Introduction to Auctions

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1 Introduction to Auctions Lots of good theory and empirical work. game is simple with well-specified rules actions are observed directly payoffs can sometimes be inferred Also, lots of data. government sales (timber rights, mineral rights, oil and gas, treasury bills, spectrum auctions, emission permits, electricity) procurement (defense, construction, school milk) private sector (auction houses, agriculture, real estate, used cars, machinery) online auctions Many possible mechanisms. open versus sealed first-price versus second-price secret versus fixed reserve price 1

2 Four auction types: First-price sealed-bid auction: you don t see your opponents bids. Highest bid wins. Winner pays her bid, b. The winner s profit is: v b. Losers get nothing. Second-price sealed-bid auction: you don t see your opponents bids. Highest bid wins. Winner pays the second highest bid in the auction. Therefore the winner s profit is: v minus the second highest bid. Losers get nothing. 2

3 English, or Ascending-price drop-out auction: Sort of like the game of chicken. Price starts ticking upwards from 0 until...wherever. You re all paying the price until you decide to drop out! As soon as the second-to-last person drops out, the auction ends. Winner s profit is v minus the price at which the secondto-last guy dropped out. Dutch, or Descending price auction: Game of chicken in reverse! Price starts at $11, ticks down until somebody says I ll take it. The winner s profit is v minus the price at which she called I take it. 3

4 Bidding Process Open Closed Price Determination 2nd p 1st p Dutch * * English FPSB Vickrey (SPSB) * has had the most empirical work denotes strategic equivalence (under certain circumstances) Can also look at variations with multi-unit auctions and/or combinatorial auctions, with discriminatory or uniform prices. 4

5 Most often studied are FPSB or English auctions. (Recent ebay studies are an exception. They are like an English auction with a time constraint, and are often modeled as SPSB.) Goals of Theory positive: describe how to bid rationally (BNE) normative: characterize optimal or efficient selling mechanism Goals of Empirical Work positive: how do agents bid? are valuations correlated? if so, how? are observed bids consistent with BNE? are agents colluding? 5

6 normative: identify revenue-maximizing or efficient auction identify optimal reserve price Two kinds of approaches reduced form: test theory predictions, make inferences about behavior and bidding environment structural: assume the theory holds and estimate the DGP 6

7 The Usual Models (Common Values, Independent Private Values) Notation: r: reserve price n: number of potential bidders m: number of actual bidders U i : u(x i, v) utility function of bidder (i). V may or may not be observed by bidder. It gives heterogeneity across auction items. X i : private signal of bidder i F : CDF of (X 1,..., X n, V ) Y i : max X j, j i w: winning bid 7

8 Assumptions: 1) Each bidder wants only one object 2) u is non-negative, continuous and increasing 3) X i R (one-dimensional signals) 4) F is exchangeable across bidder indices (no bidder has more precise information than another) 5) (X 1,..., X n, V ) are affiliated (positively correlated) 6) F, n, u are common knowledge 8

9 Independent Private Values (IPV) U i = X i i = 1,...n X i are all i.i.d. F (X 1,..., X n ) = Π n i=1 F x i (x) Pure Common Value (CV) U i = V i = 1,...n F (X 1,..., X n, V ) Affiliated Private Values (APV) U i = X i i = 1,...n F (X 1,..., X n ) Affiliated Values (AV) U i = u(x i, V ) F (X 1,..., X n, V ) Also, Conditionally Independent PV and Conditionally Independent CV 9

10 The bottom line: Your signal is your pay- Private Values: off. (Other bidder s signals (X j s) are uninformative to you.) Common Values: Uncertainty about V. Bottom line: If you see the signals of the other bidders (or their bids) and it causes you to revise your own valuation of the object, then it is CV. Otherwise, it s PV. 10

11 Theoretical setup for IPV N bidders draw valuations, v i, from a joint distribution F (v 1,.., v N ) Simple setting: valuations are draw independently from the same distribution F (v). Let s take F (v) = v, i.e. distribution on [0, 1]. the uniform 11

12 Bidding equilibria for Second Price Sealed Bid auction: Claim: Regardless of what my competitors do, I can do no better than bidding my valuation. Reason 1: My payment does not depend on what I do. My payoff in every contingency is either I win, and make v i b 2 I lose, and make zero. My bid only affects whether I win or lose. 12

13 If I bid my value, I do not regret winning or losing in any contingency. If I bid below my value, then I don t win in some situations where I would want to win. If I bid above my value, I also win in some situations where I do NOT want to win. What is the expected revenue in 2nd price auction with N bidders, with uniform distributed valuations? N 1 Answer: N+1. Reasoning: N bidders with uniformly distributed valuations partition the interval of valuations to N + 1 equal segments; in expectation, of course. 13

14 Bidding equilibria for English or Ascending auction: Claim: Regardless of what my competitors do, I can do no better than dropping out at my valuation. Reason: My payment does not depend on what I do. If I drop out below my value, then I don t win in some situations where I would be willing to win. If I drop out above my value, well, I make a loss. What is the expected revenue in ascending price auction with N bidders, with uniform distributed valuations? 14

15 Bidding equilibria for First price Sealed Bid auction: Won t bid valuation. Will want to shade your bid. How to optimally shade? max b (v i b) Pr{win b} How to calculate Pr{win b}? 15

16 Nash equilibrium: Suppose everybody plays the strategy b(v), where b(v) is an increasing function in v. Let s look at the strategy, b i, of bidder i, who has valuation v i, assuming that all other bidders are using the strategy b(v). max bi (v i b i ) Pr{b(v j ) b i, j i} max bi (v i b i )F N 1 (b 1 (b i )) 16

17 Take derivative w.r.t. b i and set to zero F N 1 (b 1 (b i )) = (v i b i )(N 1)F N 2 (b 1 (b i ))f(b 1 1 (b i )) b (b 1 (b i )) Now, v i is not necessarily equal to b 1 (b i ), but now, let s see what the above optimality condition tells us about the shape of b(v) if bidder i also plays b(v) (i.e. we re looking at a symmetric equilibrium of the game). In this case b 1 (b i ) = v i, so v i b(v i ) = 1 F (v i )b (v i ) N 1 f(v i ) This is a first-order linear differential equation, with initial condition b(0) = 0. Solving this, we get: b(v i ) = v i vi 0 F N 1 (θ)dθ F N 1 (v i ) 17

18 With uniform distribution, F (v) = v. So: b(v) = N 1 N v Expected revenue: E[Revenue] = N 1 N E[max{v i}] Remembering the equal partitioning of the interval trick: E[Revenue] = ( N 1 N )( N N + 1 ) Observe that this is equal to the expected revenue from the second price auction! This was first observed by William Vickrey in 1961, and eventually got him a Nobel prize. 18

19 Bidding equilibria for Dutch or Descending price auction: Claim: This is entirely equivalent to the first-price auction. Why? Think of your strategy as pre-determining a price level at which you ll take the good. Will you update this pre-determined pricelevel during the auction? No, the only thing you see is that nobody has taken the good until now, but even if somebody takes it, it s too late! 19

20 Call this strategy b(v). But then the optimization problem solved by b(v) is exactly the same as we did in the firstprice auction. 20

21 Summary: Although we have 4 different auctions, theory says they all yield the same expected revenue to the auctioneer. Also, theory says you should bid in the same way in the second-price and ascending auction And the same way in the first-price and descending auction 21

22 Bidding Equilibria (in a nutshell) Private Values: Dutch Auction (First price, open bidding), FPSB: Your bid should shade down from your signal, and should be increasing in N. Example: X i = x. Suppose valuations have support [0, x] and are uniformly distributed. (Rule out lots of stuff like risk aversion...) Then, E(next highest x) = N 1 N x. Equlibrium bid: N 1 N x. 22

23 English Auction (Second price, open bidding): Bid just higher than the next highest bid, up to your valuation. Vickrey Auction (Second price, sealed bidding): Bid your valuation. (Winner pays secondhighest price.) No strategic behavior for English or SPSB auctions with private values. 23

24 Common Values: Bidders should shade down to account for the winner s curse: E(V X i = x, Y i < x) < E(V X i = x) Therefore, bidder should bid less than E(V X i = x). 24

25 Common Value Auctions Everybody derives the same value from the object, v. Unfortunately, we do not know the value of the object for certain. Simplest setting: We all have the same information about the object. So we bid E(v). Big problem with this: suppose there are two types of used cars, lemons (low v) and non-lemons (high v). 25

26 Non-lemons will be driven out of the market but then, nobody will want to bid! This is the classic adverse selection problem. Now let s consider a setting where there are two types of bidders Those who know what v is, and those who don t. Suppose there are two guys, one who knows whether v is high or low, the other doesn t. We run a second price auction. Let s say v is high with probability π. 26

27 Claim: the guy who knows, bids his valuation. The guy who doesn t, bids the low value of v. Corollary 1: The uninformed guy makes zero profit upon winning the auction. Corollary 2: The uninformed guy bids less than his expected value. Corollary 3: the expected revenue of the seller is less than E(v). 27

28 Real-life examples of the above phenomenon: 1. Oil drilling 2. Sniping: when would an informed bidder bid on ebay? 3. IPO discount 28

29 The Winner s Curse: In a common value auction, the winner of the auction has the highest estimate of the value of the good. Hence, the winner runs into the danger of overestimating the value of the good. The rational response by bidders is to reduce their bids by taking into account the fact that they win conditional on being the most optimistic. Claim 1: Bidders will act more conservatively when they face more competitors 29

30 Claim 2: Bidders will act more conservatively when they face higher uncertainty about the value of the object. Both of these claims have found some backing in empirical tests. 30

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