Lecture 3
The Revenue Equivalence Theorem
Gian Luigi Albano, Ph.D.
Head of R&D
Italian Public Procurement Agency - Consip S.p.A.
Set-up
Class A of auction mechanisms with 4 properties:
1.
A buyer can make any bid above a “reserve” price announced by
the seller;
2.
The winner is the highest-bid buyer;
3.
Each bidder is alike (anonymity of auction rules);
4.
Each bidder follow the same equilibrium bidding strategy
b i = β (v i ), i = 1,..., N,
which is strictly increasing in its argument
Expected profit for each buyer is simply:
(reservation value)(probability of winning)-(expected payment)
Suppose that N
N-1
1 bidders follow the equilibrium bidding strategy β((·))
whereas bidder 1 bids b1 = β(x). Then β(v1) is an equilibrium bidding
function if bidder 1 maximizes his expected profit by choosing x = v1
(Assume private values to be drawn from the cdf F (⋅) on [v,v ])
Any auction rule must specify how much bidder 1 is going to pay given
(b1,…,b
bN),
) that is,
is
p1 = p(b1,,b2,,...,b
, N ) = p(β((x),
),β((v2 ),
),...,,β((vN )).
Given b1= β(x), the expected payment by bidder 1 is
P ( x) =
E
v 2 ,..., v N
[p(β ( x), β (v 2 ),..., β (v N ))].
Since β(⋅) is assumed to be strictly increasing, bidder 1 is the winner if
all other valuations are less than x, thus the probability of winning is
N 1(x).
FN-1
( ) Bidder
Bidd 1’s
1’ expected
d payoff
ff writes
i
Π 1 ((v1, x)) = v1F N −1 ( x)) − P ( x).
)
If v* has to be the equilibrium bidding function, then bidder 1 must
select
l t x = v1, that
th t is,
i ∏1(v
( 1 ,v1) is
i the
th maximal
i l payoff
ff for
f bidder
bidd 1.
1
The following FOC must then be satisfied:
∂ Π 1 (v1, x)
dF ( x)
= v1
− P ' ( x)) = 0 at x = v1,
∂x
dx
N −1
for all v1 ≥ v*, where v* is the reservation value for which any bidder is
indifferent between submitting the bid β(v*) and not entering the
auction That is,
auction.
is v* satisfies
Π i (v * ,v * ) = v * F N −1 (v * ) − P (v * ) = 0.
At x = v1 it has to be true that
v1
dF
N −1
(v1 )
dv 1
= P ' (v1 ).
This is a rather standard fist-order differential equation. Integrating
both sides yields
y
v1
v1
∫ P ( x) = ∫
v*
'
v*
dF N −1 ((u))
u
+c⇔
du
P(v1 ) − P (v * ) = v1F N −11 (v1 ) − v * F N −11 (v * ) −
v1
∫F
N −11
(u)du + c
v*
v1 = v*⇒ c = 0 and P(v*) = v* FN-1(v*) ⇒
P (v1 ) = v1F N −1 (v1 ) −
v1
∫
v*
F N −1 (u)du .
Let us consider the seller’s problem. From her viewpoint, both v1 and
P(v1) are random variables. The seller’s expected payment from bidder
1 is
i then
h the
h expected
d value
l off P(v
P( 1)
p 1 ≡ E v1 [ p 1 ] =
v
∫
P ( v1 ) F ' ( v1 )
v*
v
= ∫ ⎡ v1 F
⎢
* ⎣
v
v
=
∫ v1 F
v*
N −1
N −1
( v1 ) −
∫
v1
v*
F
N −1
( u ) du ⎤ F ' ( v 1 )
⎥⎦
v1
⎡
( v1 ) F ( v1 ) − ∫ * ∫ * F
v ⎢
⎣ v
'
v
N −1
( u ) du ⎤ ⋅F ' ( v 1 ).
⎥⎦
By changing the order of integration we get
⎡ v1 F N −1 ( u ) du ⎤ F ' ( v ) = v ⎡ v F ' ( u ) du ⎤ F N −1 ( v ) dv .
1
1
1
∫v* ⎢⎣ ∫v*
∫v* ⎢⎣ ∫v1
⎥⎦
⎥⎦
v
The intuition of interchanging the order of integration
j
i
∞
i
∞
∞
∑ ∑ v(i, j) = ∑ ∑ h(i, j)
i= 0 j = 0
1
42 43
vertical
j = 0 i= j
1
42 43
horizontal
∫
∫ [v F (v ) + F (v ) − 1]F
p1 =
=
N −1
'
N −1
v
F
(v
)F
(v
)
+
F
(v
)
−
1
F
(v1 )}dv 1
[ 1 ]
*{ 1
1
1
v
v
v
v*
'
1
1
1
N −1
(v1 )dv 1 .
Main result
Assume IID values and suppose
pp
that bidders are risk-neutral.
The common equilibrium bidding strategy of the family A of
auction mechanisms yields the seller an expected revenue of
N
∫
'
N −1
vF
(v)
+
F
(v)
−
1
F
] (v)dv,
* [
v
v
where v* is the reservation value below which it is
unprofitable to submit a bid.
This result does not include any reference to the equilibrium bidding
function β((x).
) In a first-price
p
sealed-bid auction β((x)) is easilyy derived.
Suppose the seller publicly announces a reserve price r. Then any
bidder i with value vi has an incentive to participate, that is, v* = r. In
this mechanism, for a bidder with value v
P(v) = Pr {β (v) is the highest bid }β (v)
= F N −1 (v)β (v) ⇒
β (v) = v −
1
F
N −1
∫
(v)
v
r
F N −1 (u)du , if v ≥ r.
The first-price and the second-price auctions belong to the class of
auctions A.
The next step is to show that the class A is optimal (max seller’s
expected revenue) given the appropriate choice of the reserve
price. For any auction mechanism there is some implied reservation
value below v* which buyers will choose not to bid.
Therefore there is a probability FN(v*) that all N buyers decide not to
bid. The total expected return for the seller then writes
v 0 F (v ) + N
N
*
∫
'
N −1
vF
(v)
+
F
(v)
−
1
F
] (v)dv,
* [
v
v
where v0 is the seller’s
seller s value for the object.
object The optimal v* must
satisfy the FOC:
Nv 0 F N −11 (v
( * )F
) ' (v
( * ) − [Nv * F ' (v
( * ) + F (v
( * ) − 1]F N −11 (v
( *) = 0 ⇒
*
1
−
F
(v
)
v* = v0 +
.
'
*
F (v
( )
A sufficient condition for a maximum is
d ⎡1 − F (v
( * )⎤
⎥ < 0.
* ⎢
'
*
dv ⎣ F (v ) ⎦
Proposition
Assume IID values and bidders’ risk neutrality. The members of
the family A of auctions mechanisms, which maximize the
expected gain of the seller, are those for which the reservation
value v* below which it is not worthwhile bidding satisfies
*
1
−
F
(v
)
*
v = v0 +
F ' (v
( *)
((1)) ,
independent of the number of bidders!
Main implications:
• both the first- and the second-price auctions are optimal if the
reserve is chosen according to (1);
• the optimal reserve price is strictly higher than the seller’s
valuation.
To see the underlying intuition of the last point consider a secondprice auction with 2 bidders…
The revenue effect of the optimal reserve price: An example
Two bidders; F(v) = v on [0,1]; v0 = 0.
According to the last proposition, it is optimal for the seller to adopt
an auction mechanism so that only bidders with reservation values
exceeding
*
*
*
*
v : v =1− v ⇔ v = 1
2
find worthwhile bidding. Therefore r = 1/2. The optimal bidding
strategy in this case becomes
1
11
β (v) = v + , ∀v ≥1/2.
2
8v
The seller’s expected
p
revenue when adopting
p g the optimal
p
reserve
price writes
⎛1
1 1⎞
5
P ≡ E[β (v (1:N ) )] = ∫ ⎜ v +
.
⎟ 2v
{ dv =
1/ 2⎝ 2
8 v ⎠ Fv'
12
I
1
( 1:N
1N)
If instead,
If,
instead the seller chooses r = 0 (that is,
is she always sells the good)
then β(v) = (1/2)v, and the expected revenue is 1/3 < 5/12.
Another look at the Revenue Equivalence Theorem
P II (v)
( ) = Pr( win
i ) ⋅ E [2 nd highest bid | β (v)
( ) highest bid ]
= Pr( win ) ⋅ E [2 nd highest value | v highest value ]
= Pr(( win ) ⋅ E [Y˜1 | Y˜1 < v]]
= G(v) ⋅ E [Y˜1 | Y˜1 < v]
v
= G(v)
( )⋅
=
∫ yg( y)dy
0
G(v)
v
y)dyy
∫ yg( y)
⎛
⎞
where Y˜ (1:N −1)≡ max ⎜⎜ v˜ 2 ,..., v˜ N ⎟⎟ and G(⋅) ≡ F N −1 (⋅).
1
424
3
⎝ N −1 rv' s ⎠
0
P I (v) = Pr( win ) ⋅ β I (v)
= G(v)
( ) E [Y1 | Y1 < v ],
which is the same as in a second-price auction.
1
G(v)
G( y)
Expected profit
Expected payment
0
v
1
y→
Because the seller’s expected revenue is just the sum of the ex ante prior to knowing their values - expected payments made by the
bidders, the expected revenues in the two auctions must be the same.
Let’s see this last point in more detail
The ex ante expected payment of a particular bidder in either
auction is
v
⎛
⎞
A
A
E[P
[ (v)]
(v˜ )] = ∫ P (v) f (v)dv = ∫ ⎜ ∫ yg(y)dy ⎟ f (v)dv,
⎠
0
0 ⎝ 0
1
1
where A = {{I,, II}.
} Interchanging
g g the order of integration
g
we g
get
⎛1
⎞
( )d ⎟vg(v)dv
( )d =
∫ ⎜⎝ ∫ f (y)dy
⎠
0
v
1
1
1 F(v)
F( ))g(v)dv.
( )d
∫ v(1−
0
The seller’s expected revenue is then
1
E [ R A ] = N ⋅ E [ P A ( v~ )] = N ∫ v (1 − F ( v ) )g ( v ) dv .
0
Note that the density of Y(2:N), the second-highest of N values
f ( 2:N ) (v) = N (1 − F (v)) f (1:N −1) ,
and since
f (1:N −1) (v) = g(v)
we can write
it
E[R A ] =
1
∫ vf
0
( 2:N )
(v)dv = E [Y˜( 2:N ) ].
The Simple Economics of Optimal Auctions
(An asymmetric) Set-up
A seller faces N individuals interested in acquiring the good on sale,
seller’s value is nought.
From the seller’s viewpoint, each buyer i independently draws his
value from a distribution function
Fi ((v i ) on [[v i ,,v i ]].
Main Question
What sales mechanism will maximize the seller’s expected
profits?
The
h solution
l
in 3 steps
1
1.
Draw the “demand
demand function
function” for each buyer i where
vi is on the price axis
1-F
1
Fi(vi) ≡ q is on the quantity axis.
axis
vi
MR i (v i )
Di (v i )
vi
1
q ≡ 1 − Fi (v i )
1. Draw the “demand function” for each buyer i where
vi is on the price axis
1-Fi(vi) ≡ q is on the quantity axis.
2. Calculate and draw the “marginal revenue” curve in the standard
fashion that is,
fashion,
is
MR i (v
( i) ≡
d
1
−1
1
−1
1
q
⋅
F
1
−
q
=
F
1
−
q
−
q
⋅
.
)] i ( )
[
i (
−1
dq
f i (Fi (1 − q ))
Rewriting the marginal revenue as function of vi we get
⎧
1 − Fi (v i )
⎪v i −
, if v i ∈ [v i ,v i ]
MR i (v i ) = ⎨
f i (v i )
otherwise
⎪⎩
−∞
3. Consider the following “second marginal revenue auction”:
a) each bidder announces his value (we will prove that there is no
incentive to lie);
);
b) translate each bidder’s value into marginal revenue (assume it
to be monotonically increasing);
c) award the object to the bidder with the highest marginal
revenue;
d) the price paid is the lowest value this bidder could have
announced without losing the auction
Remarks
• If no bidder
bidd has
h a positive
iti MR,
MR then
th no sale
l
• If only one bidder, say bidder j, has a positive MRj, then the selling
price p is equal to v*, where v* solves MRj (v*) = 0
• If more than one bidder announces a value that translates into
positive MR, then the one with the highest MR, say 1, gets the object
and pays MR-1(MR(2)), where MR(2) is the second-highest MR
• Highest value does not necessarily translate into highest MR (see
Myerson (1981))
• “Honest report” is a (weakly) dominant strategy since if a bidder the
auction, the amount paid is independent of his report
A Numerical Example
1
( A ) = 1 − v A ⇒ p A = 10 − 10q
10 A .
v˜ A iis U [ 0,10 ] → q A ≡ 1 − FA (v
10
vB − 5
ṽv B is U [ 5,20 ] → q B ≡ 1 − FB (v B ) = 1 −
⇒ p B = 20 − 15q B .
15
⎛ 10 − v A ⎞
MR A (q A ) = 10 − 20q A → MR A (v A ) = 10 − 20⎜
⎟ = 2v A − 10.
⎝ 10 ⎠
⎛ 20 − v B ⎞
MR B (q B ) = 20 − 30q B → MR B (v B ) = 20 − 30⎜
⎟ = 2v B − 20.
⎝ 15 ⎠
MR A ≥ 0 ⇔ v A ≥ 5, MR B ≥ 0 ⇔ v B ≥ 10, and MR A ≥ MR B ⇔ v A ≥ v B − 5.
Suppose
pp
A announces vA = 8 and B announces vB = 12.
Then MRA = 6 > 4 = MRB. A wins despite vA < vB and pays p = 7 since 7 =
MRA-1 (MRB).
Price Discriminating Monopolist and Optimal Auctions
Assumptions:
p
• N different markets;
• each consumer is interest at most in one unit;
• consumers in market i have private values as in the first example;
• MRi is downward sloping for all i = 1,
1 …, N.
N
• marginal cost is zero up to a certain quantity
Q˜ r.v. defined on [ Q, Q], with density h(⋅).
• the monopolist can set prices (and implicitly quantities) contingent
˜ and seeks to maximize her profits.
on the realization of Q
Call pi(v, Q) the probability that consumer i with value v will acquire
a unit if the monopolist’s capacity is Q
Q
p i (v) ≡
∫ p (v,Q)h(Q)dQ.
i
Q
The monopolist’s objective function can be formulated as the
difference between the expected social value of units sold and the
expected consumers’ surplus. The former writes
Q
E[SV ] =
N vi
N vi
(v Q)dvdQ = ∑ ∫ vf (v) p (v)dv.
(v)dv
∫ h(Q)∑ ∫ vf (v) p (v,Q)dvdQ
i
Q
i=1 v i
i
i
i
i=1 v i
What about consumers’ surplus? Note that a buyer with the lowest
possible value will always get zero surplus since the monopolist will
always
y set a p
price higher
g
that the lowest p
possible p
private value.
Suppose that a buyer’s value is distributed between 0 and 100 and
that monopolist charges a price of 50 with probability 0.5 and a
price of 100x or less, 0.5 < x ≤ 1, with probability x, that is,
Pr(( p˜ ≤ 100 x)) = x ⇔ Pr(( p˜ ≤ y) =
y
, 50 < y ≤ 100 .
100
Food ffor Thought
g 1 ((*))
Show that this situation might happen if the monopolist sold in a
market with demand curve p = 100 - q, and the monopolist’s
capacity were uniformly distributed between zero and 100.
100
Thus a buyer with v = 80 can expect to buy a unit at p = 50 with
probability 0.5 and at 50 < p ≤ 80 uniformly distributed with
probability 0.3. (**)
For buyer i we can write
⎧⎪ 0, if v < 50 (lowest possible price)
1
p i (v) = ⎨
v, if 50 < v ≤ 100 .
⎪⎩
100
Integrating along the value axis we see that consumer I’s expected
surplus can be written as
E[CS i (v)] =
v
∫ p ( x)dx .
i
vi
The expected surplus for all N buyers writes
E[CS ] =
Q
N
Q
i=1 v i
vi
∫ h(Q)∑ ∫
v
N
vi
i=1 v i
f i (v) ∫ p i ( x,Q)dx dvdQ = ∑
vi
∫
v
f i (v) ∫ p i ( x)dx dv .
vi
Let’s visualize buyer i’s expected surplus when his value v = 80
v i = 100
v = 80
E[CSi (v = 80)]
v i = 50
0
0.5 = pi (v i )
1 = pi (v i )
Th monopolist’s
The
li ’ expected
d revenue iis now
N
N
vi
vi
v
E[R] = ∑ ∫ vff i (v)
( ) p i (v)
( )dv
d − ∑ ∫ f i (v)
( ) ∫ p i ( x))dxdv
d d .
vi
i=1 v i
i=1 v i
1
44
42 4 44
3 1
4 4 42
4 4 43
E [ SV ]
E [CS ]
The monopolist’s problem becomes finally:
Max E[R]
N
s.to
vi
∑∫
f i (v) p i (v,Q)dv ≤ Q, ∀Q.
ii=1 v i
(this is a feasibility constraint on sales relative to capacity)
(QC )
The decision variables are prices in N markets or, alternatively, the
value at which pi(v,Q) switches from zero to one. Thus there are
two additional constraints:
1.
0 ≤ pi(v,Q) ≤ 1;
2.
pi(·,Q) non-decreasing.
Since the
Si
th monopolist
li t can make
k the
th allocation
ll
ti rule
l contingent
ti
t on Q,
Q we
can maximize E[R] for each possible Q. Then the problem writes
N
max E[R(Q)] = ∑
(QC )
vi
∫ vf (v) p (v,Q)dv − ∑ ∫
i=1 v i
s.to
N
vi
i
i
i=1 v i
v
f i (v) ∫ p i ( x,Q)dxdv
vi
Interchanging the order of integration of the second term in the
maximand we get
N
vi
vi
∑ ∫ p (v,Q) ∫
i
i=1 v i
v
N
f i ( x)dxdv =∑
vi
∫ [1 − F (v)] p (v,Q)dv .
i
i
i=1 v i
The monopolist’s expected revenue becomes
N vi
⎡1 − Fi (v) ⎤
Q) =∑ ∫ MR i ((v)) f i ((v)) p i ((v,Q)dv
Q) .
∑ ∫ ⎢⎣ f (v)
⎥ f i ((v)) p i ((v,Q)dv
i( ) ⎦
i=1 v i
i=1 v i
N
vi
In order to get a flavor of the problem
problem’ss solution, note that fi(v) > 0
enters linearly both the objective function and (QC). Moreover, both
the objective function and (QC) are linear in pi(v,Q)
Since MRi is increasing in v, the constraint ∂pi(v,Q)/ ∂v ≥ 0 is never
binding. The solution of the monopolist’s problem takes the following
form.
For each possible value of Q, the monopolist should:
1
1.
Sell to all buyers in each market for whom MRi(v) ≥ 0 if the quantity
constraint is not binding.
2.
If the quantity constraint is binding, sell units to the buyers with
th highest
the
hi h t marginal
i l revenues, by
b choosing
h i
prices
i
t equate
to
t the
th
marginal revenues of the lowest-valued buyers actually supplied in
each market.
Bidders’ risk aversion
u : ℜ + → ℜ; u(0) = 0, u ' (⋅) > 0, u '' (⋅) < 0.
The second-price auction is not affected by risk aversion!
First-price auction: Suppose that the equilibrium bidding strategy is
given by an increasing and differentiable function
γ : [0,1] → ℜ + , with γ (0) = 0.
If all bidders but bidder 1 follow this equilibrium bidding strategy,
then bidder 1 will never bid more than γ(1)
Given a value v, bidder 1’s problem is to choose z in [0,1] and γ(z) to
max z [G(z)u (v − γ (z))].
FOC :
g(z) ⋅ u (v − γ (z)) − G(z) ⋅ γ ' (z) ⋅ u ' (v − γ (z)) = 0.
In a symmetric equilibrium it must be optimal for bidder 1 to choose z
= v. Then
g(v) ⋅ u (v − γ (v))
'
=
G(v)
⋅
u
(v − γ (v)) ⇔
'
γ (v)
u (v − γ (v)) g(v)
γ ((v)) = '
⋅
u (v − γ (v)
( )) G(v)
G( )
'
((1))
Under risk neutrality, u(v) = v, so that (1) can be rewritten
g(v)
β (v) = [v − β (v)] ⋅
.
G(v)
'
Next notice that if u(·) is strictly concave and u(0)=0, then
u( y)
> y, ∀y > 0,
'
u ( y)
that is, average utility > marginal utility. Hence
u (v − γ (v )) g(v )
g(v )
γ (v) = '
⋅
> (v − γ (v )) ⋅
.
u (v − γ (v )) G(v )
G(v )
'
β (v) > γ (v) ⇒ (v − γ (v)) ⋅
g(v)
g(v)
> (v − β (v)) ⋅
⇒ γ ' (v) > β ' (v)!
G(v)
G(v)
The initial conditions γ((0)) = β((0)) = 0 and the contradiction above imply
py
that
γ (v) > β (v), ∀v > 0.
Asymmetries among Bidders
Asymmetric First-Price Auctions with Two Bidders
• Two bidders: 1 and 2
• Private signals: (X1, X2) ID according to (F1, F2) defined on [0,ω1]
and
d [0,ω
[0 2]
• Suppose there exists an equilibrium in which the 2 bidders
follow the strategies (β1, β2) that are strictly increasing and
differentiable with inverses
φ1 ≡ β1−1 , φ2 ≡ β 2−1
• It must be the case that: β1(0) = 0 = β2(0), and β1(ω1) = β2(ω2).
Let
b H ≡ β1 (ω1 ) = β 2 (ω2 )
Given that bidder j = 1,2 is following the strategy βj, the expected
payoff of bidder i = 3
3-jj when his value is xi and bids an amount b < bH
is
Π i (b, xi )
=
=
F j ( φ j ( b ))( x i − b )
H j ( b )( x i − b )
where Hj((·)) ≡ Fj (ϕ ((·))
)) denotes the distribution of bidder jj’ss bids
The FOC for bidder i = 1,2 requires that for all b < bH
h j (b)(φi (b) − b) = H j (b)
for j = 3-i and, as usual,
h j (b) ≡ H 'j (b) = f j (φi (b))φ 'j (b)
is the densityy of jj’s bids. The FOC can be rearranged
g as follows
F j (φ j (b))
1
φ ( b) =
f j (φ j (b)) (φi (b) − b)
'
j
A solution to the system of differential equations above – one for each
bidder – together with the relevant boundary conditions constitutes an
equilibrium of the first-price auction.
Remarks:
• Unfortunately,
Unfortunately an explicit solution can be obtained only under
particular circumstances;
• We can nonetheless deduce certain properties of the equilibrium
bidding function by making some assumptions on the nature of
asymmetries.
Weakness Leads to Aggression
Suppose that bidder 1’s values are “stochastically higher” than those of
bidder 2. Let us make an even stronger assumption, namely that F1
dominates F2 in terms of the reverse hazard rate, that is
ω1 ≥ ω2 , andd ∀x ∈ (0, ω2 ),
)
f1 ( x ) f 2 ( x )
>
.
F1 ( x ) F2 ( x )
Remark. Reverse hazard rate dominance implies that F1 stochastically
dominates F2, that is, F1(x) ≤ F2(x).
Ex.: F1(x) = [F2(x)]θ, for some θ > 1.
Under the above assumprion, we will call bidder 1 the “strong”
bidder and bidder 2 the “weak” bidder.
Proposition
Suppose that the value distribution of bidder 1 dominates that of
bidder 2 in terms of reverse hazard rate. Then, in a first-price
auction, the “weak” bidder 2 bids more aggressively than the
“strong”
st o g b
bidder
dde 1 – tthat
at iss
β1 ( x ) < β 2 ( x ), ∀x ∈ (0, ω2 ).
Remark. Since for all b, 0 ≤ b ≤ bH, ϕ1(b) > ϕ2(b) it follows from
FOCs that
H 2 ( b)
H ( b)
= φ1 (b) − b > φ2 (b) − b = 1 ,
h2 (b)
h1 (b)
so that the distribution of bidder 1’s bids H1 dominates that of
bidder 2 H2 in terms of the reverse hazard rate.
Bits of Intuition…
• If β1(x)
( ) > β2(x)
( ) Î bidder
bidd 1’s
1’ distribution
di t ib ti
off bids
bid is
i stochastically
t h ti ll
higher than that of bidder 2
• Consider a particular b and suppose that β1(x1) = β2(x2) = b Î x1 < x2
• That is, the weak bidder faces both a stochastically higher
distribution of competing bids and has a higher value
• Since both forces cause bids to be higher, if it were optimal for the
strong bidder to bid b when his value is x1, it cannot be optimal for
the weak bidder to bid b when his value is x2 Î contradiction!
• In other words, the two forces must balance each other: The weak
bidder faces a stochastically higher of competing bids than does the
strong bidder, but the value at which any particular bid b is optimal
for the weak bidder is lower than it is for the strong bidder.
First- vs Second-Price Auctions with Asymmetric
Bidders: Revenue Ranking and Efficiency
Proposition
TBW