# The Sealed Bid Auction Experiment:

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2 Figure 1: An example of a menu 2

3 Example: Suppose that Bidder 1 bids for package ABCD at price 70 and for AB at price 50, Bidder 2 bids for package CD at price 40, and Bidder 3 bids for CD at price 30, as summarized in Table 1. Table 1: Submitted bids. Bidder 1 ABCD 70 Bidder 2 CD 40 Bidder 3 CD 30 In this example, Bidder 1 bids for items A and B twice, in package ABCD and package AB. However, the auctioneer cannot sell items A and B to Bidder 1 twice, as she has only one unit of each, so Bidder 1 cannot get both ABCD and AB. Bidder 1 might get ABCD, or AB, or nothing. Winning bid determination Once all bidders have submitted their bids, the auctioneer will choose the combination of submitted bids that yields the highest revenue. The auctioneer only has one unit of each item to sell, and will never give more than one package to each bidder. For example, the auctioneer will not give both AB and D to Bidder 1, because Bidder 1 can only win one package, and the auctioneer will not give AB to Bidder 1 and B to Bidder 2, since she only has one unit of B to sell. Example: In the example presented above, we will examine how the auctioneer would choose winning bids. Consider the three possible assignments of the four goods: 1). Sell package ABCD to Bidder 1. The revenue is 70. Bidder 1 ABCD 70 2). Sell package AB to Bidder 1 and CD to Bidder 2. The revenue is 90. Bidder 2 CD 40 3). Sell package AB to Bidder 1 and CD to Bidder 3. The revenue is 80. Bidder 3 CD 30 As the 2nd assignment yields the highest revenue ( 90), the winning bids are Bidder 1 s bid for AB and Bidder 2 s bid for CD. If two or more combinations tie for the highest revenue, one of them will be selected based on the following criteria, which are used in decreasing order of importance: 1 An assignment maximizing the number of winners; 2 An assignment chosen randomly using a fair lottery. 3

4 The auctioneer will then, one at a time, choose each of the winning bidders as a pivotal bidder. The auctioneer examines the bids again, but excludes (ignores) the bids of the pivotal bidder. The auctioneer determines the allocation of goods that maximizes revenue, using the same rules as before, but not considering any bids placed by the pivotal bidder. Once this new allocation has been determined, the auctioneer compares the revenue generated by this allocation with the revenue generated when no bids were excluded. Example: example: Suppose that bidders submit the following bids, just as in the previous Bidder 1 ABCD 70 Bidder 2* (pivotal bidder) CD 40 Bidder 3 CD 30 Let Bidder 2 be the pivotal bidder. Thus, the auctioneer will exclude all of Bidder 2 s bids when she considers all possible assignments. 1). Sell package ABCD to Bidder 1. The revenue is 70. Bidder 1 ABCD 70 2). Sell package AB to Bidder 1 and CD to Bidder 2. The revenue is 90. Bidder 2 CD 40 3). Sell package AB to Bidder 1 and CD to Bidder 3. The revenue is 80. Bidder 3 CD 30 The 2nd assignment is not available, because the auctioneer should exclude the bid of the pivotal bidder, Bidder 2. In this case, the 3rd assignment yields the highest revenue. Therefore, when Bidder 2 is pivotal, the highest revenue that the auctioneer can obtain is 80. Price Adjustments The amount that the winning bidders are required to pay at the end of the auction depends on the additional revenue that each bidder generated, which is calculated by comparing the original revenue obtained by the auctioneer, versus the revenue obtained by the auctioneer when the given bidder was pivotal. Example: Suppose that the following bids are submitted: status winning Bidder 2 CD 40 winning Bidder 3 ABCD 60 Bidder 3 AB 30 4

7 Review Questions: Remember that you are free to use the instructions to answer these questions, and that we will pay you for your correct answers. Q.1 Winning Bids Let us find winning bids, given the following submitted bids. Bidder 1 A 50 Bidder 2 B 20 Bidder 2 AB 60 Bidder 3 CD 50 Hint: There are 2 assignment patterns. These assignments are marked below, with the winning bids marked with an asterisk (*). Which assignment gives the highest revenue? Bidder 1* A 50 Bidder 1 A 50 Bidder 2* B 20 Bidder 2 B 20 Bidder 2 AB 60 Bidder 2* AB 60 Bidder 3* CD 50 Bidder 3* CD 50 Revenue =. Revenue =. Which of the following is the set of winning bids? a) Bidder 1 s A, Bidder 2 s B and Bidder 3 s CD. b) Bidder 2 s AB and Bidder 3 s CD. Q.2 Price adjustment and Payments Suppose that the auction terminates with the following bids. Bidder 1 (winning) A 15 Bidder 2 AB 15 Bidder 2 ABC 30 Bidder 3 (winning) BCD 40 Bidder 3 D 20 Q.2.1 How much is the revenue to the auctioneer without any price adjustment? The revenue to the auctioneer is + = 7

8 Q.2.2 How much is the price adjustment to Bidder 1? When Bidder 1 was a pivotal bidder and excluded, the winning bids were Bidder 2 s bid on ABC and Bidder 3 s bid on D. They were indicated by the asterisks * in the table below. Q.2.3 Bidder 1 (winning) A 15 Bidder 2 AB 15 Bidder 2* ABC 30 Bidder 3 (winning) BCD 40 Bidder 3* D 20 If Bidder 1 s bid was ignored, then the highest revenue would be + =. Then, subtract the amount from 55. That is, 55 =. This is the price adjustment to Bidder 1. How much is the final price that Bidder 1 pays in the end? Hint: He pays the price and then receives the price adjustment. He pays as the price for A and receives as the price adjustment. Thus, Bidder 1 s final price is =. Q.3 Value Determination Which of the following is true? a) My value of AB is more than the sum of my values of A and B. b) My value of CD is more than the sum of my values of C and D. c) Any package that contains AB is worth more to me than any package that does not contain AB. d) None of the above 8

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