| Auction, as an old mechanism of trading, has its history over at least two thousand years, and is now being applied in many transactions. But economists began to study them less than 100 years ago.Analyses begin with simplifying all kinds of auctions into four "standard" or basic types of auctions: English Auction (ascending-bid auction), Dutch auction (descending-bid auction), First Price Auction and Second Price Auction (Vickrey Auction). Among them, English auction is applied most, while the Second Price Auction is much less commonly used than other types of auctions. It is studied in part because of its attractive theoretical properties.Early economists conjectured that expected prices generated by different types of auctions might differ from each other. But under the assumptions of private values, independent types, symmetry and risk neutrality, we have the conclusion that different auction forms yield the same expected revenue. Formally, we express this Revenue Equivalent Theorem as follows:Revenue Equivalence Theorem (RET) Assume each of a given number of risk-neutral potential buyers has a privately-known valuation independently drawn from a strictly-increasing atomless distribution, and that no buyer wants more than one of the k identical indivisible prizes. Then any mechanism in which (i) the prizes always go to the k buyers with the highest valuations and (ii) any bidder with the lowest feasible valuation expects zero surplus, yields the same expected revenue (and results in each bidder making the same expected payment as a function of her valuation)Although RET was a remarkable achievement, these seemed to be little relationship to traditional price theory, which made the subject a difficult one for many economists. But further studies show that the problem is essentially equivalent to the analysis of standard monopoly third-degree discrimination. And in particular, it is quite helpful to focus on bidder's "marginal revenues". With the powerful RET, we can revise many issues through a new perspective. The War of Attrition, shown as an example, illustrates how RET simplified our economic analyses. We also introduce the "All-pay auction" here and its application in our real life. Despite perfect reasoning of RET, it does not match what we have observed in practice, that is, a seller can obtain the same expected revenue from English auction and Dutch auction, but less from Second Price auction, and the least from First Price auction. The reason is that RET has too strict assumptions, which make it some "idealistic". We try to loose its assumptions and observe how the result changes. It is easy to see how risk-aversion affects the revenue equivalence result: a risk-neutral seller faced by risk-averse bidders prefers the firs-price auction to second auction or ascending auction. In contrast, a risk-averse seller prefers the first-price auction to the Second Price Auction and, for a similar reason, prefers the Second Price Auction to ascending open auction. If bidder's private information is correlated, then the seller can construct a mechanism that yield herself the entire social surplus that would be feasible if bidder's information were fully public! And we now have that ascending auctions lead to higher expected prices than sealed-bid second-price auctions, which in turn lead to higher expected prices than first-price auctions. After carefully study on Private Value Auction Model, we now turn to Private Value Auction Model, in which the actual value is the same for everyone, but bidders have different private information about what that value actually is. Milgrom and Weber had developed a general model that includes as special cases the independent private values model and the common value model, as well as a range of intermediate models. It yields several testable predictions. First, the Dutch and first-price auctions are strategically equivalent in the general model. Second, when bidders are uncertain about their value estimates, the English auction gene... |
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