Chapter 18 Auctions

1. Chapter 18 Auctions • Key Concept: Honesty is the best policy in a private-value second price auction. • However in a common-value auction, the winner’s curse may occur.

2. Chapter 18 Auctions • Auctions are one of the oldest form of markets, dating back to at least 500 BC. • You are probably familiar with consumer-oriented auctions.

4. In the US, the department of the interior use auctions to sell the right to drill in coastal areas. • The FCC auctions to sell radio spectrum used by cellular phones.

6. Economists’ three roles • Designers, consultants to form bidding strategies, ?.

7. The economic classification of auctions involves two considerations: the nature of the good that is being auctioned and the rules of bidding. • On the nature of the good, economists distinguish between private-value and common-value auctions.

8. Private-value: each participant has a potentially different value for the good (ex: an art work). • Common-value: the good is worth essentially the same to every bidder (ex: oil-drilling)

9. We first look at private-value auctions. • At the end, we will mention briefly the common-value auctions.

10. We turn to the second classification on bidding rules. • Open auctions • English auctions (ascending auction): Bidders successively offer higher prices until no participant is willing to increase the bid further.

11. Dutch auction (descending auction): Due to its use in the Netherlands for selling cheese and fresh flowers. • The auctioneer (typically a clock) starts with a high price and gradually lowers it by steps until someone is willing to buy the item. It proceed quickly.

12. Sealed-bid auction: each bidder writes down a bid and seals it in an envelope. All envelopes are collected and opened. • First price. Ex: construction work. The job awarded to the lowest bidder. • Second price (philatelist auction or Vickery auction)

13. Assume we have an item to auction off. • There are n bidders with private values v1, v2, …, vn. • Suppose seller has a zero value for the object.

14. We want to design an auction (mechanism) to meet our goal. • This is a special case of a mechanism design problem. • Two natural goals are Pareto efficiency and Profit maximization.

15. Profit maximization is straightforward, the seller wants to get the highest expected profit. • What about Pareto efficiency? • Suppose v1>v2> …>vn. Then to achieve efficiency, the good should be sold to person 1.

16. If the seller knows the values, v1, v2, …, vn, the auction design problem is trivial. • If the goal is to maximize profit, the seller should award the item to 1 and charge him v1. • If the goal is to maximize efficiency, it is still the case that 1 should get the good. But the price paid does not matter.

17. The more interesting case is when the seller does not knows the values. • How can we achieve effiency? • Note that the English auction will achieve this.

18. How can we achieve effiency? • Suppose v1 =100 and v2=80. And the bid increment is 5. • Then 1 will be willing to bid to 85 where 2 will not.

19. Hence the person with the highest valuation will be the winner. • He will pay the value of the second-highest bidder plus, perhaps, the minimal bid increment. • This now makes the second price auction looking less crazy.

20. What if the goal is to maximize profit? • Suppose we have two bidders. Each could have a value of 10 or 100. • (vi, vj)=(10,10), (10,100), (100,10), (100,100), each is equally likely. • The bid increment is 1.

21. (vi, vj)=(10,10), (10,100), (100,10), (100,100). • If we still use the English auction, then the winning bid will be 10, 11, 11, 100. • The expected revenue to the seller is 33=(10+11+11+100)/4.

22. (vi, vj)=(10,10), (10,100), (100,10), (100,100). • What if the seller sets the reserve price at say 100? • The expected revenue would be (0+100+100+100)/4=75>33. • It is not Pareto optimal since ¼ of the time no one gets the good!

23. This demonstrates that we might not be able to achieve the two goals (Pareto efficiency and Profit maximization) at the same time.

24. Now let us turn to the second price sealed-bid auction. • If bidders will bid truthfully, then the item will be awarded to the bidder with the highest value, who pays the price of the second highest value. This looks similar to the English auaction. • But will bidders bid truthfully?

25. Let us look at the case with two bidders vi and vj and bids bi and bj. When i gets the good, his surplus is vi - bj. • Now, if vi > bj, then i would like to get the item. How can he achieve this? He can simply bid bi = vi > bj.

26. Let us look at the case with two bidders vi and vj and bids bi and bj. When i gets the good, his surplus is vi - bj. • On the other hand, if vi < bj, then i would not like to get the item. How can he achieve this? He can simply bid bi = vi < bj. • Honesty is the best policy.

27. When v>bj, he wants bi> bj. • When v<bj, he wants bi< bj.

28. Goethe sold his poem to a publisher in this way. He wrote a reserve price, handed it to a lawyer. The publisher had to write a bit. If the bid was lower, the poem would not be sold. If the bid was higher, the poem would be sold at Goethe’s reserve price. • The publisher with a value v should bid b=?

29. When v>r, he wants the poem. • Bidding b> r will do. • When v<r, he does not want the poem. • Bidding b< r will do.

30. Does it run contrary to your intuitions? Why? • Let us first look at Vickery auctions in practice.

31. eBay introduces an automated bidding agent. • Users tell the bidding agent the most they are willing to pay for an item and an initial bid. As the bidding progresses, the agent automatically increases a participant’s bid by the minimal bid increment whenever necessary.

32. Essentially it is a Vickery’s auction. Each user reveals to their bidding agent the maximum price he or she is willing to pay. • In theory, the highest value bidder wins and pays the second highest value. And we have shown the honesty is the best policy.

33. In practice, we see late bidding. In one study, 37% of the auctions had bids in the last minute and 12 % had bids in the last 10 seconds.

34. Story one: if you are an expert on rare stamps, you may want to hold back placing your bid so as not to reveal your interest (the common value story).

35. Story two: two bidders (both valuations at 10) are bidding for a Pez (a sweet) dispenser. The seller’s reserve price is 2. If both bid early, then end up paying 10. If both bid 10 in the last possible seconds, then maybe one of the bid won’t go through, and the winner may end up paying only 2.

36. Escalation auction: The highest bidder wins but the highest bidder and the second highest bidders both have to pay the amount they bid. • A good way to earn some money in a party… • Lobbying may be an all-pay auction.

37. A position auction is a way to auction off positions such as an advertising position on a web page. • Everyone agrees that it is better to be in the top position, the second from the top and so on. However, different advertisers sell different things, so the expected profit that they will get from a visitor to their web page may differ.

38. Let us look at a simple case. • Suppose there are two slots where ads can be displayed and x1 (x2) denotes the number of clicks an ad can receive in slot 1 (2). • Assume that slot 1 is better than slot 2 so x1> x2.

39. Two advertisers bid for the two slots. • The highest bidder gets slot 1. The second highest bidder gets slot 2. • The price an advertiser pays is determined by the bid of the advertiser below him. • Suppose the reserve price is r and bm> bn>r.

40. Then bidder m gets slot 1 and pays bn per click. Bidder n gets slot 2 and pays r per click. In other words, an advertiser pays a price determined by the bid of the advertiser below him. • Let us look at any bidder i. When bi> bj, he gets slot 1 and his payoff is (v-bj) x1. On the other hand, when bi<bj, he gets slot 2 and his payoff is (v-r) x2.

41. Bidder i would like to get slot 1 (rather than slot 2) if and only if (v-bj) x1> (v-r) x2. This is equivalent to v(x1-x2)+rx2>bjx1. • When v(x1-x2)+rx2>bjx1, he wants bi> bj. • When v(x1-x2)+rx2<bjx1, he wants bi< bj. • Thus, bidder i could just bid bix1=v(x1-x2)+rx2.

42. When (v(x1-x2)+rx2)/x1>bj, he wants bi> bj. • When (v(x1-x2)+rx2)/x1<bj, he wants bi< bj. • When v>bj, he wants bi> bj. • When v<bj, he wants bi< bj. • Similar to the proof in Vickery auction.

43. But the proof breaks down when there are more bidders. • The logic is too specific and we would not go into details. Also mind that there are errors on page 343.

44. Let us now look at the expected revenue when the number of bidders increases in a second price auction. • Since everyone bids truthfully, suppose reserve price is 0, then expected revenue will be the expected value of the second-largest valuation.

45. As the number of bidders goes up, the expected value of the second-largest valuation goes up. • Shown below is the expected revenue if the values are distributed uniformly on [0,1].

47. By the time there are 10 or so bidders, the expected revenue is pretty close to 1, illustrating that auctions are a good way to generate revenue.

48. Problems with auctions: • On the buyer side, buyers may form bidding rings. • On the seller side, sellers may take bids off the wall (take fictitious bids). (Ask your employees to place bids!)

49. Turn to the common-value auction where the good that is being awarded has the same value to all bidders, for example, the auction of off-shore drilling rights. • The bidders have the same value, but they may have different estimates of that value.

50. Let us assume the estimated value is v+ei where v is the common value and ei is the error term associated with bidder i’s estimate. • To develop intuitions, let us see what happens when bidders bid their estimated values.