A General Theory of Auctions

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1 A General Theory of Auctions Milgrom and Weber 3 November 2011 Milgrom and Weber () A General Theory of Auctions 3 November / 12

2 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

3 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

4 Motivation Received (as at 1982) literature had focused on the IPV and common values models. Some interesting results that are still around: 1 Strategic equivalence of Dutch and FPA; English and SPA. 2 English and SPA are Pareto optimal; FPA/Dutch only under symmetric bidders. 3 RET - conditional on lowest v having payment of zero and highest v certain to win (plus risk neutrality, symmetric bidders, z(0) = 0, etc.). Milgrom and Weber () A General Theory of Auctions 3 November / 12

5 Motivation Received (as at 1982) literature had focused on the IPV and common values models. Some interesting results that are still around: 1 Strategic equivalence of Dutch and FPA; English and SPA. 2 English and SPA are Pareto optimal; FPA/Dutch only under symmetric bidders. 3 RET - conditional on lowest v having payment of zero and highest v certain to win (plus risk neutrality, symmetric bidders, z(0) = 0, etc.). Milgrom and Weber () A General Theory of Auctions 3 November / 12

6 Motivation Milgrom and Weber want to answer, 1 Are the results robust to intermediate cases (between common values and IPV)? 2 What if bidders are uncertain of their own value of the good? 3 Should sellers disclose all relevant info? Milgrom and Weber () A General Theory of Auctions 3 November / 12

7 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

8 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

9 Model Assumptions 1 There exists a function u on R m+n, with u i (S, X) = u(s, X i, {X j } j i ) and where X is a vector of observed informative (of the item s value) variables and S is additional variables, possibly not yet observed, but that influence the value to the bidder, e.g. drop out timing of others in English auction. 2 u is nonnegative, and is continuous and nondecreasing in its arguments. 3 E[V i ] < 4 Joint density f(s, x) is symmetric in its last n arguments. 5 Variables S 1,...S m, X 1,...X n are affiliated large value for a variable makes it more likely others will also be large. Milgrom and Weber () A General Theory of Auctions 3 November / 12

10 Model What does all this mean? IPV model, S is empty and all Xs are statistically independent, with V i = X i. Common values, m = 1 and for each V i = S 1. Affiliation allows for a bidder that finds the object particularly attractive to consider that others will find it attractive too can allow for consideration of some unknown resale value, absent in IPV model. Milgrom and Weber () A General Theory of Auctions 3 November / 12

11 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

12 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

13 Key Results Key Result 1 In the SPA, FPA and English auction suppose that the seller can choose to publicly reveal some information X 0 that s affiliated with all the other elements (e.g. the value of a timberland tract). Then the expected selling price is greater when the information is made public. Key Result 2 The expected price in the English auction is not less than that in the SPA. Intuition: in phase 1 n 2 bidders drop out and in so doing reveal X 0 to the remaining 2 bidders, who then compete in a SPA. The expected selling price in the SPA is at least as large as in the FPA. Ordering due to the linkage of prices to information revealed (the more is revealed the higher is the price). Results are robust to including a reserve price. Milgrom and Weber () A General Theory of Auctions 3 November / 12

14 Key Results Key Result 1 In the SPA, FPA and English auction suppose that the seller can choose to publicly reveal some information X 0 that s affiliated with all the other elements (e.g. the value of a timberland tract). Then the expected selling price is greater when the information is made public. Key Result 2 The expected price in the English auction is not less than that in the SPA. Intuition: in phase 1 n 2 bidders drop out and in so doing reveal X 0 to the remaining 2 bidders, who then compete in a SPA. The expected selling price in the SPA is at least as large as in the FPA. Ordering due to the linkage of prices to information revealed (the more is revealed the higher is the price). Results are robust to including a reserve price. Milgrom and Weber () A General Theory of Auctions 3 November / 12

15 Key Results Key Result 3 If bidders are risk averse then the expected revenue in a FPA exceeds that of the SPA. Intuition: still weakly dominant to bid value in SPA with risk aversion, but now is optimal to increase the bid in the FPA as if decrease bid and still win then gain, but utility cost of potentially losing the auction is greater than if risk averse. Key Result 4 Revealing info to risk averse bidders increases expected prices so long as full uncertainty is resolved and/or bidders have CARA utility. This might explain why we see reputable, independent appraisals. And combined with result 2 hints at why we tend to see English auctions. Note: by releasing info some bidders lose an informational advantage and this reduces their WTP. Hence result is not a general finding. Milgrom and Weber () A General Theory of Auctions 3 November / 12

16 Key Results Key Result 3 If bidders are risk averse then the expected revenue in a FPA exceeds that of the SPA. Intuition: still weakly dominant to bid value in SPA with risk aversion, but now is optimal to increase the bid in the FPA as if decrease bid and still win then gain, but utility cost of potentially losing the auction is greater than if risk averse. Key Result 4 Revealing info to risk averse bidders increases expected prices so long as full uncertainty is resolved and/or bidders have CARA utility. This might explain why we see reputable, independent appraisals. And combined with result 2 hints at why we tend to see English auctions. Note: by releasing info some bidders lose an informational advantage and this reduces their WTP. Hence result is not a general finding. Milgrom and Weber () A General Theory of Auctions 3 November / 12

17 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

18 Outline 1 Motivation 2 Model 3 Key Results 4 Further Research Milgrom and Weber () A General Theory of Auctions 3 November / 12

19 Further Research Milgrom and Weber end with a short discussion of open questions, as at Many remain active areas of research. Post hoc renegotiation, e.g. in procurement contracts for roads. Bajari et al (WP) combine procurement auctions with incomplete contracts. Still open questions. Correlated auctions, e.g. spectrum, timberland. Has given rise to the scoring auction literature, e.g. Asker and Cantillon. Collusion in auctions (e.g. Ohioan milk auctions by Porter; FL/TX by Pesendorfer). Syndicates of bidders - what is the effect on bidding strategy/prices? Milgrom and Weber () A General Theory of Auctions 3 November / 12

Introduction to Auctions

Introduction to Auctions Common to Optimal in Item Cheriton School of Computer Science University of Waterloo Outline Common Optimal in Item 1 2 Common Optimal 3 in 4 Item 5 Common Optimal in Item Methods for allocating goods,