卖方是风险中性时的拍卖设计。在过去基于解析形式的数学模型得到的关于对称独立私有价值拍卖的理论基础上,使用数值模拟的方法,通过模拟拍卖过程,并使用Monte Carlo方法。绘制出了卖方的期望收入随着保留价格变化的曲线图。本文计算并绘制了四种竞买人估值信息,即左峰分布、中峰分布、右峰部分和均匀分布;同时还考虑了两种竞买人数量的情况,即5个竞买人参与拍卖和10个竞买人参与拍卖。这些曲线显示卖方从拍卖中获得的期望收入在正常的保留价格区间里对保留价格的变化不敏感。这个特征可以大大简化拍卖的设计。
②对称私有价值条件下,考虑卖方是风险厌恶型时的拍卖设计。在卖方是风险中性的结论基础上,还是借助模拟拍卖过程,并使用Monte Carlo方法,计算并绘制出卖方收入的标准差随着保留价格变化的曲线图。也是绘制出四种竞买人估值信息,和两种竞买人数量时对应的曲线。这些曲线显示,第一价格拍卖产生的风险总是小于第二价格拍卖产生的风险。从而立刻得出第一价格拍卖优于第二价格拍卖的结论。最后本文还利用Monte Carlo模拟的数据给出了卖方收入分布的概率密度图。
③不对称私有价值拍卖的均衡报价策略。由于不对称条件下,关于卖方收入的“收入等价性”原则已经不成立了,所以要想计算拍卖给卖方带来的期望收入就必须知道竞买人的均衡报价策略。本文研究了两种常用拍卖规则下竞买人的均衡报价策略,第一价格拍卖和第二价格拍卖。不对称条件下,第二价格拍卖的均衡报价策略很简单,仍然是报价等于估值;而对于第一价格拍卖的均衡报价策略,本文使用变分方法求出解的一阶条件,根据该条件发现无法求出解析形式的解,于是本文从一阶条件出发,给出了均衡报价策略数值形式的解。
④不对称私有价值拍卖的设计。由于不对称条件下“收入等价性原则”不成立,所以不同的拍卖规则产生的卖方期望收入不同。因此,拍卖设计只能是在可选的拍卖规则中,通过在给定的具体拍卖环境下的数值模拟计算来做比较。本文用已经得到的第一价格拍卖和第二价格拍卖下的均衡报价策略,分别计算并绘制了两种拍卖规则下,卖方的期望收入关于保留价格变化的曲线。通过对比两种曲线可以决定选择哪种拍卖规则,以及如何设定保留价格。本文得到的结论是,不对称条件下设置保留价格的规则与对称条件下差别不大。另外,本文为不对称第一价格拍卖的均衡报价策略设计了新的数值算法。
⑤竞买人的合谋行为。本文考虑的是最简单的拍卖环境中的竞买人合谋行为—对称私有价值拍卖中的合谋。即使在这样简单的拍卖环境中,合谋行为也会导致不对称竞买人情况的出现。因此必须借助不对称拍卖的分析结论。本文模拟了在这两种拍卖规则下的合谋行为。绘制出了不同合谋规模下,卖方的期望收入随保留价格变化的曲线。结果显示在第二价格拍卖下,合谋行为给卖方的期望收入带来的损失更大。这表明,第一价格拍卖规则更有利于卖方对抗合谋行为。本文的研究还显示,在第二价格拍卖中,合谋行为给卖方造成的损失具有单调性,即合谋集团的规模越大,给卖方带来的损失越大。但是在第一价格拍卖中,这种单调性不成立。因此,合谋有时反而会给卖方带来更高收益。
⑥数值分析方法在组合拍卖中的简单应用。本文分析了同时升价拍卖在土地分块拍卖中的应用。附加垄断限制条件的同时升价拍卖可以作为土地分块拍卖的一种交易形式,并且这种交易形式可以便利地限制交易结果中强势竞买人对标的物分配的垄断。本文研究这样的土地拍卖结果会有和特征。通过在给定的具体拍卖环境中的模拟拍卖,显示这样的土地交易方式相对于整块土地出售的方式来说,首先可以灵活地限制强势开发商对一个地区土地开发权的垄断;同时还会给政府带来更高的土地拍卖收入。这表明,带垄断约束条件的组合拍卖既可以提高政府的土地出售收入,又可以实现其他的福利目标。
总分的来说,本文给出的数值模拟方法可以解决具体环境条件下的私有价值拍卖的规则选择。但是,本文的研究尚未涉及共同价值拍卖,这是本文的一个不足之处。
Auction is a transaction way that has a very long history, and be the most institution used in the property transaction in the overseas. Now property transaction in China is becoming more and more, and the related value is increasing firmly. The transacted properties include forestry resource, ore-mining right, land-developing right, spectrum license and pollutants discharging rights. The survey shows that the transactions of these nation-owned properties are mainly in the form of negotiation. So these transactions have strong color of planned economy. The government often give direction to the them at will, not on the base of scientific analysis. Many negative effects are caused by this kind of transactions. For example, there is often some entanglements on the property left after the transaction was done, because of the unreasonable pricing under the government direction. Therefore, the auction must be proposed to the transaction of the nation-owned property in the future, and the construct the normal transaction institution should be constructed to eliminate the social contradiction and promoted the harmony in the society. Thus, the research on the auction should be strengthened. The Chinese enterprises has begun the internationalized running. They have paid costly expense for the purchasing the overseas property, such as Ping An Insurance (Group) Company Of China purchasing the equity of Fortis Haitong Annuities, or Chinalco purchasing the equity of Rio Tinto. Thus, better understanding of the auction behavior can offer better reference for the overseas acquisition to avoid paying high price.
In this paper, the following topics are include.
①The risk-neutral seller’s auction design under the condition of symmetric private value. On the base of the past theories derived from the analytic model, this paper makes the numerical simulation of auction by Monte Carlo method, and draws the diagram of the seller’s‘Expected Income - Reserve Price’curve. Four kinds of valuation information are included, that is, left-hump, central hump, right-hump and uniform distribution. At the same time, two different buyer’s quantity are considered, that is, 5 buyers and 10 buyers. These curves show that the seller’s expected income is not sensitive to the reserve price. This result can simplify the auction design largely.
②The risk-averse seller’s auction design under the condition of symmetric private value. On the base of the conclusion about the risk-neutral seller, employ Monte Carlo method to draw the diagram of“Standard Deviation of Income– Reserve Price”curve. The diagram also include the cases corresponding to those four distributions and two buyer’s quantities. This paper use the data generated by Monte Carlo simulation to show the distribution of the seller’s income.
③The buyer’s equilibrium bidding strategy. The‘Revenue Equivalence Principle’fails for the buyers are not symmetric. So the buyers’equilibrium bidding strategies are necessary for the calculation of the seller’s income. This paper considers two kinds of auction rule, that is, the second-price sealed-bid auction and the first-price sealed-bid auction. The buyer’s equilibrium bidding strategy in the second-price sealed-bid auction is very simple, bidding price equal to the value of the item. But things are much more complicated for the first-price sealed-bid auction. In this paper, the first-order condition of the equilibrium is found by variation method, and it tells that the buyer’s equilibrium bidding strategy can not be got in the analytic form. So the equilibrium bidding strategy must be calculated in the numerical form from the first-order condition.
④The auction design under the condition of asymmetric private value. For the‘Revenue Equivalence Principle’fails under the condition of asymmetric buyers, different auction rules will produce different expected income for the seller. Then the so-called auction is design is to select a optimal auction rule from several given ones, and the selection is according to the comparison of the seller’s expected income calculated in the concrete environments. In this paper, the seller’s‘Expected Income– Reserve Price’curves of the two auction rules are calculated by the buyers’equilibrium bidding strategies. According these curves, the seller can determine which rule is better, and how to set the reserve price. The conclusion is that the setting reserve price under the condition of asymmetric buyers is like the one under the condition of symmetric buyers.
⑤The collusion behavior of the buyers. This paper considers the simplest collusion, the collusion in the auction with symmetric buyers. Even in this simple case, the collusion behavior will cause the asymmetric condition. So the analysis on the asymmetric auction design relies the results about the buyer’s equilibrium strategy in the asymmetric environments. In this paper, the collusion behavior in the second-price sealed-bid auction and the first-price sealed-bid auction are considered. The seller’s‘Expected Income– Reserve Price’curve are calculated under different auction rule and with different coalition dimensions. The conclusion is that the first-price sealed-bid auction can defend collusion better than the second-price sealed-bid auction. The curves also show that the loss of the seller’s income increases monotonically as the coalition dimensions in the second-price sealed-bid auction, but not in the first-price sealed-bid auction. The collusion behavior of the buyers could sometimes increase the seller’s expected income.
Generally, the numerical simulation method used in this paper can solve the auction design in the concrete environments with private value. The shortcoming of this paper is that the common value auction had not be included.
本文的主要研究了如下问题。
①对称私有价值条件下,考虑卖方是风险中性时的拍卖设计。在过去基于解析形式的数学模型得到的关于对称独立私有价值拍卖的理论基础上,使用数值模拟的方法,通过模拟拍卖过程,并使用Monte Carlo方法。绘制出了卖方的期望收入随着保留价格变化的曲线图。本文计算并绘制了四种竞买人估值信息,即左峰分布、中峰分布、右峰部分和均匀分布;同时还考虑了两种竞买人数量的情况,即5个竞买人参与拍卖和10个竞买人参与拍卖。这些曲线显示卖方从拍卖中获得的期望收入在正常的保留价格区间里对保留价格的变化不敏感。这个特征可以大大简化拍卖的设计。
②对称私有价值条件下,考虑卖方是风险厌恶型时的拍卖设计。在卖方是风险中性的结论基础上,还是借助模拟拍卖过程,并使用Monte Carlo方法,计算并绘制出卖方收入的标准差随着保留价格变化的曲线图。也是绘制出四种竞买人估值信息,和两种竞买人数量时对应的曲线。这些曲线显示,第一价格拍卖产生的风险总是小于第二价格拍卖产生的风险。从而立刻得出第一价格拍卖优于第二价格拍卖的结论。最后本文还利用Monte Carlo模拟的数据给出了卖方收入分布的概率密度图。
③不对称私有价值拍卖的均衡报价策略。由于不对称条件下,关于卖方收入的“收入等价性”原则已经不成立了,所以要想计算拍卖给卖方带来的期望收入就必须知道竞买人的均衡报价策略。本文研究了两种常用拍卖规则下竞买人的均衡报价策略,第一价格拍卖和第二价格拍卖。不对称条件下,第二价格拍卖的均衡报价策略很简单,仍然是报价等于估值;而对于第一价格拍卖的均衡报价策略,本文使用变分方法求出解的一阶条件,根据该条件发现无法求出解析形式的解,于是本文从一阶条件出发,给出了均衡报价策略数值形式的解。
④不对称私有价值拍卖的设计。由于不对称条件下“收入等价性原则”不成立,所以不同的拍卖规则产生的卖方期望收入不同。因此,拍卖设计只能是在可选的拍卖规则中,通过在给定的具体拍卖环境下的数值模拟计算来做比较。本文用已经得到的第一价格拍卖和第二价格拍卖下的均衡报价策略,分别计算并绘制了两种拍卖规则下,卖方的期望收入关于保留价格变化的曲线。通过对比两种曲线可以决定选择哪种拍卖规则,以及如何设定保留价格。本文得到的结论是,不对称条件下设置保留价格的规则与对称条件下差别不大。另外,本文为不对称第一价格拍卖的均衡报价策略设计了新的数值算法。
⑤竞买人的合谋行为。本文考虑的是最简单的拍卖环境中的竞买人合谋行为—对称私有价值拍卖中的合谋。即使在这样简单的拍卖环境中,合谋行为也会导致不对称竞买人情况的出现。因此必须借助不对称拍卖的分析结论。本文模拟了在这两种拍卖规则下的合谋行为。绘制出了不同合谋规模下,卖方的期望收入随保留价格变化的曲线。结果显示在第二价格拍卖下,合谋行为给卖方的期望收入带来的损失更大。这表明,第一价格拍卖规则更有利于卖方对抗合谋行为。本文的研究还显示,在第二价格拍卖中,合谋行为给卖方造成的损失具有单调性,即合谋集团的规模越大,给卖方带来的损失越大。但是在第一价格拍卖中,这种单调性不成立。因此,合谋有时反而会给卖方带来更高收益。
⑥数值分析方法在组合拍卖中的简单应用。本文分析了同时升价拍卖在土地分块拍卖中的应用。附加垄断限制条件的同时升价拍卖可以作为土地分块拍卖的一种交易形式,并且这种交易形式可以便利地限制交易结果中强势竞买人对标的物分配的垄断。本文研究这样的土地拍卖结果会有和特征。通过在给定的具体拍卖环境中的模拟拍卖,显示这样的土地交易方式相对于整块土地出售的方式来说,首先可以灵活地限制强势开发商对一个地区土地开发权的垄断;同时还会给政府带来更高的土地拍卖收入。这表明,带垄断约束条件的组合拍卖既可以提高政府的土地出售收入,又可以实现其他的福利目标。
总分的来说,本文给出的数值模拟方法可以解决具体环境条件下的私有价值拍卖的规则选择。但是,本文的研究尚未涉及共同价值拍卖,这是本文的一个不足之处。
Auction is a transaction way that has a very long history, and be the most institution used in the property transaction in the overseas. Now property transaction in China is becoming more and more, and the related value is increasing firmly. The transacted properties include forestry resource, ore-mining right, land-developing right, spectrum license and pollutants discharging rights. The survey shows that the transactions of these nation-owned properties are mainly in the form of negotiation. So these transactions have strong color of planned economy. The government often give direction to the them at will, not on the base of scientific analysis. Many negative effects are caused by this kind of transactions. For example, there is often some entanglements on the property left after the transaction was done, because of the unreasonable pricing under the government direction. Therefore, the auction must be proposed to the transaction of the nation-owned property in the future, and the construct the normal transaction institution should be constructed to eliminate the social contradiction and promoted the harmony in the society. Thus, the research on the auction should be strengthened. The Chinese enterprises has begun the internationalized running. They have paid costly expense for the purchasing the overseas property, such as Ping An Insurance (Group) Company Of China purchasing the equity of Fortis Haitong Annuities, or Chinalco purchasing the equity of Rio Tinto. Thus, better understanding of the auction behavior can offer better reference for the overseas acquisition to avoid paying high price.
In this paper, the following topics are include.
①The risk-neutral seller’s auction design under the condition of symmetric private value. On the base of the past theories derived from the analytic model, this paper makes the numerical simulation of auction by Monte Carlo method, and draws the diagram of the seller’s‘Expected Income - Reserve Price’curve. Four kinds of valuation information are included, that is, left-hump, central hump, right-hump and uniform distribution. At the same time, two different buyer’s quantity are considered, that is, 5 buyers and 10 buyers. These curves show that the seller’s expected income is not sensitive to the reserve price. This result can simplify the auction design largely.
②The risk-averse seller’s auction design under the condition of symmetric private value. On the base of the conclusion about the risk-neutral seller, employ Monte Carlo method to draw the diagram of“Standard Deviation of Income– Reserve Price”curve. The diagram also include the cases corresponding to those four distributions and two buyer’s quantities. This paper use the data generated by Monte Carlo simulation to show the distribution of the seller’s income.
③The buyer’s equilibrium bidding strategy. The‘Revenue Equivalence Principle’fails for the buyers are not symmetric. So the buyers’equilibrium bidding strategies are necessary for the calculation of the seller’s income. This paper considers two kinds of auction rule, that is, the second-price sealed-bid auction and the first-price sealed-bid auction. The buyer’s equilibrium bidding strategy in the second-price sealed-bid auction is very simple, bidding price equal to the value of the item. But things are much more complicated for the first-price sealed-bid auction. In this paper, the first-order condition of the equilibrium is found by variation method, and it tells that the buyer’s equilibrium bidding strategy can not be got in the analytic form. So the equilibrium bidding strategy must be calculated in the numerical form from the first-order condition.
④The auction design under the condition of asymmetric private value. For the‘Revenue Equivalence Principle’fails under the condition of asymmetric buyers, different auction rules will produce different expected income for the seller. Then the so-called auction is design is to select a optimal auction rule from several given ones, and the selection is according to the comparison of the seller’s expected income calculated in the concrete environments. In this paper, the seller’s‘Expected Income– Reserve Price’curves of the two auction rules are calculated by the buyers’equilibrium bidding strategies. According these curves, the seller can determine which rule is better, and how to set the reserve price. The conclusion is that the setting reserve price under the condition of asymmetric buyers is like the one under the condition of symmetric buyers.
⑤The collusion behavior of the buyers. This paper considers the simplest collusion, the collusion in the auction with symmetric buyers. Even in this simple case, the collusion behavior will cause the asymmetric condition. So the analysis on the asymmetric auction design relies the results about the buyer’s equilibrium strategy in the asymmetric environments. In this paper, the collusion behavior in the second-price sealed-bid auction and the first-price sealed-bid auction are considered. The seller’s‘Expected Income– Reserve Price’curve are calculated under different auction rule and with different coalition dimensions. The conclusion is that the first-price sealed-bid auction can defend collusion better than the second-price sealed-bid auction. The curves also show that the loss of the seller’s income increases monotonically as the coalition dimensions in the second-price sealed-bid auction, but not in the first-price sealed-bid auction. The collusion behavior of the buyers could sometimes increase the seller’s expected income.
Generally, the numerical simulation method used in this paper can solve the auction design in the concrete environments with private value. The shortcoming of this paper is that the common value auction had not be included.
引文
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[114]廖兰心,2007,《地产大盘遇上政策紧箍咒》,红网人居频道,2007-05-15
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[65] Levin, Dan, Smith, James L. Optimal Reservation Prices in Auctions[J]. The Economic Journal. 1996, 106: 1271-1283.
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