1. KEY CONCEPTS: (20 points - 5 points each) These questions are aimed to test how many of the core concepts stated in class you have interi- orized and are able to use comfortably. You should give...

1. KEY CONCEPTS: (20 points - 5 points each)

These questions are aimed to test how many of the core concepts stated in class you have interi- orized and are able to use comfortably. You should give clear definitions and explanations when you answer the questions.

1. (a) Is the following statement true? “5 bidders with private values uniformly distributed be- tween 0 and 1 enter a 1st price auction. Assuming that everyone is playing the symmetric equilibrium bidding strategy, the optimal bid for a bidder who makes a draw of 0.75 is 0.7.”

2. (b) A ticket to a newly staged show is on sale through a second-price sealed-bid auction. There are three bidders, A, B and C. A values the ticket at $10, B at $20, and C at $30. The bidders are free to submit a bid of any positive amount. Show that “everyone bids his or her own valuation” is a Nash Equilibrium.

3. (c) What is the winner’s curse? When does it occur? Provide examples.

4. (d) Is it true that in platforms, when one group multihomes and the other singlehomes, the platform “favors” the group that is multihoming? Is yes, why? If not, why not?

2. PRESENTATIONS: (20 points - 10 points each)

   1. (a) What evidence presented in the reading of Thaler is not supporting the assumption of rationality?

   2. (b) What is the empirical evidence about reserve prices? Does it support or contradicts the theory seen in class?

3. QUANTITATIVE EXERCISES: (60 points)

   These questions are aimed to test how comfortable you are in solving quantitative problems on the three topics seen in class during the semester. State clearly your answer in a box.

   1. (a) (20 points) Suppose a two-bidder asymmetric independent private value first-price sealed bid auction in which the first bidders value is 400, the second bidders value is either 300 or 800, each with probability one-half, the seller has an announced reservation price of 300, and the auction rules are that in the case of ties, then among those tied, the item goes first to the first bidder, second to the second bidder, and last to the seller. Show that in this case, strategies in which each bidder bids his conditional second value is not a Nash equilibrium.

   2. (b) (20 points) Consider a first-price sealed bid auction of a single object with two bidders j = 1,2 and no reservation price. Bidder 1’s valuation is v1 = 2, and bidder 2’s valuation is v2 = 5. Both v1 and v2 are known to both bidders. Bids must be in whole dollar amounts. In

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the event of a tie, the object is awarded by a flip of a fair coin. Is there a Nash equilibrium? What is it? Is it unique? Is it efficient?

(c) (20 points) Consider a Common Value auction with two bidders who both receive a signal

X that is uniformly distributed between 0 and 1. The (common) value V of the good

the players are bidding for is the average of the two signals, i.e. V = (X1+X2). Compute 2

the symmetric Nash equilibrium bidding strategy for the second-price sealed-bid auction assuming that players are risk-neutral and have standard selfish preferences. Furthermore, you may assume that the other bidder is following a linear bidding strategy. Make sure to explain your notation and the steps you take to derive the result.