Approximating welfare in large efficient markets

1.3 MB·

by  Ellen V. Muir

Approximating welfare in large efficient markets
Ellen V. Muir∗ and Konstantin Borovkov†
Abstract
We consider the efficient outcome of a canonical economic market model
6
which involves N buyers with unit demand and i.i.d. random valuations,
1
0 and M sellers with unit supply whose costs are also i.i.d. random variables
2 independent of the valuations. We approximate the joint distribution of the

n quantity K of units traded and the gains W from trade when there is a
α α
a
large number of market participants, i.e. both components of α := (N,M)
J
tend to infinity. The problem is reduced to studying a process expressed in
5
1 terms of two independent empirical quantile processes which, in large mar-
kets, can be approximated by appropriately weighted independent Brownian
]
R bridges. That allows us to approximate the distribution of (K ,W ) by that
α α
P of a functional of a Gaussian process. Moreover, we give upper bounds for
. the approximation rate.
h
t
a Keywords: Efficient market outcomes, Bayesian mechanism design, Walrasian
m
quantity, Walrasian welfare, empirical quantile processes, Brownian bridge
[
2010 Mathematics Subject Classification: Primary 60F15
1
Secondary 91B26; 60F17; 60G15
v
2
1
8 1 Introduction and main results
3
0
. Consider a canonical economic model for a market in which units of a homogeneous,
1
0 indivisible good are traded among N buyers {i ,...,i } and M sellers {j ,...,j }.
1 N 1 M
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Each buyer is interested in purchasing one unit of the good, and each seller has the
1
: capacity to produce and sell one unit of the good. Buyers are willing to purchase the
v
i good at a price not exceeding their respective reservation valuations V ,...,V that
X 1 N
are assumed to be independent and identically distributed (i.i.d.) random variables
r
a with a common distribution function (d.f.) F. Sellers are willing to produce at a
price that is not less than their respective production costs C ,...,C , which are
1 M
∗Thecorrespondingauthor. SchoolofMathematicsandStatistics,TheUniversityofMelbourne,
Parkville VIC 3010, Australia; e-mail: [email protected].
†School of Mathematics and Statistics, The University of Melbourne, Parkville VIC 3010, Aus-
tralia; e-mail: [email protected].
1
also assumed to be i.i.d. random variables, with a d.f. G. We assume that buyers
valuations and sellers costs are independent of each other and call the (N+M)-tuple
R := (V ,...,V ;C ,...,C )
(N,M) 1 N 1 M
a realization of the unitary market M := (cid:104)N,M,F,G(cid:105).
Such markets have been studied extensively in the literature on Bayesian mech-
anism design, see [9], [16] and [15]. Introductions to mechanism design can be found
in Ch. 5 of [10] or in [2].
Within a market, the welfare generated by trade is defined as the sum of trading
buyer valuations less the sum of trading seller costs:
(cid:88) (cid:88)
V − C ,
i j
i∈N j∈M
where N ⊂ {i ,...,i } and M ⊂ {j ,...,j } are, respectively, the subsets of
1 N 1 N
buyers and sellers who trade in the market, so that |N| = |M|. Market welfare can
be thought of as the total gain from trade experienced by market participants.
A market is said to be efficient if the level of market welfare is always maximised
forgivenbuyervaluationsandsellercosts. Thequantitytradedinanefficientmarket
is known in economics as the efficient quantity, the welfare-maximising quantity, or
the Walrasian quantity. To compute it, order buyer valuations V from highest
i
to lowest: V ≥ ··· ≥ V , and order seller costs C from lowest to highest:
[1] [N] j
C ≤ ··· ≤ C . The Walrasian quantity is then given by
(1) (M)
k
(cid:88) (cid:12) (cid:12)
K := argmax (V −C ) ≡ (cid:12){k ∈ {1,...,N ∧M} : V −C ≥ 0}(cid:12). (1)
[i] (i) [k] (k)
0≤k≤N∧M
i=1
The Walrasian welfare
K
(cid:88)
W := (V −C ) (2)
[i] (i)
i=1
is the level of welfare generated by an efficient market. Figure 1 provides an illus-
tration for the quantities K and W.
Note that W is given by a dependent random sum of random variables and there
seems to be no simple way to compute its distribution. A closed form expression
for the distribution of W was given in [15], but it is cumbersome and unsuitable
for further analysis. Calculating the joint distribution of (K,W) is an even more
difficult problem which is, nonetheless, of significant applied interest, as knowing
that distribution (or at least the covariance of K and W) is needed to determine
the behaviour of quantities such as revenue (see e.g. [14]).
However, in many applications in economics one also assumes that the market
is large, i.e. both N and M tend to infinity, and so determining the asymptotic
distribution of (K,W) in such a situation is of substantial interest. In much of
2
the previous mechanism design literature, only the leading term behaviour of the
mechanism outcomes was determined (which is a law of large numbers type result;
e.g. it was shown that, in large markets, K is asymptotically equivalent to N times
the quantity defined in (5) below) and heuristic methods (which often exploited
known order statistics results) were used to compute the order of magnitude of the
error term. The important problem of a higher order distributional approximation
(of a central limit theorem type) to (K,M) has remained so far open.
In this paper, we solve that problem using a strong approximation approach,
establishing a normal approximation to the quantities of interest, together with a
bound for the approximation rate. We note that our approach readily extends to the
analysis of revenue maximisation mechanisms and constrained optimal mechanisms,
through appropriate transformations of the functions F and G (for the problem
formulation, see e.g. [13]).
Now we turn to stating additional assumptions on the model that we will need.
First of all, in our large market setup, the d.f.’s F and G remain fixed and satisfy
the following standard mechanism design assumption:
(A1)The d.f.’s F and G are absolutely continuous, with respective densities f(x)
and g(x) bounded and bounded away from zero on their respective supports [a,b] and
[c,d], such that (a,b)∩(c,d) (cid:54)= ∅ (in other words, a < d and c < b).
It is natural and convenient to consider, for our fixed d.f.’s F and G, a net of
markets
M := (cid:104)α := (M,N),F,G(cid:105), α ∈ A,
α
indexed by the directed set A about which we will make the following assumption:
(A2) We assume that
A := {α = (M,N) ∈ N2 : λ := MN−1 ∈ I},
α
where I := [1−F(d)+(cid:15),1/(G(a)+(cid:15))] for a fixed (cid:15) > 0.
Here (cid:15) is assumed to be small enough so that I (cid:54)= ∅; note that F(d) > 0 and
G(a) < 1 by virtue of the assumption a < d, see (A1). Note also that infI > 0.
The set A is endowed with the natural preorder: for α = (N,M) and α(cid:48) =
(N(cid:48),M(cid:48)), one has α ≤ α(cid:48) iff N ≤ N(cid:48) and M ≤ M(cid:48).
We will be interested in the limiting distributions of the Walrasian quantities K
α
and Walrasian welfares W for the respective markets from the net {M } .
α α αA
The assumption λ ∈ I from (A2) is natural for economic applications (see
α
e.g. p.8 of [12]) and is basically necessary for excluding trivial cases. Indeed, with
probability tending to one, for large α the number of V ’s exceeding d will be equal
i
to (1−F(d)+o(1))N. So if λ < 1−F(d)−(cid:15) for a fixed (cid:15) > 0, it would mean that
α
the total number of sellers M = λ N would be less than the number of buyers with
α
valuations higher than the maximum possible productions cost d, meaning that all
sellers trade and there is rationing on the demand side of the market. Likewise, the
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Costs&Values
C
(M)
V
[1]
C V
(1) [N]
0 1 K N M
Figure 1: The Walrasian quantity K is given by the abscissa of the intersection of the
plots of the buyer and seller order statistics, and the value of Walrasian welfare W is equal
to the area of the shaded region.
situation when λ > 1/G(a) corresponds to a market with excess supply (all buyers
α
trade and there is rationing on the supply side of the market).
Tostatethemainresults, weneedsomefurthernotation. Wewillfrequentlydeal
withvariablescaledfunctionsoftheformh(λ−1t). Forconvenience,forh : [0,1] → R
α
we define
(cid:98)h(s) := h(s∧1), s ≥ 0.
Note that the function (cid:98)h(λ−1t), t ∈ [0,1], is well-defined for all λ > 0.
α α
Using notation h(−1) for the inverse of function h (to avoid confusion with the
reciprocal h−1), introduce the functions
(cid:91)
E (t) := F(−1)(1−t)−G(−1)(λ−1t), t ∈ [0,1], (3)
α α
and put
H(h) := sup{t ∈ (0,1) : h(t) ≥ 0} (4)
(which is well-defined for any h : [0,1] → R with h(0) > 0) and
t := H(E ) ∈ (0,λ ∧1), (5)
α α α
where the right relation holds due to the assumption on I from (A2).
Due to the a.s. convergence of empirical quantile functions (e.q.f.’s) to the the-
oretical ones, for large markets the function E (t/N) approximates the difference
α
between the step-functions whose plots are depicted, respectively, by the dashed
and solid lines in Figure 1 (cf. (12) below). So t N will be the “first order approxi-
α
mation” to K , while the integral of E over (0,t ) will specify such deterministic
α α α
4
approximation to W . The “second order approximation” to both K and W will
α α α
be obtained in this paper using the second order approximation to e.q.f.’s provided
by the sum of the theoretical quantile function and a Brownian bridge process.
Now observe that, provided that f,g are continuous inside their respective sup-
ports, the function E (t) is continuously differentiable for t ∈ (0,1)\{λ }, and
α α
1 1
E(cid:48)(t) = − − , t ∈ (0,λ ∧1). (6)
α f(F(−1)(1−t)) λ g(G(−1)(λ−1t)) α
α α
We will need one more technical assumption on the d.f.’s F and G:
(A3) The densities f and g are differentiable on (a,b) and (c,d), respectively.
Moreover, the functions
d 1 f(cid:48)(F(−1)(t)) d 1 g(cid:48)(G(−1)(t))
= − and = −
dtf(F(−1)(t)) f3(F(−1)(t)) dtg(G(−1)(t)) g3(G(−1)(t))
are bounded on (0,1).
Finally, let
1 (cid:20) t (1−t ) t (1−λ−1t ) (cid:21)
σ2 := α α + α α α , (7)
α (E(cid:48)(t ))2 f2(F(−1)(1−t )) λ2g2(G(−1)(λ−1t ))
α α α α α α
and
(cid:90) tα
ς2 := 2 S (t)dt, (8)
α α
0
where the function S (t), t ∈ (0,t ), is given by
α α
1−t (cid:90) b 1−λ−1t (cid:90) G(−1)(λ−α1t)
(1−F(x))dx+ α G(x)dx.
f(F(−1)(1−t)) g(G(−1)(λ−1t))
F(−1)(1−t) α c
Our main result is the following strong approximation theorem. It implies that,
under the above assumptions, both the Walrasian quantity and Walrasian wel-
fare are asymptotically normal for large markets with approximate distributions
N(Nt ,Nσ2) and N(cid:0)N (cid:82)tαE (t)dt,Nς2(cid:1), respectively. Moreover, their joint distri-
α α 0 α α
butionisalsoasymptoticallynormal. Wealsogiveupperboundsfortheconvergence
rates.
Theorem 1. Under assumptions (A1)–(A3), for the net of markets {M }
α α∈A
there exist a net {R } of realizations of these markets on a common probability
α α∈A
space together with a net of bivariate normal random vectors {(Z(1),Z(2))} with
α α α∈A
zero means and
Var(Z(1)) = σ2, Var(Z(2)) = ς2, Cov(Z(1),Z(2)) = κ := −S (t )/E(cid:48)(t ),
α α α α α α α α α α
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such that
(cid:12) (cid:12)
limsup N1/4 (cid:12)(cid:12)Kα −Ntα −Z(1)(cid:12)(cid:12) < ∞ a.s. (9)
(lnN)1/2 (cid:12) N1/2 α (cid:12)
α
and
N1/2 (cid:12)(cid:12) 1 (cid:18) (cid:90) tα (cid:19) (cid:12)(cid:12)
limsup (cid:12) W −N E (t)dt −Z(2)(cid:12) < ∞ a.s.
lnN (cid:12)N1/2 α α α (cid:12)
α 0
As an illustration to the assertion of the theorem, Figure 2 shows the scatterplot
of 104 independent realizations of the centered and scaled (as per the statement of
Theorem 1) random vector (K ,W ) in the case when α = (250,500), F(x) = x and
α α
G(x) = x2 for x ∈ [0,1], together with the 0.25, 0.5 and 0.95 ellipsoidal quantiles
for the sample. Figure 3 displays these empirical ellipsoidal quantiles together with
the respective (theoretical) ellipsoidal quantiles for the approximating distribution
of (Z(1),Z(2)). Both plots were generated using Mathematica 10.
α α
N-1/2(Wα-N1/2∫0tαEα(t)ⅆt)
0.4
0.2
N-1/2(Kα-Ntα)
-1.0 -0.5 0.5 1.0
-0.2
-0.4
SimulatedData SimulatedEllipsoidalQuantiles
Figure 2: The scatterplot of simulated 104 independent scaled copies of (K ,W ) and
α α
its ellipsoidal quantiles (see the paragraph following Theorem 1 for more detail).
Remark 1. Note that N (cid:16) M under assumption (A2), so one could state the
assertions of the theorem in a similar way using M rather than N as well. Note also
that κ > 0 (as one would expect, of course) since S (t ) > 0 and E(cid:48)(t ) < 0.
α α α α α
Remark 2. The true rate of convergence of the distribution of K to the normal
α
one is most likely N−1/2, as indicated by the bounds (30) below established in the
special case when F = G. So the crudeness of the bound in (9) seems to be due to
the method of proof employed.
6
z
2
0.4
0.2
z
1
-1.0 -0.5 0.5 1.0
-0.2
-0.4
TheoreticalEllipsoidalQuantiles SimulatedEllipsoidalQuantiles
Figure 3: The empirical ellipsoidal quantiles from Figure 2 together with the ellipsoidal
quantiles of the approximating bivariate normal distribution.
The rest of the paper is structured as follows. The proof of Theorem 1 is given
in Section 2. The special case when F = G is further discussed in Section 3.
2 Proof of the main result
The proof will use the common probability space method and transformed uniform
e.q.f.’s, sowewillbeginwithintroducingbasicnotationandrecallingsomekeyfacts.
For a sample X ,...,X with order statistics X ≤ ··· ≤ X , the respective
1 n (1) (n)
e.q.f. X (t), t ∈ [0,1], is defined by
n
X (0) := X , X (t) := X , (i−1)n−1 < t ≤ in−1, i = 1,...,n,
n (1) n (i)
Forthee.q.f.U (t)constructedfromasampleofU(0,1)-distributedi.i.d.U ,...,U ,
n 1 n
√
R (t) := n(U (t)−t), t ∈ [0,1],
n n
denotes the respective uniform quantile process. Recall that both U and R are
n n
random elements of the Skorokhod space D[0,1] of ca`dla`g functions on [0,1].
By the well-known Donsker’s theorem (see e.g. Section 14 of [1]), as n → ∞,
the distribution of R in D[0,1] converges weakly to that of the Brownian bridge
n
process B. A sharp bound on the rate of convergence is given in [5]. A corollary
of that result states that there exists a probability space carrying a sequence of
processes R∗ =d R , n = 1,2,..., and a Brownian bridge process B such that, as
n n
7
n → ∞,
(cid:107)R∗ −B(cid:107) = O(cid:0)n−1/2lnn(cid:1) a.s.,
n ∞
where (cid:107)h(cid:107) := sup |h(t)|, h ∈ D[0,1].
∞ t∈[0,1]
Therefore, as n → ∞,
U (t) = t+n−1/2R (t)=d t+n−1/2R∗(t) = t+n−1/2B(t)+θ (t)n−1lnn (10)
n n n n
for t ∈ [0,1], where (cid:107)θ (cid:107) = O(1) a.s.
n ∞
Proof of Theorem 1. It is easy to see from (1), (4) and (5) that
δ := K N−1 −t = H(E )−H(E ), (11)
α α α α α
where
E (t) := V (1−t)−C(cid:100)(λ−1t), t ∈ [0,1], (12)
α N M α
and V and C denote the e.q.f.’s for the samples of buyer valuations V ,...,V
N M 1 N
and seller costs C ,...,C , respectively. Now we will analyze the behavior of δ for
1 M α
“large” α when market realizations R are defined using the common probability
α
space construction (10).
To that end, we will assume that
(cid:0) (cid:1) (cid:0) (cid:1)
V = V := F(−1) UV and C = C := G(−1) UC , (13)
i i,N i,N j j,M j,M
(cid:8) (cid:9) (cid:8) (cid:9)
where UV ,...,UV and UC ,...,UC are independent triangular
1,N N,N N≥1 1,M M,M M≥1
arrays of row-wise independent U(0,1)-distributed random variables, which are de-
fined on a common probability space with Brownian bridges BV and BC that are
independent of each other so that, as N,M → ∞, for the respective e.q.f.’s one has
UV (t) = t+N−1/2BV(t)+θV(t)N−1lnN, t ∈ [0,1],
N N
(14)
UC (t) = t+M−1/2BC(t)+θC (t)M−1lnM, t ∈ [0,1],
M M
(cid:13) (cid:13) (cid:13) (cid:13)
where (cid:13)θV(cid:13) = O(1) and (cid:13)θC (cid:13) = O(1) a.s.
N ∞ M ∞
In this construction, the e.q.f.’s V and C in (12) are the ones for the samples
N M
in (13), i.e. are given by the compositions
V (t) := (cid:0)F(−1) ◦UV (cid:1)(t) and C (t) := (cid:0)G(−1) ◦UC (cid:1)(t). (15)
N N M M
So to analyse the asymptotic distribution of K , we will now turn to the asymptotic
α
behaviour of E specified in (12) where the terms on the right-hand side are given
α
by (15).
Using (15), (14) and conditions (A1), (A3) to take Taylor series expansions
with two terms and Lagrange’s form of the remainder gives, after elementary trans-
formations,
V (1−t) = F(−1)(cid:0)1−t+N−1/2BV(1−t)+θV(1−t)N−1lnN(cid:1)
N N
N−1/2BV(1−t)+ϑV (t)N−1lnN
= F(−1)(1−t)+ N , t ∈ (0,1), (16)
f(F(−1)(1−t))
8
and, for t ∈ (0,1∧λ ),
α
C (λ−1t) = G(−1)(cid:0)λ−1t+M−1/2BC(λ−1t)+θC (λ−1t)M−1lnM(cid:1)
M α α α M α
M−1/2BC(λ−1t)+ϑC (t)M−1lnM
= G(−1)(λ−1t)+ α M , (17)
α g(G(−1)(λ−1t))
α
(cid:13) (cid:13) (cid:13) (cid:13)
where (cid:13)ϑC (cid:13) +(cid:13)ϑV (cid:13) = O(1) a.s.
M ∞ N ∞
Substituting (16) and (17) into the representation for E in (12) and using (3)
α
gives
E (t) = E (t)+N−1/2Z (t)+ϕ (t)N−1lnN, (18)
α α α α
where (cid:107)ϕ (cid:107) = O(1) a.s. and
α
BV(1−t) B(cid:99)C(λ−1t)
Z (t) := − α , t ∈ (0,1). (19)
α f (F(−1)(1−t)) λ1/2g(cid:16)G(cid:91)(−1)(λ−1t)(cid:17)
α α
One can see from (11) and (18) that if the functional H were differentiable in a
suitable sense at the “point” E , one could derive the desired asymptotic normality
α
of K using a suitable version of the delta method. So we will now turn to studying
α
the local properties of H at E .
α
Since f, g and λ are bounded, we see from (6) that
α
−γ := sup sup E(cid:48)(t) < 0.
α
α∈A t∈(0,λα∧1)
As E (t ) = 0, the above implies that
α α
E (t) ≥ γ(t −t), 0 ≤ t < t ; E (t) ≤ γ(t −t), t < t ≤ λ ∧1.
α α α α α α α
Hence, for any v ∈ D[0,1], setting η := (cid:107)v(cid:107) , one has
∞
inf (E (t )+v(t)) > γη/γ −η = 0,
α α
0≤t<tα−η/γ
sup (E (t)+v(t)) < −γη/γ +η = 0,
α
tα+η/γ<t≤λα∧1
so that the plot of E (t )+v(t) must “dive” under zero within η/γ of t :
α α α
|H(E +v)−H(E )| ≡ |H(E +v)−t | ≤ η/γ = (cid:107)v(cid:107) /γ.
α α α α ∞
Thus, the functional H is Lipschitz continuous in the uniform topology at the
point E . Using this, (11) and (18), we have
α
δ = O(N−1/2) a.s. (20)
α
Further, E is right continuous, it follows from the definition of H that
α
E (t +δ ) ≤ 0. (21)
α α α
9
E (t)
α
V −C
[i−1],N (i−1),M
1 2 Kα N−1 1 t
N N N N
V −C
[i],N (i),M
Figure 4: A graphical illustration of inequalities (21), (22). Recall that K N−1 =
α
H(E (t)) = t +δ .
α α α
On the other hand, using the order statistics V (descending) and C (as-
[i−1],N (i),M
cending) for our samples (13) (cf. the definitions just before (1)), we obtain (see
Fig. 4 for an illustration of the first inequality)
E (t +δ ) ≥ − max (cid:2)(cid:0)V −V (cid:1)+(cid:0)C −C (cid:1)(cid:3)
α α α [i−1],N [i],N (i),M (i−1),M
2≤i≤Kα
(cid:0) (cid:1) (cid:0) (cid:1)
≥ − max V −V − max C −C . (22)
[i−i],N [i],N (i),M (i−1),M
2≤i≤N 2≤i≤M
From a well-known result regarding maximal uniform spacings (see e.g. [8]) and
representation (13), we have
(cid:16) (cid:17)
max UV −UV
(cid:0) (cid:1) 2≤i≤N [i−1],N [i],N
max V −V ≤ = O(N−1lnN) a.s.
[i−1],N [i],N
2≤i≤N mint∈[a,b]f(t)
Sincethesecondtermontheright-handsideof (22)hasthesameorderofmagnitude,
(21) and (22) now yield
E (t +δ ) = O(N−1lnN) a.s.
α α α
Therefore (18) implies that
E (t +δ )+N−1/2Z (t +δ ) = O(N−1lnN). (23)
α α α α α α
Under assumption (A3) we also have
E (t +δ ) = E (t )+δ E(cid:48)(t )+O(δ2) = δ E(cid:48)(t )+O(δ2) a.s. (24)
α α α α α α α α α α α α α
Combining (23) with (24) and using (20) gives
δ E(cid:48)(t ) = −N−1/2Z (t +δ )+O(N−1lnN)
α α α α α α
= −N−1/2Z (t )+N−1/2ψ ω (δ )+O(N−1lnN) (25)
α α α Zα α
10