Auctions Basic Theory Applications Chapter 23 Auctions 1

Auctions: Basic Theory & Applications Chapter 23: Auctions 1

Introduction • Auctions have long history – Roman Empire sold at auction in 193 AD – Colonial auctions to sell shipwreck salvage – Slave auctions – Treasury Bond auctions – FCC radio spectrum auctions – e. Bay • Auction Theory has Insights for Industrial Organization – Bertrand Competition – Asymmetric advantages of Incumbency Chapter 23: Auctions 2

Auction Taxonomy • 4 Main Auction Types – English or Ascending Auction—Bids publicly called out and rise until only one bidder is left – Dutch or Descending Auction—Public price is lowered until a buyer comes forth – 1 st Price Sealed Bid Auction • Bids Submitted in Secret • Highest Bidder Wins and Pays Amount Bid – 2 nd Price Sealed Bid Auction • Bids Submitted in Secret • Highest Bidder Wins but Pays 2 nd Highest Amount Bid Chapter 23: Auctions 3

Auction Taxonomy (2) • A Further Taxonomy of Auctions • Private Value Auctions—each bidder has her own value of the auction object unrelated to other bidders’ values • Common Value Auction—each bidder has her own estimate of the auction object’s true market value common to all • Usual assumption is that values/estimates are uncorrelated across bidders but bidders may in fact have affiliated valuations Chapter 23: Auctions 4

Private Values and Revenue Equivalence (1) • Auction Bidders are in a game of strategic interaction —What is the equilibrium of this game? —At what price at will the item be sold? • Revenue Equivalence Theorem (Vickrey): Equilibrium Price (Revenue) Independent of Auction Type • Proof Reflects Optimal Bidding Strategies (Best Response Functions) in Different Auction Designs Chapter 23: Auctions 5

Private Values and Revenue Equivalence (2) • Revenue/Winning Bid Equivalence in English Auction and 2 nd Price-Sealed Bid Auction —Assume Bidders with Private Values from 0 to $500 —Is it a Nash equilibrium for each to Bid her True Valuation? • English Auction: Consider Bidder with vi = $400 and suppose that the current bid is $399. If he drops out, he gains 0. If he raises the bid by a small amount ε to $399 + ε: —He either wins and gains $1 – ε —Or he is overbid and gains $0 —Raising his bid does as well if not better So he will bid up to vi = $400 —Of course, bidding above vi = $400 can only bring a loss • All Bidders bidding their Valuation is a Nash equilibrium Chapter 23: Auctions 6

Private Values and Revenue Equivalence (3) • Intuition Behind the Foregoing Result: • 2 nd Price Sealed bid Auction Shares a Common Essential Feature with the English Auction: Bidding your true valuation affects only the chance of winning—Not What You Pay —Winning Bidder will Pay second highest bid which is exactly the point at which all but one bidders drop out in an English auction —As the mechanism is essentially the same in both an English and 2 nd Price auction, the Nash equilibrium in both is for each to bid vi Chapter 23: Auctions 7

Private Values and Revenue Equivalence (4) • For English and 2 nd price sealed bid: • Expected Price with n bidders is expected value of 2 nd-highest bid or valuation —e. g. , if there are n = 5 bidders, we are looking for the expected value of the n-1 th or 4 th lowest bid—This is the n-1 th order statistic —For the uniform distribution between Vmin and Vmax this implies that the winning bid has expected value: —So, if n = 5 and Vmin = 0 while Vmax = $500, then the expected winning bid is $400 Chapter 23: Auctions 8

Private Values and Revenue Equivalence (5) • Dutch and 1 st Price Auction share key common feature analogous to the Identity of English and 2 nd Price Auction • In both the Dutch and 1 st Price Sealed Bid cases each bidder must choose a “buy price” or “jumping in” bid point • This bid will determine both —Chance of Winning and —Price Paid • Because Higher Bidding also leads to Higher Price Paid, Bidders need to shade their bids relative to their true values • bi = λvi ; What is optimal value of λ or optimal shading? Chapter 23: Auctions 9

Private Values and Revenue Equivalence (6) • Of course, no bidder will bid less than minimum Vmin • Optimal shading in Dutch and 1 st Price —Nash equilibrium bidding strategy is bi = Vmin + λvi —So, With n bidders, winning bidder will have nth lowest vi or again, assuming a uniform distribution. Chapter 23: Auctions 10

Private Values and Revenue Equivalence (7) • Combining the above results the expected winning bid in the Dutch and 1 st Price Sealed Bid cases is • Exactly the same as in the English and 2 nd Price auctions! • Revenue Equivalence Theorem: In auctions with risk neutral bidders with independent private values drawn from a continuous distribution, and where the auction always goes to the highest valued-bidder (and 0 -value bidders bidding 0) the expected auction price or revenue is the same regardless of auction type Chapter 23: Auctions 11

Common Value Auctions (1) • Many values, e. g. , bidding for broadcast or franchise rights, have a common (market) value —In this case, each bidder’s valuation vi is an estimate of the true common value that will only be revealed after the auction —vi = V* + εi where εi has mean of 0 • The Winner’s Curse —vi is an unbiased estimate of the true value V* unconditionally —But a bidder is interested in the expectation for V* conditional on her winning the bidding —Winning Bidder likely has the highest εi and would pay too much unless the potential for this “curse” is foreseen Chapter 23: Auctions 12

Common Value Auctions (2) • Protecting Against the Winner’s Curse requires shading one’s bid below the observed estimated value vi • What is optimal shading? —For uniform εi distribution centered on 0 and ranging to optimal bid bi is equal to vi less an amount that rises with the number of bidders n — —e. g. , As n gets very large, each bidder assumes (correctly) that if she wins she had the highest value estimate vi possible, namely, V* and so reduces her bid by Chapter 23: Auctions 13

Common Values, Affiliated Values, and Revenue Equivalence • Revenue Equivalence can hold in Common Value Auctions • It will fail if: —Bidders are not risk neutral; —Draw value estimates from different distributions; —Have affiliated value estimates • In such cases, a rule of thumb is that English Auction and 2 nd Price Auction generally yield more revenue —English auction limits Winner’s Curve as bidders extract information from others’ decisions to stay in or exit—So, less bid shading —Again, in 2 nd Price Auction, bid only affects chance of winning not amount paid and so again less shading Chapter 23: Auctions 14

Auction Insights for Industrial Organization (1) • Bertrand Competition —n firms compete in price bids —Each has cost ci that is known only to duopolist i and drawn from a uniform distribution between 0 and cmax —What is the equilibrium bidding function? • In this auction, bidders are selling not buying —But analogy to earlier bidding rule works —Equilibrium price bidding rule is: Chapter 23: Auctions 15

Auction Insights for Industrial Organization (2) • Bertrand Competition Insights —Numbers matter now: As n grows larger firms bid closer to marginal cost ci —From Revenue Equivalence, expected winning sale price bid is expected value of 2 nd lowest ci—the 2 nd Order Statistic • Asymmetries and Incumbent Advantage —Recall that value of maintaining monopoly power for an incumbent exceeds the value of breaking into the market as a duopolist for the entrant —Auction theory makes clear that such small asymmetries can have large implications in bidding wars, etc. —Think of resources firms spend in a fight for the market as their “bids” for market position Chapter 23: Auctions 16

Auction Insights for Industrial Organization (3) • A Symmetric Incumbent/Entrant Game —Let the market have value G 1 +G 2 the same for both the entrant and the incumbent • Incumbent knows G 1 but not G 2 • Entrant knows G 2 but not G 1 —In fighting or “bidding” for the market, the symmetric Nash equilibrium “bid” is • 2 G 1 for the incumbent • 2 G 2 for the entrant Proof: If firm 1 wins it does so when firm 2 drops out at a bid of 2 G 2 In this case, firm 1’s net gain is G 1 + G 2 – 2 G 2 > 0 as G 1 > G 2. Raising its bid will not raise firm 1’s chances of winning and lowering it will only risk losses. So, firm 1 has no incentive to change. A symmetric argument applies to firm 2. Chapter 23: Auctions 17

Auction Insights for Industrial Organization (4) • The Winner’s Curse in the Incumbent/Entry game —Winner’s Curse aversion reflected in the bidding strategy bi = 2 Gi —Firm 2, for example, stops bidding once price has reach 2 G 2. Yet if the bidding reaches this point, firm 2 knows that G 1 is at least G 2. • Asymmetry and the Winner’s Curse —Now introduce asymmetry similar to usual incumbent/entry difference by letting market’s value to the incumbent be G 1 + G 2 + 1 • Firm 1 now bids more aggressively, intensifying firm 2’s curse • Firm 2 therefore bid less aggressively, weakening firm 1’s curse, which leads firm 1 to bid even more aggressively and so on —In equilibrium, firm 2 (the entrant) bids very conservatively allowing the incumbent (firm 1) to almost always win the market at a very low bid (cost) Even small asymmetries can give incumbency great advantages Chapter 23: Auctions 18

Empirical Application: Finding Collusion in Public School Milk Contract Bidding (1) • Porter and Zona (1999) reviewed bidding behavior in Ohio School Milk Procurement auctions for evidence that Trauth Dairy was part of a collusive bidding ring – Meyers Dairy and Coors Dairy had confessed and implicated Trauth – Trauth denied accusations • Porter and Zona look for evidence of common collusive behavior – “Complementary bidding”: Cartel members make many more bids in many more school districts to give appearance of aggressive competition – Irrational bidding: Price-colluding firms submit bids that are “irrationally” high because they are not meant to be competitive but only give the appearance of competition Chapter 23: Auctions 19

Empirical Application: Finding Collusion in Public School Milk Contract Bidding (2) • Evidence on Complementary Bidding – Porter and Zona first estimate a model of the likelihood that a noncolluding dairy will submit a bid in any district – Compare model predictions with behavior of Meyer, Coors, and Trauth – Finding: Meyer, Coors and Trauth submit significantly more bids, especially in nearby markets, e. g. , submit 20% more bids in markets 2030 miles away Propensity of Alleged Conspirators to Submit Milk Bids Relative to Competing Dairies Distance of School District in Miles from Dairy Coors Meyer Trauth 0 – 10 24. 1%* 5. 6* 7. 0%* 10 – 20 42. 9%* 8. 2% 15. 2%* 20 – 30 22. 9%* 18. 5%* 20. 6%* *Significant at 5% level Chapter 23: Auctions 20

Empirical Application: Finding Collusion in Public School Milk Contract Bidding (3) • Porter and Zona also find that, when they make bids, Meyers, Coors, and Trauth bids have unusual property • All the bidding rules derived earlier have the common sense implication that the optimal bid bi rises as does the firm’s cost ci – The Distance that the milk has to be transported is a critical cost in delivery to schools—Costs rise as this distance rises – For Competing firms, bid also rises with school distance from dairy – For Meyers, Coors, and Trauth however, bids fall as distance rises Consistent with collusive arrangement of submitting high bids to allow cartel members to win contracts at high prices but to submit lots of bids to make their behavior look competitive Chapter 23: Auctions 21

Empirical Application: Finding Collusion in Public School Milk Contract Bidding (4) • Porter and Zona find that – Cartel raised school lunch milk prices by 6% - 7% overall – In some specific school districts, however, the effect was nearly 50% • Careful work with auction theory predictions and empirical evidence has important policy payoffs Chapter 23: Auctions 22