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Auction Theory

Auction Theory. Single Object. Auction Explosion of the Last Two Decades. Art Auctions Foreclosed properties and goods E-bay Fish, Flowers Stamps Bonds to fund government debt Portfolio’s of Mortgage Backed Securities. Common Auction Formats. Open Dynamic

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Auction Theory

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  1. Auction Theory Single Object

  2. Auction Explosion of the Last Two Decades • Art Auctions • Foreclosed properties and goods • E-bay • Fish, Flowers • Stamps • Bonds to fund government debt • Portfolio’s of Mortgage Backed Securities

  3. Common Auction Formats Open Dynamic • Price increases or decreases over time • English Auction (Sotheby’s, E-bay) • Dutch Auction(Fish, Tulips) Sealed Bid Static • Bidders submit sealed bids a the same time • First Price (Procurement) • Second Price (Stamp’s and other collectibles)

  4. Basic Auction Questions • Strategic Behavior – What are the optimal bidder strategies? Are these dominant strategies or part of a Nash Equilibrium? • Auction Design • Social Planner: Does the auction achieve a Pareto Optimal Outcome • Auctioneer: Does the auction maximize the auctioneers profit or utility?

  5. Why Auctions? • Auctions are used because the seller is unsure about the values that bidders attach to the object(s) being sold. • The (monetary) value of an object is the highest price a bidder would accept if he had all relevant information. • If the seller knew the bidders‘ valuations, she would make a take-it-or-leave-it offer to the bidder with the highest value. • If the seller does not know the values, she can exploit competition between bidders to raise the price.

  6. Valuation Paradigms • Private Values • Each bidder knows her valuation for the object. Different bidders in general will have different valuations. • Interdependent Values • Precise value is unknown to a bidder. It depends on own information as well as on information held by other bidders and/ or the seller. • Examples include: Cars, Used items of uncertain quality, spectrum licenses • Common Values • Special case of interdependent values where everyone has the same valuation, such as Oil fields

  7. English Auction with Private Values Oral version: • Auctioneer sets starting price • Bidders make verbal bids and raise the price • Auction ends if bidding stops, highest bidder wins and pays his bid Continuous version: • The price raises with time and all bidders press a button. • If a bidder releases his button, he quits the auction. • The auction ends immediately when there is only one bidder left. This last bidder wins. • The winner pays the price at which the auction ends.

  8. English Auction with Private Values Solution Dominant Strategy to exit the auction at your valuation: • A bidder keeps pressing the button if the price is below his valuation and there is at least one other bidder who still presses her button • As soon as a bidder is the only one left, he releases the button • If the price is above his valuation, the bidder releases the button, i.e. he quits the auction Outcome (continuous version): • The bidder with the highest valuation wins which is Pareto Optimal • The winner pays a price equal to the second-highest valuation

  9. Second Price Sealed Bid Auction – or the Vickery Auction Auction formulation: • All bidders submit a (sealed) bid to the auctioneer • The auctioneer opens all bids at the same time and awards the object to the highest bidder • The highest bidder pays the second highest bid Solution: The Dominant Strategy is always bid your value • Why does bidding your value weakly dominate bidding over your value? • Because the second highest bid could be between your value and your bid, and your payoff is negative. • Why does bidding your value weakly dominate bidding under your value? • Because the second highest bid will could be higher than your bid and but below your value, and your payoff is zero when it could have been positive

  10. Second Price Sealed Bid Auction Continued Outcome • The bidder with the highest valuation wins • The bidder pays a price equal to the second highest valuation Second Price Sealed Bid Auction and English Auction are strategically equivalent. • Consider the example with (v1, v2, v3) = (99, 21, 19) • SPSB price and English Auction price are both 21

  11. First Price Sealed Bid Auctions Auction formulation: • All bidders submit a (sealed) bid to the auctioneer • The auctioneer opens all bids at the same time and awards the object to the highest bidder • The highest bidder pays his bid There is no dominant strategy to always bid your value • Bidding your value yields a payoff of zero for all strategy profiles • Decreasing your bid below value increases the payoff when you when, at the same time decreases the probability you win. • So your best response depends your beliefs about the strategies and the distribution of other bidders’ values.

  12. First-price auction with private values • A game of incomplete information • A Strategy is a function the gives a bid for each possible valuation. bi(vi) • It is never optimal to submit a bid that is higher than the own valuation • If the second highest bid was known, it would be optimal to bid “just above” that bid • As bids depend on private information, the bids of other bidders are unknown at the time of bidding • A bidder needs to form a belief on other bidders‘ valuations and strategies • Given these expectations, a bidder can only bid optimal conditional upon beliefs about private information

  13. The symmetric independent private values model • The bidders’ values are independent draws from a uniform distribution on [v,v], example [0,1] • Each bidder i knows the distribution of valuations and the realization of his own valuation vt • Bidders are risk-neutral • The utility (payoff) of a bidder i who receives the object and has to pay t is given by vi – t

  14. Equilibrium of the first-price auction • Each bidder draws his value from an independent draw from the uniform distribution on [0, 1] • Suppose that there are 2 bidders and bidder 2 bids always a fraction f of his valuation, i.e. b2(v2) = av2. • • If bidder 1 bids above a he will win for sure, as b2(v2) = av2 < a if v2 <1 • • If bidder 1 bids below a he will win with probability • P(win) = P(b2<b1) = P(av2 <b1) = P(v2<b1/ a) = b1/a • Hence the expected payoff of such a bid is p1(b1(v1), b2(v2))=(v1- b1)P(win) = (v1 - b1)(b1/a) • This is maximized for b1 = v1/2, hence does not depend on a

  15. Equilibrium of the first-price auction • Therefore equilibrium strategy for bidder i is: bi(vi) = vi/2 • For the n bidder case the symmetric Nash Equilibrium strategy is bi(vi) = [(n-1)/n] vi • Behavioral interpretation: A bidder bids the expected valuation of the 2nd highest bidder conditional on his valuation being the highest. • Therefore equilibrium strategy for bidder i is: bi(vi) = vi/2 and vi/2 is the expected valuation of bidder j conditional on being lower than vi vi/2 vi 0 1

  16. First Price Auction Performance • If equilibrium bid functions are increasing in value, then the outcome is Pareto Efficient (the bidder with the highest value receives the object) • The seller expects to receive a price that is equal to the expected second highest valuation (More on this in a moment)

  17. Dutch Auction • Isomorphic to the First Price Sealed Bid Auction • Symmetric Nash Equilibrium where each bidders stopping price is bi(vi) = [(n-1)/n] vi • Once again Pareto Optimal • Expected price is equal to the expected second highest value. • Notice that the expected revenue is the same for all four auction types!

  18. Revenue Equivalence Theorem • Vickery (1962) and Myerson (1981) – In the symmetric private values case, the four basic auction formats have the same expected price. If the following are true • Sellers are risk neutral • Sellers’ costs are independent • Sellers are symmetric • Payment is a function of bid alone

  19. Intuition Behind Revenue Equivalence Example • Three bidders • Each draws a value according to a uniform distribution on [0, 20] • Expected value of order statistics 15, 10, and 5 • Expected Price is 10 and the support of prices is [0, 20]

  20. Intuition Behind Revenue Equivalence FPSB • NE bid strategy- submit a price that is the expected 2nd lowest value conditional upon your value is the highest • B(v) = (2/3)*v • Expected Price is 10 same as Reverse Auction • support of prices is [0, 13.33] • Lower price volatility Jason Shachat First Year Review

  21. Does Revenue Equivalence Really Hold?

  22. Does Revenue Equivalence Really Hold?

  23. Maximizing Revenue and Reserve Prices • Seller cares about maximizing revenue not Pareto optimality • What auction will maximize a seller’s expected revenue? • An English Auction with an optimal reserve price. Myserson 1981 • A reserve price is a minimum auction price the auctioneer will consider • Prevelant on E-bay and many other auctions

  24. Optimal Reserve Price example • Consider a seller wanting to sell an object he values at zero to a single bidder who value is determined by mother nature’s draw from a uniform distribution on [0, 1] • In an auction it is trivial the bidder bids zero, and captures all the gains from exchange • The auctioneer can set a minimum price it will accept p(reserve price r) = r* Prob(buyers value v > r) = r * (1 – r) • First order condition is 1 – 2 r = 0 or r = .5 • The sellers expected revenue is now .25 • Notice this is not Pareto Optimal because 50% of the time the object is not sold

  25. Common Value and the Winner’s Curse • In common value auctions, bidder’s typically must estimate the value of the object. The realized value will be the same for everyone, but the estimates will not. • We can think of the estimate as a draw from random variable whose expected value is equal to the true common value • Think of wildcat contractors deciding how to bid on the mineral rights to a land area • Every bidder has an unbiased estimate of the value, however the highest of the bidders estimate is going to be biased above the true value. • If bidder’s treat their estimate the same as they do in the private value case, then the winner of the auction will expect to lose money – ie the winner’s curse.

  26. The Winner’s Curse example • I have a twelve sided die • The expected value of a roll is 6.5 • Imagine the value of an unknown object is 6.5 • The roll of the die is an unbiased signal of this value • However, imagine that four different bidders each receive and independent signal (I,e, a roll of the die) • Let’s look at what the highest estimated value from 6 different auctions will be.

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