Intermediate Microeconomics Midterm (50%) (4/27) Final (50%) (6/22) Term grades based on relative ranking. Mon 1:30-2:00 ( 社科 757) or by appointment Course TA 何宗祐,

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Chapter 17 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.

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

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

Assume there are n bidders with private values v 1, v 2, …, v n. 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. Profit maximization is straightforward, the seller wants to get the highest expected profit.

Suppose v 1 >v 2 > …>v n. Then to achieve efficiency, the good should be sold to person 1. How can we achieve this? Note that the English auction will achieve this. Suppose v 1 =100 and v 2 =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.)

Suppose (v i, v j )=(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 about the seller setting the reserve price say 100? The expected revenue would be ( )/4=75.

This demonstrates that we might not be able to achieve the two goals (Pareto efficiency and Profit maximization) at the same time. Now let us turn to the second price 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?

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

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.

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.

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). 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.

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.

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 x 1 (x 2 ) denotes the number of clicks an ad can receive in slot 1 (2). Assume that slot 1 is better than slot 2 so x 1 > x 2. Two advertisers bid for the two slots. The reserve price is r. Suppose b m > b n >r.

Then bidder m gets slot 1 and pays b n 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 b i > b j, he gets slot 1 and his payoff is (v-b j ) x 1. On the other hand, when b i <b j, he gets slot 2 and his payoff is (v-r) x 2.

Bidder i would like to get slot 1 (rather than slot 2) if and only if (v-b j ) x 1 > (v-r) x 2. This is equivalent to v(x 1 -x 2 )+rx 2 >b j x 1. When v(x 1 -x 2 )+rx 2 >b j x 1, he wants b i > b j. When v(x 1 -x 2 )+rx 2 <b j x 1, he wants b i < b j. Thus, bidder i could just bid b i x 1 =v(x 1 - x 2 )+rx 2. Notice how similar this is to the proof where we show that honesty is the best policy in Vickery auction.

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).

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

The person with the highest value of  i or  max gets the good. But as long as  max >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.

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.

Thinking of things this way, mechanism design is pretty much 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.

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.