Auctions: Types and Equilibriums

Transcription

1 Auctions: Types and Equilibriums Emrah Cem and Samira Farhin University of Texas at Dallas April 25, 2013 Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

2 Common Auction Forms English Auction: In one variant of the English auction, the sale is conducted by an auctioneer who begins by calling out a low price and raises it, typically in small increments, as long as there are at least two interested bidders. Dutch Auction: In Dutch auction the auctioneer begins by calling out a price high enough so that presumably no bidder is interested in buying the object at that price. This price is gradually lowered until some bidder indicates her interest. The object is then sold to this bidder at the given price. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

3 Common Auction Forms First-price Auction: The sealed-bid first-price auction is another common form where bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object and pays what he bid. Second-price Auction: In sealed-bid second-price auction bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object but pays not what he bid but the second-highest bid. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

4 Equivalent Auctions The Dutch open descending price auction is strategically equivalent to the first-price sealed-bid auction. When values are private, the English open ascending auction is also equivalent to the second-price sealed-bid auction, but in a weaker sense. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

5 Private Value Auctions We will analyze equilibrium bidding behavior in the two common auction forms : First-price Auctions and Second-price Auctions in an environment with independently and identically distributed private values. It is sufficient to consider the two sealed-bid auctions. Each auction format determines a game of incomplete information among the bidders and, keeping the informational environment fixed, we determine a Bayesian-Nash equilibrium for each resulting game. When there are many equilibria, we usually select one on some basis dominance, perfection, or symmetry. We will typically be interested in comparing the outcomes of a symmetric equilibrium an equilibrium in which all bidders follow the same strategy of one auction with a symmetric equilibrium of the other. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

6 The Symmetric Model There is a single object for sale, and N potential buyers are bidding for the object. Bidder i assigns a value of X i to the object the maximum amount a bidder is willing to pay for the object. Each X i is independently and identically distributed on some interval [0, ω] according to the increasing distribution function F. It is assumed that F admits a continuous density f F and has full support. A strategy for a bidder is a function β i : [0, ω] R +, which determines his or her bid for any value. The distribution F is common knowledge, as is the number of bidders. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

7 Assumptions Independence - the values of different bidders are independently distributed. Risk neutrality - all bidders seek to maximize their expected profits. No budget constraints - all bidders have the ability to pay up to their respective values. Symmetry - the values of all bidders are distributed according to same distribution function F. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

8 Second-price Auctions In a second-price auction, each bidder submits a sealed bid of b i, and given these bids, the payoffs are: x i max j i b j if b i > max j i b j Π i = 0 if b i < max j i b j If there is a tie, so b i = max j i b j, the object goes to each winning bidder with equal probability. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

9 Second-price Auctions Proposition 1 In a second-price sealed-bid auction, it is a weakly dominant strategy to bid according to β II (x) = x. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

10 Second-price Auctions It should be noted that the argument in Proposition 1 relied neither on the assumption that bidders values were independently distributed nor the assump- tion that they were identically so. Only the assumption of private values is important, and proposition holds as long as this is the case. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

11 Second-price Auctions Proof: Consider bidder 1, say, and suppose that p 1 = max j 1 b j is the highest competing bid. By bidding x 1, bidder 1 will win if x 1 > p 1 and not if x 1 < p 1 (if x 1 = p 1, bidder 1 is indifferent between winning and losing). Suppose, however, that he bids an amount z 1 < x 1. If x 1 > z 1 p 1, then he still wins, and his profit is still x 1 p 1. If p 1 > x 1 > z 1, he still loses. However, if x 1 > p 1 > z 1, then he loses, whereas if he had bid x 1, he would have made a positive profit. Thus, bidding less than x 1 can never increase his profit but in some circumstances may actually decrease it. A similar argument shows that it is not profitable to bid more than x 1. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

12 Second-price Auctions Let us ask how much each bidder expects to pay (N-1) in equilibrium. Fix a bidder say, 1 and let the random variable Y 1 Y 1 denote the highest value among the N - 1 remaining bidders. In other words, Y 1 is the highest-order statistic of X 2, X 3,..., X N. Let G denote the distribution function of Y 1. Clearly, for all y, G(y) = F (y) N 1. In a second-price auction, the expected payment by a bidder with value x can be written as m II (x) = Prob[Win] E[2nd highest bid x is the highest bid] = Prob[Win] E[2nd highest value x is the highest value] = G(x) E[Y 1 Y 1 < x] Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

13 First-price Auctions In a first-price auction, each bidder submits a sealed bid of b i, and given these bids, the payoffs are: x i b i if b i > max j i b j Π i = 0 if b i < max j i b j If there is more than one bidder with the highest bid, the object goes to each such bidder with equal probability. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

14 First-price Auctions Suppose that bidders j 1 follow the symmetric, increasing, and differentiable equilibrium strategy β I β. Suppose bidder 1 receives a signal, X 1 = x, and bids b. We wish to determine the optimal b. It can never be optimal to choose a bid b > β(ω) We will only consider bids b β(ω). Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

15 First-price Auctions Bidder 1 wins the auction whenever he submits the highest bid that is, whenever max i 1 β(x i ) < b. Since β is increasing, max i 1 β(x i ) = β(max i 1 X i ) = β(y 1 ) where, as before, Y 1 Y (N 1) 1, the highest of N - 1 values. Bidder 1 wins whenever β(y 1 ) < b or equivalently, whenever Y 1 < β 1 (b). His expected payoff is therefore G(β 1 (b))x(x b) where, again, G is the distribution of Y 1. Maximizing this with respect to b yields the first-order condition: g(β 1 (b)) β (β 1 (b)) (x b) G(β 1 (b)) = 0 Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

16 First-price Auctions At a symmetric equilibrium, b = β(x), and putting the value in the differential equation yields, G(x)β (x) + g(x)β(x) = xg(x) or equivalently, d (G(x)β(x)) = xg(x) dx and since β(0) = 0, we have β(x) = 1 yg(y)dy G(x) 0 = E[Y 1 Y 1 < x] Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

17 First-price Auctions Proposition 2 Symmetric equilibrium strategies in a first-price auction are given by β I (x) = E[Y 1 Y 1 < x] where Y 1 is the highest of N - 1 independently drawn values. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

18 First-price Auctions Proof: Suppose that all but bidder 1 follow the strategy β I β. We will argue that in that case it is optimal for bidder 1 to follow β also. First, notice that β is an increasing and continuous function. Thus, in equilibrium the bidder with the highest value submits the highest bid and wins the auction. It is not optimal for bidder 1 to bid a b > β(ω). The expected payoff of bidder 1 with value x if he bids an amount b β(ω) is calculated as follows. Denote by z = β 1 (b) the value for which b is the equilibrium bid that is, β(z) = b. Then we can write bidder 1s expected payoff from bidding β(z) when his value is x as follows: Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

19 First-price Auctions Π(b, x) = G(z)[x β(z)] = G(z)x G(z)E[Y 1 Y 1 < x] = G(z)x z 0 yg(y)dy = G(z)x G(z)z + z = G(z)(x z) + 0 z 0 G(y)dy G(y)dy Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

20 First-price Auctions We thus obtain that Π(β(x), x) Π(β(z), x) = G(z)(x z) + G(y)dy 0 z x regardless of whether z x or z x. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

21 Revenue Comparison Proposition 3 With independently and identically distributed private values, the expected revenue in a first-price auction is the same as the expected revenue in a second-price auction. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

22 Revenue Comparison When values are uniformly distributed and there are only two bidders, the equilibrium strategy in a first-price auction is β I (x) = 1 2 x. If the realized values are such that 1 2 x 1 > x 2, then the revenue in a first-price auction is greater than that in a second-price auction. If 1 2 x 1 < x 2 < x 1, the opposite is true. Depending on the realized values, we have argued that on average the revenue to the seller will be the same. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

23 Revenue Comparison Let L I denote the distribution of the equilibrium price in a first-price auction let L II be the distribution of prices in a second-price auction. L II is a mean-preserving spread of L I Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

24 Revenue Comparison The figure depicts the two distributions in the case of uniformly distributed values with two bidders. Since the two distributions have the same mean, the two shaded regions are, as they must be, equal in area. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

25 Revenue Comparison Proposition 4 With independently and identically distributed private values, the distribution of equilibrium prices in a second-price auction is a mean-preserving spread of the distribution of equilibrium prices in a first-price auction. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

26 Standard Auction An auction is standard if the rules of the auction dictate that the person who bids the highest amount is awarded the object. Both first and second-price auctions are standard auctions. Given a standard auction form, A, and a symmetric equilibrium β A of the auction, let m A (x) be the equilibrium expected payment by a bidder with value x. The expected payment of a bidder with value 0 is 0. The expected payment function m A (.) does not depend on the particular auction form A. The expected revenue in any standard auction is the same, a fact known as the revenue equivalence principle. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

27 Standard Auction Proposition 5 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. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28

28 Qualifications and Extensions The revenue equivalece principle is based on four assumptions discussed before. The revenue equivalence principle is affected when some of these assumptions are relaxed. With symmetric, independent private values, the expected revenue in a first-price auction is greater than that in a second-price auction for risk averse bidders with same utility function. Suppose bidders are subject to financial constraints. If the first-price auction has a symmetric equilibrium of the form B I (x, w) = min(β(x), w), then the expected revenue in a first-price auction is greater than the expected revenue in a second-price auction. With asymmetric bidders, the expected revenue in a second-price auction may exceed that in a first-price auction. Emrah Cem and Samira Farhin (UTD) Auctions April 25, / 28