Auction. Li Zhao, SJTU. Spring, Li Zhao Auction 1 / 35

Transcription

1 Auction Li Zhao, SJTU Spring, 2017 Li Zhao Auction 1 / 35

2 Outline 1 A Simple Introduction to Auction Theory 2 Estimating English Auction 3 Estimating FPA Li Zhao Auction 2 / 35

3 Background Auctions have been used since antiquity for the sale of a variety of objects. Herodotus reports that auctions were used in Babylon as early as 500 b.c. Today, both the range and the value of objects sold by auction have grown to staggering proportions. Art objects, antiques Commodities Securities, treasure bonds Rights to use natural resources Transfer of assets from public to private hands Internet auction Government purchases Li Zhao Auction 3 / 35

4 Why Study Auctions? Understand how bidder bid. Whether or not the object ends up in the hands of the person who values it the most? Design the mechanism that maximizes revenue by Choosing the bidding format; Setting the optimal reserve price. Detect potential problems Are bidders collude? Li Zhao Auction 4 / 35

5 Types of Auctions By auction format: English (ascending) auction First-price auction Second-price auction Dutch auction Private value, common value All pay auction Unit-good, multi-good auction Li Zhao Auction 5 / 35

6 The Benchmark Model (SIPV) Single object for sale. n potential buyers. Bidder i assigns a value of v i to the object. vi i.i.d. distributed (private value) according to distribution with cdf F and pdf f. Bidder i knows the realization v i (the value of his or her own value) and F. Bidders are risk neutral. No liquidity or budget constraints. Li Zhao Auction 6 / 35

7 Second Price Auction { vi max π i = j i b j if b i > max j i b j (i wins) 0 if b i < max j i b j (i loses) Theorem: In a second-price sealed-bid auction, it is a weakly dominant strategy to bid according to β(v) = v. (Krishna, page 13, Proposition 2.1) Doesn't rely on the assumption that values are independently distributed. Doesn't rely on the assumption that values are identically distributed. The only assumption that is important is the assumption of private values. Li Zhao Auction 7 / 35

8 English Auction What is the optimal bidding strategy? What if β(v) < v? What if β(v) > v? Any concerns? Haile and Tamer (JPE 2003). Oral ascending auction in reality vs. English auction in theory. Li Zhao Auction 8 / 35

9 First Price Auction - Setup v i i.i.d. distributed (private value) on interval [0,ω] according to distribution with cdf F and pdf f. { vi b π i = i if b i > max j i b j (i wins) 0 if b i < max j i b j (i loses) We look for symmetric, increasing, and dierentiable equilibrium strategy b = β(v). A bidder with value 0 would never submit a positive bid, β(0) = 0. Let Y 1 max j / i v j denote the highest value of n 1 bidders. Let H( ) be the distribution of Y 1, H(c) = (F (c)) n 1. Bidder i wins whenever b i > max j / i β(v j ), or equivalently, if β 1 (b i ) > max j / i v j. The probability of winning by submitting bid b is therefore H(β 1 (b)) Li Zhao Auction 9 / 35

10 First Price Auction - Equilibrium Suppose bidder i submit a bid b, the expected payo is Π = H(β 1 (b)) (v b). Take rst order of this objective function with respect to b. The optimal strategy is to bid b = β(v) therefore the rst order condition evaluated at b = β(v) is 0. d db (Π) b=β(v) = 0. We can get (H(v)β(v)) = h(v) v. Together with the boundary condition β(0) = 0, the optimal bidding strategy is β(v) = 1 v yh(y)dy H(v) 0 1 v = v F (y) n 1 dy. F (v) n 1 0 Li Zhao Auction 10 / 35

11 First Price Auction - With Reserve Price The seller could set a small serve price r > 0. The object is not sold if bids is lower than r. β(r) = r. As before, we have (H(v)β(v)) = h(v) v, but with the condition β(r) = r. Equilibrium bidding strategy is β(v) = 1 H(v) H(r)r + 1 v yh(y)dy. H(v) r It can be veried that bidder bid more aggressively if the auction has reserve price. However, with probability H(r) the item will not be sold. Li Zhao Auction 11 / 35

12 Outline 1 A Simple Introduction to Auction Theory 2 Estimating English Auction 3 Estimating FPA Li Zhao Auction 12 / 35

13 Goal of Empirical Work We observe bids b i and we want to recover valuations v i. Why? We can evaluate the market power" of bidders. Interesting policy question: how fast does margin decrease as n (number of bidders) increases? Useful for the optimal design of auctions (auction format / optimal reserve price). Methodology: identication, non-parametric estimation Li Zhao Auction 13 / 35

14 Estimating English Auction Athey, S. and Haile, P.A., Identication of standard auction models. Econometrica, 70(6), pp Private value: (v 1,..v n ) F v1,v 2,...v n. Independent private value v i i.i.d. F. Equilibrium bid: β(v i ) = v i. Li Zhao Auction 14 / 35

15 Order Statistics Theorem: The ith order statistic from an i.i.d. sample of size n from an arbitrary distribution F ( ) has distribution F (i:n) (z) = n! F (z) t i 1 (1 t) n i dt. (n i)!(i 1)! 0 Example, the highest value, the nth order statistics, has distribution F (n:n) (z) = n! F (z) t n 1 (1 t) n n dt. (n n)!(n 1)! 0 = F (z) n. Because F (i:n) (z) is strictly increasing in F (z), F (z) is uniquely determined by F (i:n) (z) for any (i : n). Li Zhao Auction 15 / 35

16 Estimating and Testing SIPV (Ahey and Haile 2002) In the symmetric IPV model, F ( ) is identied from the transaction price. The model is testable if either (a) more than one bid per auction is observed or (b) transaction prices are observed from auctions with exogenous varying numbers of bidders. Extensions: Asymmetric private value model: identied if bids and bidders' identities are observed. Deviating from button auction. Correlated private value. Li Zhao Auction 16 / 35

17 Correlated Private Value Aradillas-López, A., Gandhi, A. and Quint, D., Identication and inference in ascending auctions with correlated private values. Econometrica, 81(2), pp IPV model is mis-specied if there is dependence among bidder valuations. The authors exploit the fact that both seller prots and bidder surplus depend only on the marginal distributions of the two highest valuations, and not on any other features of the joint distribution. π n (r) = max(r,v)df n 1:n (v) F n:n (r) (r v 0 ) v 0, BS n (r) = 0 0 max(r,v)df n:n (v) 0 max(r,v)df n 1:n (n). Li Zhao Auction 17 / 35

18 Correlated Private Value - Bounding F n:n F n:n (v) is bounded by F n:n (v) [φ n (F n 1:n (v)) n,f n 1:n (v)] where φ n is the inverse of the function nx n 1 (n 1)x n. The lower bound on F n:n is consistent with a model of independent private values. The upper bound is consistent with perfectly-correlated values. Li Zhao Auction 18 / 35

19 Outline 1 A Simple Introduction to Auction Theory 2 Estimating English Auction 3 Estimating FPA Li Zhao Auction 19 / 35

20 The SIPV model of FPA We consider symmetric independent private value model. Single object for sale. n potential buyers, risk neutral. Bidder i assigns a value of v i to the object. vi i.i.d. distributed (private value) according to distribution with cdf F and pdf f. Bidder i knows the realization v i and submit a sealed bid b i = β(v i ). In a First-Price sealed bid auction, the winning bidders pays his own bid. Li Zhao Auction 20 / 35

21 Equilibrium of FPA The payo of bidder i is { vi b π i = i if b i > max j i b j (i wins) 0 if b i < max j i b j (i loses) Let G( ) denote the CDF of the bids. The probability of winning is G(b i ) n 1. Bidder i maximizes Π = G(b) n 1 (v b). Take FOC, the rst order condition evaluated at b = β(v) is 0. We will get v = b + G(b) (n 1)g(b). Li Zhao Auction 21 / 35

22 GPV Method (1) Guerre, E., Perrigne, I. and Vuong, Q., Optimal Nonparametric Estimation of First-Price Auctions. Econometrica, 68(3), pp Assuming a data set consisting of T n-bidder auctions. v = b + Empirical distribution function: Ĝ(b) = 1 T n G(b) (n 1)g(b). T t=1 Kernel density estimate of bid density: ĝ(b) = 1 T n T t=1 n i=1 n i=1 1(b it b). 1 h k(b it b ). h Li Zhao Auction 22 / 35

23 GPV Methods (2) GPV recommend a two-step approach to estimating the valuation distribution f (x) In rst step, estimate G(b) and g(b) nonparametrically. In second step, estimate f (x) by using kernel density estimator of recovered valuations. Li Zhao Auction 23 / 35

24 Estimating SPIV FPA: Quantile-Based Approach Marmer, V. and Shneyerov, A., Quantile-based nonparametric inference for rst-price auctions. Journal of Econometrics, 167(2), pp Recall in GPV, we have shown v = b + 1 G(b) n 1 g(b). Consider the τ-th quantile of valuation Q(τ) and the τ-th quantile of bids q(τ), the equation above implies that Q(τ) = q(τ) + τ (n 1)g(q(τ)). Li Zhao Auction 24 / 35

25 Are Structural Estimation Reasonable? Bajari, P. and Hortacsu, A., Are structural estimates of auction models reasonable? Evidence from experimental data. Journal of Political economy, 113(4), pp The structural approach is based on three strong assumptions. The rst is that bidders' goal is to maximize their expected utility. Second, bidders are able to compute the relationship between their bid and the probability of winning the auction. Third, given their beliefs, bidders are able to correctly maximize expected utility. Many applied researchers are not comfortable with the strict rationality assumptions imposed in the econometric analysis. This paper structurally estimates rst-price auction models using data from laboratory experiments. Li Zhao Auction 25 / 35

26 The Experiments Four alternative rst-price auction models using bidding data from the experiments of Dyer, Kagel, and Levin (1989). (1) Risk-neutral Bayes-Nash, (2) Risk-averse Bayes-Nash, (3) An adaptive model of learning, and (4) Quantal response equilibrium. This paper measures the distance between the estimated and true valuations. Li Zhao Auction 26 / 35

27 Findings The authors nd that the symmetric CRRA Bayes-Nash model is able to recover the distribution of valuations better than the risk-neutral Bayes-Nash model and adaptive learning model. The QRE model with risk aversion uncovers a risk aversion parameter very similar to that of the Bayes-Nash model, though the CRRA Bayes-Nash model yields more accurate estimates of the underlying distribution of bidder valuations. Li Zhao Auction 27 / 35

28 Experiment Design Bidders were assigned i.i.d. valuations v drawn from a uniform distribution on [$0, $30]. Each subject participated in 28 auctions over the course of two hours. Either n = 3 or n = 6. The number of bidders is random. After each auction, bids and corresponding private values were publicly posted on a blackboard. Li Zhao Auction 28 / 35

29 Assessing Goodness of Fit Suppose we have estimated ˆv and ˆF (v) using one of the models. We can assess the goodness of t of our model by comparing the estimated distribution of valuations with the actual (assigned) distribution of valuations. A Kolmogorov-Smirnov statistics takes the form KS T = T sup ˆF T (v) F (v). v Li Zhao Auction 29 / 35

30 The Risk-Neutral Model Prot takes the form π = F (β 1 (b)) n 1 (v b). FOC evaluates at b = β(v) equals zero. Let G( ) and g( ) be the be the distribution and density of the bids, respectively. We get G(b) = F (β(v)), g(b) = g(β (v))β(v). v = b + G(b) (n 1)g(b). Li Zhao Auction 30 / 35

31 Estimate the RN Model GPV (2000): v = b + G(b) (n 1)g(b). Step 1: Estimate ĝ(b) and Ĝ(b) nonparametrically. Step 2: estimate ˆv. Li Zhao Auction 31 / 35

32 The Risk-Averse Model Assume CRRA utility function u(c) = c θ where θ [0,1]. Prot takes the form π = F (β 1 (b)) n 1 (v b) θ. FOC evaluates at b = β(v) equals zero. We get G(b) v = b + θ (n 1)g(b). Li Zhao Auction 32 / 35

33 Estimate the R-A Model Let v α denote the α'th percentile of the distribution of v. Take dierence, we get α = b α (3) + θ v (3) v (6) α = b α (6) + θ b α (3) b α (6) G(b α (6) ;6) = θ( (6 1)g(b (6) G(b (3) α ;3) (3 1)g(b (3) α ;3) ; G(b (6) α ;6) (6 1)g(b (6) α ;6). (3) α ;3) α ;6) G(b (3 1)g(b α (3) ;3) ). Li Zhao Auction 33 / 35

34 Estimate the R-A Model (2) The Risk-Averse model can be estimated by three steps: Step 1: Estimate ĝ(b) and Ĝ(b) nonparametrically. Step 2: Estimate ˆθ using b α (3) b α (6) G(b (6) (3) α ;6) = θ( (6 1)g(b α (6) ;6) G(bα ;3) (3 1)g(b α (3) ;3) ). Step 3: Estimate ˆv. Li Zhao Auction 34 / 35

35 The Adaptive Model Bidders learn rather than know G(b). Bidders form beliefs using previously submitted bids: G(b history t ). Prot takes the form π t = G(b t history t ) n 1 (v t b t ). We get v t = b t + G(b history t ) (n 1)g(b history t ). Li Zhao Auction 35 / 35