- Email: [email protected]
A note on the continuity of the optimal auction
Economics Letters 137 (2015) 127–130
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
A note on the continuity of the optimal auction Paulo Klinger Monteiro Escola Brasileira de Economia e Finanças (FGV/EPGE), Praia de Botafogo 190, Rio de Janeiro, RJ, Brazil
highlights • I study the continuity of the optimal auction revenue as a function of the types distribution. • Revenue is lowersemicontinuous for convergence in distribution. • Revenue is continuous if the limit distribution is a continuous distribution.
article
info
Article history: Received 20 August 2015 Received in revised form 14 October 2015 Accepted 23 October 2015 Available online 29 October 2015
abstract In the independent private values auction model I study the continuity properties of the optimal auction revenue as a function of the valuations or types distribution. I show that the optimal revenue is lowersemicontinuous for convergence in distribution. If the limit distribution is continuous, the optimal auction revenue is continuous in distribution as well. © 2015 Elsevier B.V. All rights reserved.
JEL classification: D44 Keywords: Optimal auction Convergence in distribution Independent private values
1. Introduction The independent private values auction model is the most used auction model in practice and in theory. For example under symmetry, the symmetric equilibrium bidding function of the firstprice auction has a formula that is simple to use. The optimal auction for the independent private values model was obtained in Myerson (1981) for valuations (or types) distribution that have continuous strictly positive densities. In a very general model, optimal auction existence is studied in Page (1988). However, the convenient characterization obtained by Myerson is lost in the general model. Monteiro and Svaiter (2010) obtained a characterization of the optimal auction, similar to Myerson’s, valid for general distributions. This characterization will be very useful for the results in this note. Let I be the number of Bidders. If Fi is Bidder’s i = 1, . . . , I valuations distribution and F = ΠiI=1 Fi is the joint distribution, let R(F ) denote the optimal revenue. The first result I prove is the lower-semicontinuity of R(F ). That is if F n converges to F in distribution then lim infn→∞ R (F n ) ≥ R (F ). It remains to prove the upper-semicontinuity. For this I need the
additional assumption that F is continuous. The general case must await further research. 2. Preliminaries A function f : R → R is increasing if x < y implies f (x) ≤ f (y). The limits f (x+) := lim infy↓x f (y) and f (x−) := lim supy↑x f (y) always exist. We also define f (∞) := sup f (R) and f (−∞) := inf f (R). The point x is a point of jump of f if f (x−) < f (x+). It is well known that the only discontinuities of an increasing function are points of jump. Moreover the set of discontinuities of an increasing function is at most countably infinite. Definition 1. A function F : R → [0, 1] is a distribution if it is increasing, right-continuous, F (−∞) = 0 and F (∞) = 1. If A ⊂ R is a Borelean set, define F (A) = A dF for the integral of A under the Borel–Stieltjes measure associated with F . In particular F ({x}) = F (x) − F (x−). I now define convergence in distribution.
Definition 2. Let G and Gn ; n ≥ 1 be distributions. We say that Gn d
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.econlet.2015.10.030 0165-1765/© 2015 Elsevier B.V. All rights reserved.
converges in distribution to G and write Gn → G if for every x ∈ R, point of continuity of G, then limn→∞ Gn (x) = G(x).
128
P.K. Monteiro / Economics Letters 137 (2015) 127–130
Definition 4 (Constraints). The direct mechanism (q, P ):
The following lemma will be useful soon. Lemma 1. Let φ : R → R be an increasing, bounded function. Suppose φ has no discontinuity points in common with G. Then
d
Gn → G H⇒
φ (u) dGn (u) →
φ (u) dG (u) .
(i) satisfies the incentive compatibility constraints if for every si , s′i , si Qi (si ) − Pi (si ) ≥ si Qi (s′i ) − Pi (s′i ); (ii) satisfies voluntary participation if si Qi (si ) − Pi (si ) ≥ 0.
Proof. See Portmanteau Lemma 5.2.5 page 358 in HoffmanJørgensen (1994).
The expected revenue is R (F ) =
d
Lemma 2. Suppose Gn → G. Let F be a closed subset of the real line containing only points of continuity of G. Then the convergence is uniform in F . Proof. Let us prove by contradiction. If (Gn )n does not converge uniformly in F then there is an ϵ > 0 and a sequence (xn )n of points in F such that |Gn (xn ) − G (xn )| ≥ ϵ . Now note that for N large enough, G (−N ) < ϵ/2 and G (N ) > 1 − ϵ/2. Thus there exists an n0 such that for every n ≥ n0 ,
The following is proved in Monteiro and Svaiter (2010): Theorem 1. There is a family of increasing functions li (si ), ai ≤ si ≤ bi , 1 ≤ i ≤ I and l0 (·) ≡ 0 such that if H (s) = i ≥ 1 : li (si ) = maxj≥0 lj (sj ) , qi (s) =
sup {|Gn (x) − G (x)| : x ≤ −N } ≤ Gn (−N ) + G (−N ) < ϵ.
1
#H (s) 0
if
i ∈ H (s);
if
i ̸∈ H (s),
(1)
and
Also sup {|Gn (x) − G (x)| : x ≥ N } ≤ (1 − Gn (N )) + (1 − G (N )) < ϵ.
(xn )n is bounded and there is a converging subsequence Therefore x′n n . Let x′ = lim x′n ∈ F . The continuity of G at x′ implies that
there is a δ > 0 such that G (x) − G x′ < ϵ if x − x′ ≤ δ . For
′ ′ n large enough G xn − G x < ϵ . Moreover for n large enough, Gn x′ − δ ≤ Gn x′n ≤ Gn x′ + δ . Thus ′ Gn x − G x′ n n ′ ≤ max Gn x + δ − G x′ , Gn x′ − δ − G x′ n
n
is smaller than ϵ . A contradiction. 3. The independent private values model An object is to be sold at an auction with I Bidders. The risk neutral seller maximize expected revenue. Bidder i = 1, . . . , I has valuation si ∈ R. The distribution of i’s valuation is Fi : R → [0, 1]. I suppose that 0 ≤ ai < bi < ∞ where ai = inf {x : Fi (x) > 0} and bi = sup {x : Fi (x) < 1}. Thus the support of Fi is contained in the interval Si := [ai , bi ]. Let S = ΠiI=1 Si . I suppose the distributions, F1 , F2 , . . . , FI , independent. Thus the joint distribution is F (s) = ΠiI=1 Fi (si ). Let F−i (s−i ) = Πj̸=i Fj (sj ) be the distribution of valuations s−i := (s1 , . . . , si−1 , si+1 , . . . , sI ). Each Bidder knows his valuation si . Definition 3. A direct mechanism is a family of functions (q, P ) = (qi , Pi )Ii=1 , such that 1. qi : S → [0, 1] , 2. Pi : S → R.
I
i=1
qi (s) ≤ 1;
Remark 1. qi (s) should be interpreted as the probability of Bidder i receiving the object and Pi (s) is the corresponding (expected) payment. Define
Pi (s)dF−i (s−i )
pi (s) = si qi (s) −
si
qi (y, s−i )dy
qi (s)dF−i (s−i ).
(2)
ai
then the optimal auction mechanism is (q, p). Moreover the optimal revenue is max{lj (sj ) : j ≥ 0}dF (s). To describe the virtual valuation I need a couple of definitions. If G is a distribution and 0 ≤ a = inf{x : G(x) > 0} < b = sup {x : G (x) < 1} < ∞ let H (x) = a − x (1 − G (x)), a ≤ x ≤ b, and H (x) = 0 if x < a, H (x) = a if x ≥ b. I now define the auxiliary set
Γ = (α, β) ∈ R2 : α + β G (·) ≤ H (·)
(3)
and the generalized convex hull,
φ (x) = sup {α + β G (x) : (α, β) ∈ Γ } .
(4)
Definition 5 (Subgradient). The number β is a subgradient of φ at x, denoted β ∈ ∂φ (x), if for every z, φ (z ) ≥ φ (x)+β (G (z ) − G (x)). Definition 6. The virtual valuation l (x) := inf ∂φ (x) if x is a point of discontinuity of G and l (x) ∈ ∂φ (x) if G is continuous at x. Thus, if G is continuous, any selection of the subgradient correspondence, x → ∂φ (x), is a virtual valuation. Proposition 1. The generalized convex hull, φ , is right-continuous and the subgradient correspondence is increasing with G(x). Proof. That φ is right-continuous is Prop. 1 in Monteiro and Svaiter (2010). Suppose G(x) < G(y). If β ′ ∈ ∂φ(x) and β ′′ ∈ ∂(y) then
φ(y) − φ(x) ≥ β ′ (G(y) − G(x)) and
φ(x) − φ(y) ≥ β ′′ (G(x) − G(y)). Adding both equations we get 0 ≥ (β ′ − β ′′ )(G(y) − G(x)). Thus β ′′ ≥ β ′ . It might be useful to present an example1 and its auxiliary calculations.
and Qi (si ) :=
i i=1 P (s) dF (s).
4. Characterization of the optimal auction for general distributions
The following lemma generalize Pólya (1920):
Pi (si ) :=
I
1 I thank the referee for this suggestion.
P.K. Monteiro / Economics Letters 137 (2015) 127–130
Example 1. Let 0 ≤ a < b and 0 < p < 1 be given numbers. If a occurs with probability p and b with probability q := 1 − p the distribution is given by 0 p 1
G(x) =
x < a, a ≤ x < b, b ≤ x.
if if if
if and only if α ≤ min {0, a − bq − β p, a − β}. Let β˜ = a ≤ x < b,
a−bq . p
If
˜ b]. Hence l (x) = β˜ . It is Thus φ (x) = a − bq and ∂φ (x) = [β, easy to see that l (b) = b. To conclude:
5. Lower-semicontinuity I now prove the lower-semicontinuity of the revenue. d
Theorem 2. If Fin → Fi for i = 1, . . . , I, then lim infn→∞ R (F n ) ≥ R (F ). Proof. Using Eq. (2) we see that the optimal revenue is R(F n ) =
xQin (x) dFin (x) −
1 − Fin (x) Qin (x) dx
xQi (x) dFi (x) −
f
< ϵ . For each d ∈ Di there is an open interval Iid containing d and ϵd > 0 such that ϵ < ϵ , F Iid \ Iid = 0 d d and F Iid < F ({d}) + ϵd . The open set d∈Di Iid \ d∈Df Iid can be i that F Di \ Di
written as,
m≥1 Jim , the union of a pairwise disjoint family of open Jim . Define l0 x 0 Jim , m 1. Fix for each m, a xim
∈
≥
li(d) l
≥
xim
li (x)
if if
( ) ≡
f
x ∈ Iid , d ∈ Di ; x ∈ Jim , m ≥ 1; otherwise.
f
The function li (·) ̸= li (·) only in the set n=1 Ini \ Di . This set has F measure smaller than ϵ + d ϵd < 2ϵ . Thus Qi differs from the optimal Qi by at most 2(I − 1)ϵ . Since ϵ is arbitrary we get lim infn→∞ Ri (F n ) ≥ R(F ), finishing the proof.
I will prove the continuity of the revenue under the additional assumption that the limit distribution is continuous. d
The revenue is lower-semicontinuous. Thus it suffices to prove the revenue upper-semicontinuity. For each i = 1, . . . , I the sequence lni (x) of increasing functions has, by Helly’s lemma, a subsequence that converges pointwise. Without loss of generality suppose that lni (x) → ti (x) for every x ∈ [ai , bi ]. For G = Fin define Γin and φin as in (3) and (4). Similarly define Γi and φi for G = Fi . Lemma 3. ti (x) ∈ ∂φi (x) for every i and x.
we have that αi + ti (x) Fi (u) ≤ ai − u (1 − Fi (u)), for every u and therefore (αi , ti (x)) ∈ Γi . Thus φi (x) ≥ αi + ti (x) Fi (x) = lim sup φin (x). Let δn = supx |Fin (x) − Fi (x)|. Suppose now that (α, β) ∈ Γi . If M = max{|β − ai |, |β − bi |} then for every u
α − M δn + β Fin (u) ≤ ai − u (1 − Fin (u)) . Thus α − M δn + β Fin (x) ≤ φin (x) and in the limit, α + β Fi (x) ≤ lim infn φin (x). Therefore
φin (z ) − φin (x) ≥ lni (x) (Fin (z ) − Fin (x)) . Thus in the limit when n → ∞, φi (z ) − φi (x) ≥ ti (x) Fi (z )
H (s) = i ≥ 1 : li (si ) = max li (sj ) .
− Fi (x) . Therefore ti (x) ∈ ∂φi (x).
j ≥0
We now want to prove that limn R (F n ) =
Define qˆ as in (1):
Qi (x) dx. (1 − Fi (x))
ending the proof that φi (x) = lim φin (x). Now, for every z,
qi (s) =
φi (x) = sup {α + β Fi (x) : (α, β) ∈ Γ } ≤ lim inf φin (x)
It is clear that li (·) is increasing. Let
x Qi (x) dFi (x) −
αin + lni (x) Fin (u) ≤ ai − u (1 − Fin (u)) , ∀u
(1 − Fi (x)) Qi (x) dx .
Let Di be the set of discontinuity points of Fi , i = 1, . . . , I. Take ϵ > 0. Since d∈Di Fi ({d}) ≤ 1 there is a finite set Dfi ⊂ Di such
i
n→∞
αin + lni (x) Fin (x) = φin (x) .
i=1
li (x) =
Without loss of generality αin → αi and φin (x) → lim sup φin (x). Thus since
I
intervals and
1 − Fin (x) Qi (x) dx.
lim inf Ri F n
and
Proof. Let αin , lni (x) ∈ Γin be such that
i =1
R(F ) =
x Qi (x) dFin (x) −
Theorem 3. Suppose Fin → Fi and that Fi is continuous for i = 1, . . . , I. Then limn→∞ R (F n ) = R (F ).
a ≤ x < b, x = b.
I
6. Continuity
α + β p ≤ min {β p, a − bq, a − β q} ≤ a − bq.
if if
Ri (F n ) ≥
∞
α ≤ 0, α + β p ≤ a − bq, α + β ≤ a,
l(x) = β˜ l(x) = b
We have that
The function Qi is increasing and has no discontinuity points in common with Fi thus
Suppose (α, β) ∈ Γ . If x < a then α = α + β G (x) ≤ a − x (1 − G (x)) = a − x. Thus α ≤ 0. If x ≥ b we get α + β ≤ a. Finally if a ≤ x < b, α + β p ≤ a − x (1 − p). Thus α + β p ≤ a − bq. Therefore (α, β) ∈ Γ if and only if
129
1 # H ( s) 0
if if
i∈ H (s) ; i ̸∈ H (s) .
lim n
(5)
=
max lni (xi ) : i ≥ 0 dF n (x)
max {ti (xi ) : i ≥ 0} dF (x) .
130
P.K. Monteiro / Economics Letters 137 (2015) 127–130
Let ti+ (x) = ti (x+). Then ti+ is increasing and right-continuous. Also lni (x) converges to ti (x) = ti+ (x) at every point of continuity of ti+ . Thus if we take an open set U containing every point of discontinuity of ti+ and such that F (U ) < ϵ we see, using the uniform convergence of lni on the closed set U c , that
max lni (xi ) : i ≥ 0 dF n (x)
lim n
= lim n
max {ti (xi ) : i ≥ 0} dF n (x) .
Since ti (x) ≤ ti+ (x) and ti+ (x) is upper-semicontinuous we have that
lim n
≤
max lni (xi ) : i ≥ 0 dF n (x)
max ti+ (xi ) : i ≥ 0 dF (x) .
From ti (xi ) being a subgradient of φi at xi we easily check, using Proposition 1, that ti+ (xi ) ∈ ∂φi (x). Thus the right-hand side above is R(F ). This finishes the upper-semicontinuity proof. Acknowledgments I thank E. Balder for asking the question studied in this note. I acknowledge the financial support of CNPq-Brazil (Processo 301363/2010-2). References Hoffman-Jørgensen, J., 1994. Probability with a View Towards Statistics Vol. I. Chapman & Hall/CRC. Monteiro, P.K., Svaiter, Benar Fux, 2010. Optimal auctions with a general distribution: Virtual valuation without densities. J. Math. Econom. 46, 21–31. Myerson, R.B., 1981. Optimal auction design. Math. Oper. Res. 6 (1), 58–73. Page Jr., F.H., 1988. Existence of optimal auctions in general environments. J. Math. Econom. 29, 389–418. Pólya, G., 1920. Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und Momentenproblem. Math. Z. 8, 171–181.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
A note on the continuity of the optimal auction Paulo Klinger Monteiro Escola Brasileira de Economia e Finanças (FGV/EPGE), Praia de Botafogo 190, Rio de Janeiro, RJ, Brazil
highlights • I study the continuity of the optimal auction revenue as a function of the types distribution. • Revenue is lowersemicontinuous for convergence in distribution. • Revenue is continuous if the limit distribution is a continuous distribution.
article
info
Article history: Received 20 August 2015 Received in revised form 14 October 2015 Accepted 23 October 2015 Available online 29 October 2015
abstract In the independent private values auction model I study the continuity properties of the optimal auction revenue as a function of the valuations or types distribution. I show that the optimal revenue is lowersemicontinuous for convergence in distribution. If the limit distribution is continuous, the optimal auction revenue is continuous in distribution as well. © 2015 Elsevier B.V. All rights reserved.
JEL classification: D44 Keywords: Optimal auction Convergence in distribution Independent private values
1. Introduction The independent private values auction model is the most used auction model in practice and in theory. For example under symmetry, the symmetric equilibrium bidding function of the firstprice auction has a formula that is simple to use. The optimal auction for the independent private values model was obtained in Myerson (1981) for valuations (or types) distribution that have continuous strictly positive densities. In a very general model, optimal auction existence is studied in Page (1988). However, the convenient characterization obtained by Myerson is lost in the general model. Monteiro and Svaiter (2010) obtained a characterization of the optimal auction, similar to Myerson’s, valid for general distributions. This characterization will be very useful for the results in this note. Let I be the number of Bidders. If Fi is Bidder’s i = 1, . . . , I valuations distribution and F = ΠiI=1 Fi is the joint distribution, let R(F ) denote the optimal revenue. The first result I prove is the lower-semicontinuity of R(F ). That is if F n converges to F in distribution then lim infn→∞ R (F n ) ≥ R (F ). It remains to prove the upper-semicontinuity. For this I need the
additional assumption that F is continuous. The general case must await further research. 2. Preliminaries A function f : R → R is increasing if x < y implies f (x) ≤ f (y). The limits f (x+) := lim infy↓x f (y) and f (x−) := lim supy↑x f (y) always exist. We also define f (∞) := sup f (R) and f (−∞) := inf f (R). The point x is a point of jump of f if f (x−) < f (x+). It is well known that the only discontinuities of an increasing function are points of jump. Moreover the set of discontinuities of an increasing function is at most countably infinite. Definition 1. A function F : R → [0, 1] is a distribution if it is increasing, right-continuous, F (−∞) = 0 and F (∞) = 1. If A ⊂ R is a Borelean set, define F (A) = A dF for the integral of A under the Borel–Stieltjes measure associated with F . In particular F ({x}) = F (x) − F (x−). I now define convergence in distribution.
Definition 2. Let G and Gn ; n ≥ 1 be distributions. We say that Gn d
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.econlet.2015.10.030 0165-1765/© 2015 Elsevier B.V. All rights reserved.
converges in distribution to G and write Gn → G if for every x ∈ R, point of continuity of G, then limn→∞ Gn (x) = G(x).
128
P.K. Monteiro / Economics Letters 137 (2015) 127–130
Definition 4 (Constraints). The direct mechanism (q, P ):
The following lemma will be useful soon. Lemma 1. Let φ : R → R be an increasing, bounded function. Suppose φ has no discontinuity points in common with G. Then
d
Gn → G H⇒
φ (u) dGn (u) →
φ (u) dG (u) .
(i) satisfies the incentive compatibility constraints if for every si , s′i , si Qi (si ) − Pi (si ) ≥ si Qi (s′i ) − Pi (s′i ); (ii) satisfies voluntary participation if si Qi (si ) − Pi (si ) ≥ 0.
Proof. See Portmanteau Lemma 5.2.5 page 358 in HoffmanJørgensen (1994).
The expected revenue is R (F ) =
d
Lemma 2. Suppose Gn → G. Let F be a closed subset of the real line containing only points of continuity of G. Then the convergence is uniform in F . Proof. Let us prove by contradiction. If (Gn )n does not converge uniformly in F then there is an ϵ > 0 and a sequence (xn )n of points in F such that |Gn (xn ) − G (xn )| ≥ ϵ . Now note that for N large enough, G (−N ) < ϵ/2 and G (N ) > 1 − ϵ/2. Thus there exists an n0 such that for every n ≥ n0 ,
The following is proved in Monteiro and Svaiter (2010): Theorem 1. There is a family of increasing functions li (si ), ai ≤ si ≤ bi , 1 ≤ i ≤ I and l0 (·) ≡ 0 such that if H (s) = i ≥ 1 : li (si ) = maxj≥0 lj (sj ) , qi (s) =
sup {|Gn (x) − G (x)| : x ≤ −N } ≤ Gn (−N ) + G (−N ) < ϵ.
1
#H (s) 0
if
i ∈ H (s);
if
i ̸∈ H (s),
(1)
and
Also sup {|Gn (x) − G (x)| : x ≥ N } ≤ (1 − Gn (N )) + (1 − G (N )) < ϵ.
(xn )n is bounded and there is a converging subsequence Therefore x′n n . Let x′ = lim x′n ∈ F . The continuity of G at x′ implies that
there is a δ > 0 such that G (x) − G x′ < ϵ if x − x′ ≤ δ . For
′ ′ n large enough G xn − G x < ϵ . Moreover for n large enough, Gn x′ − δ ≤ Gn x′n ≤ Gn x′ + δ . Thus ′ Gn x − G x′ n n ′ ≤ max Gn x + δ − G x′ , Gn x′ − δ − G x′ n
n
is smaller than ϵ . A contradiction. 3. The independent private values model An object is to be sold at an auction with I Bidders. The risk neutral seller maximize expected revenue. Bidder i = 1, . . . , I has valuation si ∈ R. The distribution of i’s valuation is Fi : R → [0, 1]. I suppose that 0 ≤ ai < bi < ∞ where ai = inf {x : Fi (x) > 0} and bi = sup {x : Fi (x) < 1}. Thus the support of Fi is contained in the interval Si := [ai , bi ]. Let S = ΠiI=1 Si . I suppose the distributions, F1 , F2 , . . . , FI , independent. Thus the joint distribution is F (s) = ΠiI=1 Fi (si ). Let F−i (s−i ) = Πj̸=i Fj (sj ) be the distribution of valuations s−i := (s1 , . . . , si−1 , si+1 , . . . , sI ). Each Bidder knows his valuation si . Definition 3. A direct mechanism is a family of functions (q, P ) = (qi , Pi )Ii=1 , such that 1. qi : S → [0, 1] , 2. Pi : S → R.
I
i=1
qi (s) ≤ 1;
Remark 1. qi (s) should be interpreted as the probability of Bidder i receiving the object and Pi (s) is the corresponding (expected) payment. Define
Pi (s)dF−i (s−i )
pi (s) = si qi (s) −
si
qi (y, s−i )dy
qi (s)dF−i (s−i ).
(2)
ai
then the optimal auction mechanism is (q, p). Moreover the optimal revenue is max{lj (sj ) : j ≥ 0}dF (s). To describe the virtual valuation I need a couple of definitions. If G is a distribution and 0 ≤ a = inf{x : G(x) > 0} < b = sup {x : G (x) < 1} < ∞ let H (x) = a − x (1 − G (x)), a ≤ x ≤ b, and H (x) = 0 if x < a, H (x) = a if x ≥ b. I now define the auxiliary set
Γ = (α, β) ∈ R2 : α + β G (·) ≤ H (·)
(3)
and the generalized convex hull,
φ (x) = sup {α + β G (x) : (α, β) ∈ Γ } .
(4)
Definition 5 (Subgradient). The number β is a subgradient of φ at x, denoted β ∈ ∂φ (x), if for every z, φ (z ) ≥ φ (x)+β (G (z ) − G (x)). Definition 6. The virtual valuation l (x) := inf ∂φ (x) if x is a point of discontinuity of G and l (x) ∈ ∂φ (x) if G is continuous at x. Thus, if G is continuous, any selection of the subgradient correspondence, x → ∂φ (x), is a virtual valuation. Proposition 1. The generalized convex hull, φ , is right-continuous and the subgradient correspondence is increasing with G(x). Proof. That φ is right-continuous is Prop. 1 in Monteiro and Svaiter (2010). Suppose G(x) < G(y). If β ′ ∈ ∂φ(x) and β ′′ ∈ ∂(y) then
φ(y) − φ(x) ≥ β ′ (G(y) − G(x)) and
φ(x) − φ(y) ≥ β ′′ (G(x) − G(y)). Adding both equations we get 0 ≥ (β ′ − β ′′ )(G(y) − G(x)). Thus β ′′ ≥ β ′ . It might be useful to present an example1 and its auxiliary calculations.
and Qi (si ) :=
i i=1 P (s) dF (s).
4. Characterization of the optimal auction for general distributions
The following lemma generalize Pólya (1920):
Pi (si ) :=
I
1 I thank the referee for this suggestion.
P.K. Monteiro / Economics Letters 137 (2015) 127–130
Example 1. Let 0 ≤ a < b and 0 < p < 1 be given numbers. If a occurs with probability p and b with probability q := 1 − p the distribution is given by 0 p 1
G(x) =
x < a, a ≤ x < b, b ≤ x.
if if if
if and only if α ≤ min {0, a − bq − β p, a − β}. Let β˜ = a ≤ x < b,
a−bq . p
If
˜ b]. Hence l (x) = β˜ . It is Thus φ (x) = a − bq and ∂φ (x) = [β, easy to see that l (b) = b. To conclude:
5. Lower-semicontinuity I now prove the lower-semicontinuity of the revenue. d
Theorem 2. If Fin → Fi for i = 1, . . . , I, then lim infn→∞ R (F n ) ≥ R (F ). Proof. Using Eq. (2) we see that the optimal revenue is R(F n ) =
xQin (x) dFin (x) −
1 − Fin (x) Qin (x) dx
xQi (x) dFi (x) −
f
< ϵ . For each d ∈ Di there is an open interval Iid containing d and ϵd > 0 such that ϵ < ϵ , F Iid \ Iid = 0 d d and F Iid < F ({d}) + ϵd . The open set d∈Di Iid \ d∈Df Iid can be i that F Di \ Di
written as,
m≥1 Jim , the union of a pairwise disjoint family of open Jim . Define l0 x 0 Jim , m 1. Fix for each m, a xim
∈
≥
li(d) l
≥
xim
li (x)
if if
( ) ≡
f
x ∈ Iid , d ∈ Di ; x ∈ Jim , m ≥ 1; otherwise.
f
The function li (·) ̸= li (·) only in the set n=1 Ini \ Di . This set has F measure smaller than ϵ + d ϵd < 2ϵ . Thus Qi differs from the optimal Qi by at most 2(I − 1)ϵ . Since ϵ is arbitrary we get lim infn→∞ Ri (F n ) ≥ R(F ), finishing the proof.
I will prove the continuity of the revenue under the additional assumption that the limit distribution is continuous. d
The revenue is lower-semicontinuous. Thus it suffices to prove the revenue upper-semicontinuity. For each i = 1, . . . , I the sequence lni (x) of increasing functions has, by Helly’s lemma, a subsequence that converges pointwise. Without loss of generality suppose that lni (x) → ti (x) for every x ∈ [ai , bi ]. For G = Fin define Γin and φin as in (3) and (4). Similarly define Γi and φi for G = Fi . Lemma 3. ti (x) ∈ ∂φi (x) for every i and x.
we have that αi + ti (x) Fi (u) ≤ ai − u (1 − Fi (u)), for every u and therefore (αi , ti (x)) ∈ Γi . Thus φi (x) ≥ αi + ti (x) Fi (x) = lim sup φin (x). Let δn = supx |Fin (x) − Fi (x)|. Suppose now that (α, β) ∈ Γi . If M = max{|β − ai |, |β − bi |} then for every u
α − M δn + β Fin (u) ≤ ai − u (1 − Fin (u)) . Thus α − M δn + β Fin (x) ≤ φin (x) and in the limit, α + β Fi (x) ≤ lim infn φin (x). Therefore
φin (z ) − φin (x) ≥ lni (x) (Fin (z ) − Fin (x)) . Thus in the limit when n → ∞, φi (z ) − φi (x) ≥ ti (x) Fi (z )
H (s) = i ≥ 1 : li (si ) = max li (sj ) .
− Fi (x) . Therefore ti (x) ∈ ∂φi (x).
j ≥0
We now want to prove that limn R (F n ) =
Define qˆ as in (1):
Qi (x) dx. (1 − Fi (x))
ending the proof that φi (x) = lim φin (x). Now, for every z,
qi (s) =
φi (x) = sup {α + β Fi (x) : (α, β) ∈ Γ } ≤ lim inf φin (x)
It is clear that li (·) is increasing. Let
x Qi (x) dFi (x) −
αin + lni (x) Fin (u) ≤ ai − u (1 − Fin (u)) , ∀u
(1 − Fi (x)) Qi (x) dx .
Let Di be the set of discontinuity points of Fi , i = 1, . . . , I. Take ϵ > 0. Since d∈Di Fi ({d}) ≤ 1 there is a finite set Dfi ⊂ Di such
i
n→∞
αin + lni (x) Fin (x) = φin (x) .
i=1
li (x) =
Without loss of generality αin → αi and φin (x) → lim sup φin (x). Thus since
I
intervals and
1 − Fin (x) Qi (x) dx.
lim inf Ri F n
and
Proof. Let αin , lni (x) ∈ Γin be such that
i =1
R(F ) =
x Qi (x) dFin (x) −
Theorem 3. Suppose Fin → Fi and that Fi is continuous for i = 1, . . . , I. Then limn→∞ R (F n ) = R (F ).
a ≤ x < b, x = b.
I
6. Continuity
α + β p ≤ min {β p, a − bq, a − β q} ≤ a − bq.
if if
Ri (F n ) ≥
∞
α ≤ 0, α + β p ≤ a − bq, α + β ≤ a,
l(x) = β˜ l(x) = b
We have that
The function Qi is increasing and has no discontinuity points in common with Fi thus
Suppose (α, β) ∈ Γ . If x < a then α = α + β G (x) ≤ a − x (1 − G (x)) = a − x. Thus α ≤ 0. If x ≥ b we get α + β ≤ a. Finally if a ≤ x < b, α + β p ≤ a − x (1 − p). Thus α + β p ≤ a − bq. Therefore (α, β) ∈ Γ if and only if
129
1 # H ( s) 0
if if
i∈ H (s) ; i ̸∈ H (s) .
lim n
(5)
=
max lni (xi ) : i ≥ 0 dF n (x)
max {ti (xi ) : i ≥ 0} dF (x) .
130
P.K. Monteiro / Economics Letters 137 (2015) 127–130
Let ti+ (x) = ti (x+). Then ti+ is increasing and right-continuous. Also lni (x) converges to ti (x) = ti+ (x) at every point of continuity of ti+ . Thus if we take an open set U containing every point of discontinuity of ti+ and such that F (U ) < ϵ we see, using the uniform convergence of lni on the closed set U c , that
max lni (xi ) : i ≥ 0 dF n (x)
lim n
= lim n
max {ti (xi ) : i ≥ 0} dF n (x) .
Since ti (x) ≤ ti+ (x) and ti+ (x) is upper-semicontinuous we have that
lim n
≤
max lni (xi ) : i ≥ 0 dF n (x)
max ti+ (xi ) : i ≥ 0 dF (x) .
From ti (xi ) being a subgradient of φi at xi we easily check, using Proposition 1, that ti+ (xi ) ∈ ∂φi (x). Thus the right-hand side above is R(F ). This finishes the upper-semicontinuity proof. Acknowledgments I thank E. Balder for asking the question studied in this note. I acknowledge the financial support of CNPq-Brazil (Processo 301363/2010-2). References Hoffman-Jørgensen, J., 1994. Probability with a View Towards Statistics Vol. I. Chapman & Hall/CRC. Monteiro, P.K., Svaiter, Benar Fux, 2010. Optimal auctions with a general distribution: Virtual valuation without densities. J. Math. Econom. 46, 21–31. Myerson, R.B., 1981. Optimal auction design. Math. Oper. Res. 6 (1), 58–73. Page Jr., F.H., 1988. Existence of optimal auctions in general environments. J. Math. Econom. 29, 389–418. Pólya, G., 1920. Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und Momentenproblem. Math. Z. 8, 171–181.