Advanced Microeconomics II

Transcription

1 Advanced Microeconomics Auction Theory Jiaming Mao School of Economics, XMU

2 ntroduction Auction is an important allocaiton mechanism Ebay Artwork Treasury bonds Air waves

3 ntroduction

4 Common Auction Formats Open ascending price or English auction Auctioneer begins by calling out low price and raises it in small increments Auction ends when there is only one remaining bidder Open descending price or Dutch auction Descending counterpart to English auction

5 Common Auction Formats First-Price, Sealed Bid Auction Bidders submit bids in sealed envelopes At a pre-determined time, auctioneer opens all envelopes and ranks bids Highest bidder obtains object and pays his bid amount Second-Price, Sealed Bid Auction Same as first-price sealed bid auction except that highest bidder obtains object and pays second highest bid amount

6 Common Auction Formats All-Pay auction Bidders submit bids (open or closed) At a pre-determined time, auctioneer opens all envelopes and ranks bids Every bidder pays what they bid regardless of whether or not they have the highest bid Highest bidder obtains object Examples: Elections, almost any kind of contest or sports, R&D, wars, lobbying

7 Auctions All auctions can be interpreted as allocation mechanisms with the following ingredients: An allocation rule who gets the object Apaymentrule how much every bidder pays

8 Private Valuations Each bidder knows her own valuation v i,butnototherbidders valuations The distribution of i s valuation v i is common knowledge: F i (x) =Pr (v i < x)

9 Private Valuations v U (0, 1)

10 Private Valuations Let b i :[0, 1]!< + denote i s bidding function We will also use bi to denote i s actual bid f all bidders have the same valuation distribution F,thenthey will be using the same bidding function b They might, however, submit different bids, depending on their privately observed valuation f b (v) = v 2,thenbidderwithvaluation4willsubmit$2. Bidder with valuation 10 will submit $5 Even if bidders are symmetric in the bidding function they use, they can be asymmetric in the actual bid they submit.

11 First-Price Auctions b i < v i Bid shading : no point in bidding bi v i Assume v i U (0, 1) 8i We will be looking for equilibrium bidding functions of the form b i (v i )=a v i, a 2 (0, 1) We will prove later when we look at the general case (without the assumption of v i U (0, 1)) thattheonlysymmetric equilibrium bidding function in FPA under uniform value distribution is indeed of this form

12 FPA with 2 bidders ) EU i (b i v i ) = Pr (b i > b j )(v i b i ) = Pr (b i > av j )(v i b i ) bi = F (v i b i ) a = b i a (v i b i ) b i = arg max EU i (x v i ) x = v i 2

13 FPA with 2 bidders Bid shading in half Optimal bidding function with N = 2bidders

14 FPA with N bidders EU i (b i v i ) = Pr (b i > max {b i })(v i b i ) = Y j6=i Pr (b i > b j )(v i b i ) ) = = apple F apple bi a N 1 bi (v i b i ) a N 1 (v i b i ) b i = arg max EU i (x v i ) x = N 1 N v i

15 FPA with N bidders Optimal bidding function: b (v) = N 1v N Bid shading diminishes as N increases Optimal bidding function increases in N

16 FPA - Generalization Arbitrary distribution of v i b i = arg max x = arg max x = arg max x Pr (b i > max {b i })(v i b i ) Y F j b 1 j (x) (v i x) j6=i F b 1 (x) N 1 (vi x) FOC: F b 1 (x ) N 1 +(N 1) F b 1 (x ) N 2 f b 1 (x ) 1 b 0 (b 1 (x )) (v i x ) = 0

17 FPA - Generalization n symmetric equilibrium, b i (v i )=b (v i )=x, ) b i = 1 F N 1 (v i ) ˆ vi = v i 1 F N 1 (v i ) 0 udf N ˆ vi F N 0 1 (u) 1 (u) du When F (v) =v, b i = N 1v N i

18 FPA - Generalization Theorem f N bidders have independent private values drawn from the common distribution F, then bidding b i (v i )= 1 F N 1 (v i ) ˆ vi 0 udf N 1 (u) is the symmetric NE of a first-price, sealed-bid auction.

19 FPA with risk-averse bidders Utility function is concave in income: u (c) =c, 2 (0, 1) Expected utility of bidding: EU i (b i v i )=Pr (b i = max {b 1,...,b N })(v i b i )

20 FPA with risk-averse bidders Consider 2 bidders with v i U (0, 1) 8i b i = arg max x = v i 1 + x a (v i x) When = 1 (risk-neutral), b i = v i 2

21 FPA with risk-averse bidders Optimal bidding function: b (v) = v 1+ Bid shading is ameliorated as bidders risk aversion increases Optimal bidding function with risk-averse bidders

22 FPA with risk-averse bidders ntuition: for a risk-averse bidder: the positive effect of slightly lowering his bid, arising from getting the object at a cheaper price, is offset by... the negative effect of increasing the probability that he loses the auction. Ultimately, the bidder s incentives to shade his bid are diminished.

23 Second-price Auctions Let h i max j6=i {b j } EU i (b i v i, h i ) = Pr (b i > h i )(v i h i )

24 Second-price Auctions b i (v i )=v i EU i = Pr (v i > h i )(v i h i )

25 Second-price Auctions b i (v i ) < v i EU i = Pr (b i > h i )(v i h i ) < Pr (v i > h i )(v i h i )

26 Second-price Auctions b i (v i ) > v i EU i = Pr (b i > h i )(v i h i ) = Pr (v i > h i )(v i h i ) Pr (v i < h i < b i )(h i v i ) < Pr (v i > h i )(v i h i )

27 Second-price Auctions Bidding b i (v i )=v i is a weakly dominant strategy. Alternative argument: Since you pay hi when you win, the optimal strategy is one that guarantee you will win when v i > h i and you will lose when v i < h i This strategy is to bid vi The result is unaffected by number of bidders N their risk-aversion preferences their valuation distributions F bidders need not be symmetric: F i (v i ) can differ from F j (v j )

28 Second-price Auctions Theorem f N bidders have independent private values, then bidding one s value is the unique weakly dominant bidding strategy for each bidder in a second-price, sealed-bid auction.

29 Dutch Auctions Bidder decides at what price to raise hand. f wins, bidder pays the price at which he/she raises hand. Dutch auction is equivalent to FPA. Theorem f N bidders have independent private values drawn from the common distribution F, then raising one s hand when the price reaches 1 F N 1 (v i ) ˆ vi 0 udf N 1 (u) is the symmetric NE of a Dutch auction.

30 English Auctions Bidder decides at what price to drop out. Winner pays the price at which last remaining competitor drops out, i.e. winner pays the second highest bid. English auction is equivalent to SPA. Theorem f N bidders have independent private values, then dropping out when the price reaches one s value is the unique weakly dominant bidding strategy for each bidder in an English auction.

31 Revenue Comparisons n First-price and Dutch auctions, bidders bid less than their valuations but the seller receives the highest bid. n Second-price and English auctions, bidders bid their valuations but the seller receives the second highest bid. Which formats have the highest expected revenue? n o [1] Let v N max {v 1,...v N } and v [2] N max {v 1,...v N }\ v [1] N [1] When v N v [2] N, FPA can yield more revenue than SPA [1] When v N, v [2] N are close, SPA can yield more revenue than FPA

32 Revenue Comparisons Distribution of v [1] N F [1] N (v) =F N (v) Distribution of v [2] N n o F [2] N (v) =Pr v [2] N apple v

33 Revenue Comparisons n o The event v [2] N apple v can occur in one of two distinct (mutually exclusive) ways 1 All valuations below v: v i apple v 8i 2 N 1valuationsbelowv and 1 valuation above v This event can occur in N different ways 1 v 1 > v and v i apple v 8i 6= 1 2 v 2 > v and v i apple v 8i 6=

34 Revenue Comparisons Hence, F [2] N (v) = Pr n v [1] N apple v o + NX Pr {v 1 > v and v j apple v 8j 6= i} i=1 = F N (v)+n (1 F (v)) F N 1 (v) = NF N 1 (v) (N 1) F N (v)

35 Revenue Comparisons Assuming v is distributed according to F (v) on [0, 1], the expected revenue of FPA E r FPA = ˆ 1 0 ˆ 1 b (v) df [1] N apple (v) ˆ v 1 = udf N 1 (u) df N (v) 0 F N 1 (v) 0 ˆ 1 appleˆ v = N (N 1) uf N 2 (u) f (u) du f (v) dv = N (N 1) = N (N 1) 0 0 ˆ 1 ˆ 1 0 ˆ 1 0 u uf N 2 (u) f (u) f (v) dvdu uf N 2 (u) f (u)(1 F (u)) du

36 Revenue Comparisons The expected revenue of SPA: E r SPA = ˆ 1 0 vdf [2] N (v) = N (N 1) = E r FPA ˆ 1 0 vf N 2 (v) f (v)(1 F (v)) dv

37 Revenue Comparisons When v i U (0, 1), E r FPA = = E r SPA = ˆ 1 b (v) df [1] N (v) =ˆ 1 0 ˆ 1 vdf [2] 0 = N 1 N + 1 N 1 0 N v Nv N 1 (v) ˆ 1 N (v) =N (N 1) v N 1 (1 v) dv 0

38 Revenue Equivalence Principle Theorem Suppose that values are independently and identically distributed and all bidders are risk-neutral. Then any symmetric and increasing equilibrium of any standard auction, such that the expected payment of a bidder with value zero is zero, yields the same expected revenue to the seller. An auction is called standard if the rules of the auction dictate that the person who bids the highest is awarded the object

39 Revenue Equivalence Principle Proof. Let P (v) be the equilibrium expected payment by a bidder with value v. Nowconsiderbidderi. Supposeotherbiddersarefollowing the equilibrium bidding strategy b (v). Bidderi needs to decide whether to bid b (z) instead of the equilibrium bid b (v i ) Bidder i solves,whereg (z) F (z) N 1 max {G (z) v i P(z)} z

40 Revenue Equivalence Principle Proof. (Cont.) FOC) n symmetric equilibrium, P 0 (v i ) g (z) v i P 0 (z) =0 = g (v i ) v i ˆ vi ) P(v i ) = ug (u) du 0h i = E v [1] i v i > v [1] i Pr v i > v [1] i,wherev [1] i denotes the highest value among v i. Since the equilibrium expected payment function P (v) does not depend on any particular auction format, we prove the theorem.

41 Revenue Equivalence Principle Example (FPA) Since the equilibrium h expected i payment is P (v) =E v [1] i v i > v [1] i Pr v i > v [1] i and the winning bidder pays what she bids, the equilibrium bidding strategy must be h i b (v i ) = E v [1] i v i > v [1] i = ˆ vi 1 udg (u) G (v i ) 0 ˆ 1 vi udf N 1 (u) F N 1 (v i ) 0

42 Efficiency in Auctions The object is assigned to the bidder with the highest valuation. Otherwise, the outcome of the auction cannot be efficient, since there exist alternative reassignments that would still improve welfare. FPA, SPA, Dutch and English auctions are hence efficient, since the player with the highest valuation submits the highest bid and wins the auction. Lottery auctions are not necessarily efficient.

43 Reference The lecture slides draw from the following sources Jehle, G. A. and P. J. Reny Advanced Microeconomic Theory, Prentice Hall, 3e. Krishna, V Auction Theory, Academic Press, 2e. Munoz-Garcia, F Advanced Microeconomic Theory, lecture notes at