Optimal Nonparametric Estimation of
First-Price Auctions
by Emmanuel Guerre, Isabelle Perrigne and Quang Vuong
Presenter: Andreea ENACHE1
1 CREST-LEI
and Paris School of Economics
25 April 2012
Outline
Objectives of the paper
Economic framework
Identification issue
Estimation
Advantages of GPV’s procedure
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Objectives of the paper
• Does a theoretical auction model place any restrictions on observable
data to be tested?
• Does a structural approach require a priori parametric information about
the structural elements to identify the model?
• Propose an estimation procedure that does not rely upon parametric assumptions and that is computationally feasible.
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Economic framework
Hypothesis of the first-price auction model
• A selling auction of a single and indivisible object.
• All bids are submitted simultaneously.
• The bidder with the highest bid wins and pays its bid.
• Bids are taken into account only if they are at least as high as a reservation price p0 .
• Each bidder has a private valuation vi for the auctioned object.
• Each bidder doesn’t know other bidders’ private values, but knows that
all private values including his own have been independently drawn from
a common distribution F(·) (IPV environment).
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Economic framework
Hypothesis of the first-price auction model
• F(·) is absolutely continuous with density f (·) and support [v, v].
• F(·), I (the number of potential bidders) and the reservation price p0 are
common knowledge with p0 ∈ [v, v]
• Each bidder is assumed to be risk neutral.
• The equilibrium bid bi corresponding to the symmetric Bayesian Nash
Equilibrium is given by:
bi = s(vi , F, I, p0 ) ≡ vi −
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Zvi
1
F(vi )
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F(u)
I−1
p0
I−1
du
(1)
Economic framework
Resolution of the model
• The objective function of the bidder
max(vi − bi ) Pr(bi wins) = (vi − bi )F I−1 s−1 (bi )
bi
leads to the following FOC:
1 = (vi − s(vi ))(I − 1)
f (vi ) 1
F(vi ) s0 (vi )
(2)
with boundary condition s(p0 ) = p0 .
• (2) is a first order differential equation in s(·) whose solution gives
us the equilibrium bid in (1).
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Identification issue
"A rose by any other name may
not be a rose!"
(Gujarati, Porter, 2009, Essentials of Econometrics)
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Identification issue
What is identification?
Identification...
• Allows to verify whether the underlying structure (distribution, parameters...) can be recovered from the observed random variables =⇒ it is an existence problem.
• Precedes estimation and it is invariant to the estimation procedure.
• Is based on the population version of the stochastic system and
not on a particular sample.
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Identification issue
Some definitions...
Definition 1
The parameters a1 and a2 in F0 are observationally equivalent (a1 ∼ a2 ) iff G1 = G2 ,
where Gi = Fai ◦ s−1
ai (i = 1, 2).
Definition 2
The parameter a ∈ F0 is globally identified iff ∀a∗ ∈ F0 , a∗ ∼ a ⇒ a∗ = a.
The model (s, a) is globally identified (for a given functional s) if all a’s in F0 are globally
identified.
Definition 3
The parameter a ∈ F0 is locally identified iff there exists a neighborhood V(a) in F0
such that ∀a∗ ∈ V(a), a∗ ∼ a ⇒ a∗ = a.
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Identification issue
Nonparametric identification of first-price sealed bid auction
Assume that p0 = v ⇒ number of potential bidders (I)=number of actual bidders. Hence:
• I and bi are observed by the econometrician.
• F is the unknown structural element which needs to be identified.
Technical issue: s(·) depends also on the unknown parameter F(·), as we can see
from (1).
Solution provided by GPV: If G is the distribution function of the bids and g the probability density function of the bids, then:
G(b) = Prob(bi ≤ b) = Prob(s−1 (bi ) ≤ s−1 (b)) = Prob(vi ≤ s−1 (b)) = F s−1 (b) = F(v)
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Identification issue
Nonparametric identification of first-price sealed bid auction
Then:
g(b) =
Therefore:
f (v)
f (s−1 (b))
= 0
s0 (v)
s (v)
G(b)
F(v)s0 (v)
=
g(b)
f (v)
Using (2) we rewrite:
vi = s(vi ) +
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1 F(vi )s0 (vi )
I − 1 f (vi )
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(3)
Identification issue
Nonparametric identification of first-price sealed bid auction
If we replace the expression of
G(b)
in (3) we get the central result of the paper:
g(b)
vi = ξ(bi , G, I) ≡ bi +
1 G(bi )
I − 1 g(bi )
(4)
Theorem
Let I ≥ 2. Let the joint distribution of bids G(·) belong to the set P I with support [b, b]I .
There exists a distribution of bidders’ private values F(·) ∈ P such that G(·) is the
distribution of the equilibrium bids in a first price sealed bid auction with independent
private values and a Q
nonbinding reservation price if and only if:
C1: G(b1 , b2 , ..., bI )= Ii=1 G(bi ).
C2: The function ξ(·, G, I) defined in (4) is strictly increasing on [b, b] and its inverse is differentiable on [v, v] ≡ [ξ(b, G, I), ξ(b, G, I)]
Moreover, when F(·) exists, it is unique with support [v, v] and satisfies
F(v) = G ξ −1 (v, G, I)
for all [v, v]. In addition, ξ(·, G, I) is the quasi inverse of
the equilibrium strategy in the sense that ξ(b, G, I) = s−1 (b, F, I, for all b ∈ [b, b] .
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Identification issue
Nonparametric identification of first-price sealed bid auction
Proof :
• bi = s(vi , F, I) and vi are iid ⇒ bi are also iid and thus C1 must hold.
• s(·, F, I) is the strictly increasing differentiable and BNE corresponding
to F(·) on [v, v].
G(b) = F s−1 (b, F, I) for every b ∈ [b, b] ≡ [s(v, F, I), s(v, F, I)].
s(·, F, I) solves (2), (3) follows from (2) ⇒ ξ(s(v, F, I), G, I) = v ⇒
ξ(b, G, I) = s−1 (b, F, I) ∀b ∈ [b, b].
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Identification issue
Nonparametric identification of first-price sealed bid auction
The knowledge of the joint distribution of the private values, F, allows one to:
• Simulate outcomes under alternative market mechanisms;
• Assess efficiency and the division of surplus;
• Determine the optimal reserve price.
• Evaluate the "market power" of the bidders v − b.
• Analyze how this margin decrease as the number of bidder increases.
• Testing between CV and PV.
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Estimation
Nonparametric estimation
Two step estimation:
• Construction of a sample of pseudo private values using (3).
• Obtain the density of bidders’ private values using the pseudo
sample constructed previously.
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Estimation
Nonparametric estimation
First stage
We begin by the nonparametric estimation of G and g:
L
I
1 XX
e
G(b)
=
1[Bpl ≤b]
IL
(5)
l=1 p=1
L
I
1 XX
e
Kg
g(b) =
ILhg
l=1 p=1
b − Bpl
hg
!
(6)
where L is the number of homogeneous auctions with he same number
of bidders, I.
This will give us the following pseudo-values:
V̂pl = Bpl +
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e pl )
1 G(B
ˆ pl , I)
≡ ξ(B
I−1 e
g(Bpl )
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(7)
Estimation
Nonparametric estimation
Intermediate step (sample trimming)
ˆ pl , I) such that:
Redefine V̂pl = ∞ for each V̂pl = ξ(B
Bpl ∈ [Bmin , Bmin + hg ] ∪ [Bmax − hg , Bmax ]
Final step
Estimate the density of the private value distribution according to:
!
L
I
v − V̂pl
1 XX
Kf
(8)
f̂ (v) =
ILhf
hf
l=1 p=1
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Estimation
Some drawbacks of the sample trimming procedure
• Sample trimming involves a non random loss of data.
• The boundary effect problem is compounded at the second stage
of the estimation. Within a width of ∆(hg , hf ) from the boundaries,
the kernel density estimator will be downward biased.
• The usual bandwidth selection rule (MISE) are designed for one
stage estimator and not for two step estimator.
• Consistency of the GPV estimator is achieved only on closed subsets of the interior of private values support.
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Advantages of GPV’s procedure
Advantages of GPV’s estimation procedure
• A nonparametric procedure robust to misspecifications of the underlying
distribution.
• Each step of the structural estimation procedure consists of nonparametric techniques.
• Derivation of the best rate of uniform convergence of nonparametric estimates of the density of latent variables from the unobserved bids.
• GPV’s estimation procedure avoids numerical difficulties.
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Advantages of GPV’s procedure
Conclusions
• GPV show that the distribution of bidders’ private values within
IPV is identified from observables, without any parametric assumptions.
• They obtain a global identification result.
• Their methodology can be applied to other types of auctions (see
GPV, 1995 on Dutch auctions).
• Their methodology avoids the determination of the equilibrium
strategy.
• GPV’s identification results provide necessary and sufficient conditions for the existence of a latent distribution that "rationalizes"
the distribution of bids.
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Advantages of GPV’s procedure
La fin...
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