1. Auctions: Basic Theory & Applications
Chapter 23: Auctions

2. Introduction Auction Theory has Insights for Industrial Organization
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
eBay
Auction Theory has Insights for Industrial Organization
Bertrand Competition
Asymmetric advantages of Incumbency
Chapter 23: Auctions

3. 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
1st Price Sealed Bid Auction
Bids Submitted in Secret
Highest Bidder Wins and Pays Amount Bid
2nd Price Sealed Bid Auction
Highest Bidder Wins but Pays 2nd Highest Amount Bid
Chapter 23: Auctions

4. 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

5. 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

6. Private Values and Revenue Equivalence (2)
Revenue/Winning Bid Equivalence in English Auction and 2nd 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

7. Private Values and Revenue Equivalence (3)
Intuition Behind the Foregoing Result:
2nd 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 2nd Price auction, the Nash equilibrium in both is for each to bid vi
Chapter 23: Auctions

8. Private Values and Revenue Equivalence (4)
For English and 2nd price sealed bid:
Expected Price with n bidders is expected value of 2nd-highest bid or valuation
—e.g., if there are n = 5 bidders, we are looking for the expected value of the n-1th or 4th lowest bid—This is the n-1th 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

9. Private Values and Revenue Equivalence (5)
Dutch and 1st Price Auction share key common feature analogous to the Identity of English and 2nd Price Auction
In both the Dutch and 1st 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

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

11. Private Values and Revenue Equivalence (7)
Combining the above results the expected winning bid in the Dutch and 1st Price Sealed Bid cases is
Exactly the same as in the English and 2nd 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

12. 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

13. 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

14. 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 2nd 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 2nd Price Auction, bid only affects chance of winning not amount paid and so again less shading
Chapter 23: Auctions

15. 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

16. 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 2nd lowest ci—the 2nd 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

17. Auction Insights for Industrial Organization (3)
A Symmetric Incumbent/Entrant Game
—Let the market have value G1 +G2 the same for both the entrant and the incumbent
Incumbent knows G1 but not G2
Entrant knows G2 but not G1
—In fighting or “bidding” for the market, the symmetric Nash equilibrium “bid” is
2G1 for the incumbent
2G2 for the entrant
Proof: If firm 1 wins it does so when firm 2 drops out at a bid of 2G2 In this case, firm 1’s net gain is G1 + G2 – 2G2 > 0 as G1 > G2. 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

18. 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 = 2Gi
—Firm 2, for example, stops bidding once price has reach 2G2. Yet if the bidding reaches this point, firm 2 knows that G1 is at least G2.
Asymmetry and the Winner’s Curse
—Now introduce asymmetry similar to usual incumbent/entry difference by letting market’s value to the incumbent be G1 + G2 + 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

19. 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

20. 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 non- colluding 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 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

21. 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

22. 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