1. Chapter 18 Auctions
Key Concept: Honesty is the best policy in a private-value second price auction.
However in a common-value auction, 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.
Auctions to sell the right to drill in coastal areas and the FCC auctions to sell radio spectrum.
Let us first look at private-value auctions. At the end, we will mention briefly the common-value auctions.

3. Bidding rules
Open auctions
English auctions (ascending auction): Bidders successively offer higher prices until no participant is willing to increase the bid further.

4. Dutch auction (cheese and fresh flowers, descending auctions): The auctioneer starts with a high price and gradually lowers it by steps until someone is willing to buy the item.

5. Sealed-bid auction
First price (construction work)
Second price (philatelist auction or Vickery auction)

6. Assume there are n bidders with private values v1, v2, …, vn
Assume there are n bidders with private values v1, v2, …, vn. Suppose seller has a zero value for the object. We want to design an auction (mechanism) to meet our goal.
Two natural goals are Pareto efficiency and Profit maximization.

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

8. How can we achieve effiency
How can we achieve effiency? Note that the English auction will achieve this.
Suppose v1 =100 and v2=10. And the bid increment is 1. Then the winning price may be 11. (So the winner will pay the value of the second-highest bidder. Similar to the second price auction if bidders bid truthfully.)

9. Suppose (vi, vj)=(10,10), (10,100), (100,10), (100,100), each occurring with probability 1/4. Then the winning bid may be 10, 11, 11, 100. The expected revenue to the seller is 33=( )/4.
What if the seller sets the reserve price at say 100? The expected revenue would be ( )/4=75. It is not Pareto optimal since ¼ of the time no one gets the good!

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

11. Now let us turn to the second price sealed-bid auction
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.
But will bidders bid truthfully?

12. Let us look at the case with two bidders vi and vj and bids bi and bj
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.
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.

13. When v>bj, he wants bi> bj.

14. Does it run contrary to your intuitions? Why?
Vickery auctions in practice? 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 min bid increment whenever necessary.

15. Essentially it is a Vickery’s auction
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.
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.

16. 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).

17. Story two: two bidders (valuations at 10) are bidding for a Pez 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.

18. 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…
Analogous to international escalation?
Lobbying may be an all-pay auction.

19. A position auction is a way to auction off positions such as a position on a web page. 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.
Two advertisers bid for the two slots. The reserve price is r. Suppose bm> bn>r.

20. Then bidder m gets slot 1 and pays bn per click
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.

21. 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.

22. 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.
Notice how similar this is to the proof where we show that honesty is the best policy in Vickery auction.

23. But the proof breaks down when there are three bidders and three slots
But the proof breaks down when there are three bidders and three slots. The logic is too specific so I would not go into details. Also mind that there are errors on page 343.

24. As for the expected revenue when the number of bidders change 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.
As the number of bidders goes up, the expected value of the second-largest valuation goes up.

26. Shown is the expected revenue if the values are distributed uniformly on [0,1]. 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.

27. 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!)

28. Turn to the common-value auction where the good that is being awarded has the same value to all bidders (off-shore drilling rights).
Let us assume that 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.

29. The person with the highest value of ei or emax gets the good
The person with the highest value of ei or emax gets the good. But as long as emax>0, the bidder pays more than v, the true value of the good. This is called the winner’s curse.
So bidders should shade bids. Moreover, the more bidders there are, the lower you want your bid to be.
(Swing voter’s curse)

30. A slight detour to stable marriage problem (auctions are matching persons to goods). In general we could consider men matched to women, interns matched to hospitals, organ donors matched to recipients.

31. Consider n men and n women and we need to match them up as dancing partners. Each woman rank the men according to her preference and the same goes for the men. Suppose there are no ties and that everyone prefers to dance than to sit on the sidelines.

32. What is a good way to arrange for dancing partners
What is a good way to arrange for dancing partners? Is there a “stable” matching?
“Stable” here means no couple would prefer each other to their current partners.
The deferred acceptance algorithm could find a stable matching.

33. Step 1: Each man proposes to his most preferred woman.
Step 2: Each woman records the list of proposals she receives on her dance card.
Step 3: After all men have proposed to their most-preferred choice, each woman rejects all of the suitors except for her most preferred.
Step 4: The rejected suitors propose to the next woman on their lists.
Step 5: Continue to step 2 or terminate the algorithm when every woman has received an offer.

34. An example:
m1: w1 w w1: m1 m2
m2: w1 w w2: m2 m1
m w m1----w1
m w m w2
m1----w m1----w1
m w m2 ----w2, a stable match.

35. Auctions are examples of economic mechanisms
Auctions are examples of economic mechanisms. The idea is to design a game that will yield some desired outcome.
For instance, you may want to sell a painting. First of all, we need to make sure what your goal is (To max profit? To max efficiency?). Then we should think about which auction format (or game) may help you achieve that.

36. Thinking of things this way, mechanism design is the “inverse” of game theory. With game theory, we are given a game and we want to know what the equilibrium outcome will be. With mechanism design, we are given an outcome we want to achieve and we try to design a game so that that equilibrium of the game is the outcome.

37. Let us look at the Vickery auction again using this view.
The seller has an item and his goal is to award the item to the highest value person.
In this case, he can design a mechanism, which is the Vickery auction. Since we have shown that in Vickery auction, it is an equilibrium that bidders will bid truthfully, this equilibrium outcome will achieve what the seller wants.

38. Typically the agents participating in the mechanism will have some private information the mechanism designer does not know (for instance, in auctions, bidders know their values but the auctioneer does not). So the agents will report some message about their private information to the center.
The center then examines the messages and determines the outcome.

39. There are typically some constraints the center has to be aware of
There are typically some constraints the center has to be aware of. The resource constraint is obvious (for instance there is only one item to be sold), constraints about how agents will behave are less obvious but not hard to understand.

40. First, agents will act in their self-interest, this leads to the incentive compatible constraint (IC).
Second, agents will participate in the mechanism voluntarily. They get at least as high a payoff from participating as not participating. This leads to the individual rationality constraint (IR).

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

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