Optimal Nonparametric Estimation of First-Price Auctions
BY EMMANUEL GUERRE, ISABELLE PERRIGNE AND QUANG VUONG (2000)
Edmund Y. Lou
Abstract
The only purpose of this report is to replicate the two-step nonparametric esti- mation proposed by Emmanuel Guerre, Isabelle Perrigne and Quang Vuong (2000). The abbreviation GPV is used throughout the text.
1 Introduction
With auction data increasingly available from different sources, such as government pro- curement, eBay and Google page positioning, the modern industrial economics is facing a challenge of its empirical usefulness. Having this in mind, GPV proposes a two-step non- parametric procedure to estimate the density of private values in the first-price sealed-bid auction under the independent private value paradigm.
GPV successfully answers three fundamental quesitions associated with structural analysis: (i) Whether a theoretical model places restrictions on observed data to be testable, (ii) Whether a structural analysis requires a priori parametric information, and
(iii) Whether an estimation procedure can be free from parametric assumptions.
Our aim is to replicate such two-step nonparametric method in the following sections.
2 The First-Price Sealed-Bid Auction Model
Consider an auction consists of I risk-neutral players, who bid for a single and indivisible object. Each player i knows privately how much the object is worth to her (i.e., her private value vi for the object) but does not know other players’ valuations. Each player considers the private values of the object to the other I 1 players be independent random variables drawn from an interval v, v R with a common absolutely continuous distribution F . In the auction, each player i simultaneously submits a sealed bid bi and the object is sold to the player whose bid is the highest. The player who wins the auction must pay the amount of her bid, provided that the bid is no lower than the reserve price p0 of the owner of the object, while the other players pay nothing. The distribution F with its density f ,
the number of players I, and the reserve price p0 v, v are common knowledge among all the players.
This auction model is in nature a Bayesian game where each player’s type is given by her private value for the object. The corresponding (symmetric) Bayesian Nash equilib- rium bid bi of the i-th player, assuming I ≥ 2, is
(1) bi = s(vi; F, I, p0) = vi − ( 1 I−1 J (F (u)) du,
vi
I−1
whenever vi p0. Note that s ; F, I, p0 is strictly increasing and that bi can be any value strictly less than p0 such that vi p0. The equilibrium bid is obtained from maximising expected payoff
max
bi
(vi − bi)F (s−1(vi))I−1,
namely, from solving the (rearranged) first-order condition
(2) s′(v ) = [v − s(v )](I − 1) f (vi) .
i i i
with boundary condition s(p0) = p0.
F (vi)
Let G be the distribution of bi. Due to the strict monotonicity,
G bi P b bi P v s−1 bi F vi , bi v, s v .
It follows that f vi g bi s vi , where g is the density function of bi. Taking the ratio gives that
g(bi) = f (vi) ′ 1 .
(2’) vi = ξ(bi; G, I) = bi + (
g(bi) .
3 Identification and Estimation
In auction bids are observed but private values are commonly unobservable. Whenver we know the distribution function G and its density g, the private values can be easily recovered from equation (2’). However, they are unknown in general. Thanks to the modern development of econometric techniques, they can be estimated nonparametrically. This suggests the following two-step estimation.
First, using observed bids we estimate G and g by empirical distribution and kernel density estimator, respectively; that is, by
(3) G(b) = IL Σ Σ 1(Bpl ≤ b),
l=1 p=1
and
1 L I
b − Bpl
(4)
g˜(b) = ILh
K
g l=1 p=1
h ) ,
where Kg is a kernel with a compact support, denoted by ρg its length, and hg is a bandwidth.
Second, let Bmin and Bmax be the minimum and maximum of the observed bids
and assume that there are L auctions. Since g˜ is asymptotically unbiased on [Bmin +
ρghg 2, Bmax ρghg 2 , we can define the pseudo private value Vˆpl corresponding to its
bids Bpl by
ııBpl + 1
G˜(Bpl)
if Bmin + ρghg ≤ Bpl ≤ Bmax − ρghg
(5)
Vˆpl = ıı›+∞
I−1 g˜(Bpl)
2 2
otherwise
where p 1, . . . , I and l 1, . . . , L. The set of pseudo private values is finally utilised to estimate the density function of the private values via
4 Monte Carlo Experiments
To illustrate the two-step nonparametric procedure, we conduct a Monte Carlo experiment with 1000 replications. Every one of our replications consists of L 200 auctions, each having I 5 players. The true distribution F of private values is assumed to be log-normal with parameters zero and one, truncated at 0.055 and 2.5. Its corresponding distribution function is the following:
−0.00227682 + 0.610964 × erfc [lo√g(x)] , x ∈ [0.055, 2.5],
where erfc is the complementary error function.
the program FORTRAN used in GPV, we use Julia instead. Then we apply the following simulation and estimation procedure for each replication.
• Step 1: Simulate I L private values from F .
• Step 2: Compute numerically the corresponding “observed” bids Bpl using (1) with reserve price p0 = 0.055.
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