Homework 3 ECON 490 (J1): Auctions Due November 9, 2015 Instructions: Answer these questions neatly and show your work. Do not try to cram your answers onto a printout of these questions; use separate sheets of paper. Risk Aversion 1. Would bidders with the following utility functions be risk averse, or not? (a) u(x) = x 1 4 (b) u(x) = x 3 (c) u(x) = log(x) (d) u(x) = 5x 2. Derive the equilibrium bidding function in a first price auction with two risk-averse bidders, each of whom are maximizing their utility function u(x) = √ x. (a) First, write out the bidder’s expected profit function. (b) Find the first order condition (by taking the derivative with respect to bi and setting equal to zero), a solve for the optimal bid. Reserve Prices 3. Consider a Vickrey auction in which there are two bidders, whose valuations are drawn from the unit uniform distribution. Assume that the bidders play their dominant strategies. Give examples of scenarios for the bidders’ valuations such that: (a) The seller’s revenue is higher if he sets a reserve price b0 = 0.5 than if he sets no reserve price. (b) The seller’s revenue is lower if he sets a reserve price b0 = 0.5 than if he sets no reserve price. (c) The seller’s revenue is the same, whether he sets a reserve price b0 = 0.5 or he sets no reserve price. 1 4. Use the spreadsheet “Reserve Price Auction Simulations” to (a) Record the average auction revenue obtained by the seller in the 100 auction simulations at the following reserve prices: 0, .2, .5, .7, .9. If you wish, you may use Table 1 on the last page of this assignment to record your answers. (b) What reserve price maximizes the seller’s auction revenue, on average, with 2 bidders? What reserve price maximizes the seller’s auction revenue, on average, with 8 bidders? Does the number of bidders affect the optimal reserve price? (c) Now, change the value of the item to the seller from 0 to 0.3. (This value is in cell C3 in each spreadsheet.) When the item has some value to the seller, the seller is not concerned only with the expected auction revenue, but the net auction revenue (the price paid by the winning bidder minus the value of the item). Record the average net auction revenue obtained by the seller in the 100 auction simulations at the following reserve prices: 0, .2, .5, .7, .9. If you wish, you may use Table 2 on the last page of this assignment to record your answers. (d) What reserve price maximizes the seller’s net revenue, on average? How does it compare to the optimal reserve price when the item has no value to the seller? Spite and Behavioral Economics 5. Derive the equilibrium bidding function in a first price auction with two spiteful bidders. Assume that each bidder’s payoff from bidding bi against bj is: ∆i = ( vi − bi if bi > bj −(vj − bj ) if bj > bi (a) First, write out the bidder’s expected profit function. (b) Find the first order condition (by taking the derivative with respect to bi and setting equal to zero), and solve for the optimal bid. 6. Come up with an alternative way to model “spiteful” bidders, using a different utility function than the one used in the previous problem. (There is not one correct answer to this question, it is an opportunity to be creative!) Common Value Auctions 7. (a) Consult the data from our common value auction experiment. (The excel spreadsheet is available in the assignments folder.) For each auction that was conducted in class, record the following information: • The true value of the item (number of pennies in the jar) • The average guess of the bidders • The average bid of the bidders • The percentage of bidders whose bid was greater than their guess • The percentage of bidders whose bid was equal to their guess • The percentage of bidders whose bid was less than their guess • The highest bid in the auction • The price (the second highest bid in the auction) • The winner’s profit (the true value minus the price) You may record these results in your own table, or use the table on the last page and attach it to your homework submission. (b) Discuss bidders’ estimates: were the participants’ guesses about the items’ true values accurate, on average? How would this affect the bidders’ optimal strategy? (c) Discuss bidders’ strategies: Did any bidders change their strategies from one auction to the next? Describe any patterns in bidder’s behavior that you observe in the data. (d) Discuss auction outcomes: is there evidence for a winner’s curse? 8. Incentives to participate in a common value auction. Bidders can end up paying more than the true value of the item in an auction with common values. Does this mean that bidders should never participate in a common value auction? To answer that question, consider the following: (a) If every student in the classroom had submitted a bid of zero in one of our common value auctions, what is each bidder’s expected payoff? (b) In that scenario, would it be possible for a bidder to increase his expected payoff by changing his bid? (c) Conclude with an explanation of why it is not a Nash equilibrium for all bidders not to participate. 9. Conditional expectation of item’s true value in a common value auction. A bidder receives a signal vi = V + i , where V is the auctioned item’s true value, and i is drawn uniformly from the interval (−5, 5). Note that the cdf of i is F(x) = x−5 10 and the pdf is f(x) = 1 10 . You can refer to our notes on first order statistics to help you calculate the first order statistic of the highest draw i from a sample of n draws. What is the expected value of V, conditional on… (a) The bidder learning that her signal is the highest signal of the 11 total bidders? (b) The bidder learning that her signal is the highest signal of the 33 total bidders? Remember to show the steps you use to get your answer. 10. In a common value auction, concern about the winner’s curse should influence bidders to reduce their bids below what they would have bid in a private value setting. Will the amount of the reduction increase, or decrease, if there are more bidders participating in the auction? In other words, does the winner’s curse get better, or worse, with more bidders? Table 1: Effect of Reserve Prices when item has no value to the seller Reserve Price 0 .2 .5 .7 .9 Average Revenue with n = 2 bidders Average Revenue with n = 8 bidders Table 2: Effect of Reserve Prices when item has value of 0.3 to the seller Reserve Price 0 .2 .5 .7 .9 Average Net Revenue with n = 2 bidders Average Net Revenue with n = 8 bidders Table 3: Common Value Auction Experiment Results Round/Group 1A 1B 2A 2B 3A 3B 4 True Value Average Guess Average Bid % Bidders Bid > Guess % Bidders Bid = Guess % Bidders Bid < Guess High Bid Price Winner’s Profit
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