Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
1
Optimal Dynamic Mechanism Design
and the Virtual Pivot Mechanism
Sham Kakade
Ilan Lobel
Hamid Nazerzadeh
Microsoft Research, New York University and University of Southern California
skakade@microsoft.com, ilobel@stern.nyu.edu, hamidnz@marshall.usc.edu
Appendix
Appendix A: Proofs for Section 3
Lemma A.1 For any reporting strategy y ! z and initial type x, the partial derivative of the
expected value of agent i Viy!z (x) (see definition in Eq. (10)) with respect to x exists and is:
"1
#
X
@V y!z (x)
t @
t
=E
vi,t (a , si,0 , si,1 , ..., si,t ) si,0 =x
@x
@si,0
t=0
(where the expectation is under y ! z and T i ). Furthermore, it is bounded by
@Viy!z (x)
V̄

.
@x
1
Proof:
From Assumption 2.2, we have that for all i, t, a, x and si,1 , ..., si,t ,
@
vi,t (at , x, si,1 , ..., si,t )  V̄ < 1.
@x
y!z
Therefore, by Lebesgue’s Dominated Convergence Theorem, the partial derivative @ V̄ @x (x) exists,
"1
#
"1
#
X
X @
@V y!z (x)
@
t
t
=
E
vi,t (at , x, si,1 , ..., si,t ) = E
vi,t (at , x, si,1 , ..., si,t )
@x
@x
@x
t=0
t=0
and | @V
y!z
@x
(x)
|E
⇥ P1
t=0
t
⇤
V̄ =
V̄
1
.
Proof of Lemma 3.1 Any strategy available to the agents in the relaxed environment is a feasible strategy in the dynamic environment. Therefore, if all other agents are truthful, any profitable
deviation from the truthful strategy in the relaxed environment implies a profitable deviation in
the dynamic environment. Since no such profitable deviations exist in the dynamic environment,
we obtain that the mechanism M is incentive compatible in the relaxed environment. Therefore,
the optimal revenue in the relaxed environment provides an upper bound on the revenue in the
dynamic environment.
Proof of Lemma 3.2 For consistency with the notation used in the rest of the paper, we represent the utility of agent i with initial type si,0 = z 0 and reporting his initial type as ŝi,0 = z by
Uiz!z (z 0 ), assuming all other agents are truthful. Respectively, Viz!z (z 0 ) and Piz!z (z 0 ) represent
the expected discounted value and payment of agent i under initial type z 0 and reported initial
type z 0 .
The expected utility of agent i under reporting strategy z ! z and initial type x is
Uiz!z (z) = Viz!z (z)
Piz!z (z).
(27)
Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
2
Under the same reporting strategy z ! z, but under initial type z 0 , the utility of agent i is
Uiz!z (z 0 ) = Viz!z (z 0 )
Piz!z (z 0 ).
(28)
The payments are functions only of reported types, not true types, and therefore, Piz!z (z) =
Piz!z (z 0 ). Therefore, for any z 6= z 0 , combining Eqs. (27) and (28) yields
Uiz!z (z)
z
Uiz!z (z 0 ) Viz!z (z)
=
z0
z
Viz!z (z 0 )
.
z0
0
0
At the same time, if z > z 0 , incentive compatibility yields Uiz !z (z 0 )
0
0
Uiz!z (z) Uiz !z (z 0 ) Uiz!z (z)

z z0
z
Uiz!z (z 0 ), hence
Uiz!z (z 0 )
.
z0
z!z
Since the partial derivative @V @x (x) exists for all x (see Lemma A.1), we can take the limit as
z 0 " z and obtain that the left-hand side derivative of Uiz!z (z) satisfies
d Uiz!z (z) @Viz!z (s)

dz
@s
s=z
.
Using the same argument for z 0 > z, we obtain that the right-hand side derivative of Uiz!z (z)
satisfies
d+ Uiz!z (z) @Viz!z (s)
.
dz
@s
s=z
@V z!z (s)
Since | i @s | is bounded by 1V̄ by Lemma A.1, we get that the absolute value of both the lefthand and right-hand side derivatives of Uiz!z (z) are also bounded by 1V̄ . The function Uiz!z (z)
is, therefore, 1V̄ -Lipschitz-continuous and, thus, di↵erentiable almost everywhere. At all points
where the derivative exists,
Uix!x (x)
dUiz!z (z)
dz
=
0
0
Uix !x (x0 )
@Viz!z (s)
@s
=
Z
x
x0
s=z
. Therefore, the envelope condition follows:
dUiz!z (z)
dz =
dz
Z
x
x0
@Viz!z (s)
@s
s=z
dz.
(29)
Plugging in the result from Lemma A.1, we obtain the desired result.
Proof of Lemma 3.3
For notational convenience, we write:
@vi,t (at , si,0 , si,1 , ..., si,t )
@si,0
si,0 =si,0
=
@vi,t (at , sti )
@si,0
where the sti implicitly depends on the first signal.
Consider first the utility UiM (s) of an agent i under an initial type profile s, which is given by
#
Z si,0 "X
1
t t
M
M
t @vi,t (a , si )
Ui (si,0 , s i,0 ) Ui (0, s i,0 ) =
E
si,0 = z, s i,0 dz.
@si,0
0
t=1
from Lemma 3.2. Taking the expectation of this term over all possible first period signals
s1,0 , ..., sn,0 , we obtain
# ◆
Z 1 ✓Z si,0 "X
1
t t
@v
(a
,
s
)
i,t
i
t
E[UiM (si,0 , s i,0 ) UiM (0, s i,0 )] =
E
si,0 = z dz fi (si,0 )dsi,0 .
@s
i,0
0
0
t=1
Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
3
Inverting the order of integration,
E[UiM (si,0 , s
i,0 )
#
1
t t
X
@v
(a
,
s
)
i,t
i
t
UiM (0, s i,0 )] =
E
si,0 = z fi (si,0 )dsi,0 dz
@s
i,0
0
z
t=1
#
Z 1 "X
1
t t
@v
(a
,
s
)
i,t
i
t
=
E
si,0 = z (1 Fi (z))dz.
@s
i,0
0
t=1
Z 1Z
1
"
By multiplying and dividing the right-hand side of the equation above by the density fi (z) we
obtain an unconditional expectation,
"1
#
X 1 Fi (si,0 ) @vi,t (at , st )
i
M
M
t
E[Ui (si,0 , s i,0 ) Ui (0, s i,0 )] = E
.
fi (si,0 )
@si,0
t=1
P1 t
Now note that the discounted
sum
of
payments
E[
pi,t ] is equal to the expected discounted
t=1
P1 t
t t
valuation of agent i – E[ t=1 vi,t (a , si )] – minus her utility, which yields the claim.
Proof of Lemma 3.4
Observe that for multiplicatively-separable value functions
@vi,t (at , sti )
= A0i (si,0 )Bi,t (at , si,1 , ..., si,t )
@si,0
and, therefore, Eqs. (7) and (6) are identical. Similarly, for additively-separable functions,
@vi,t (at , sti )
= A0i (si,0 )Ci (at )
@si,0
and, therefore, Eqs. (7) and (6) are again identical.
Proof of Corollary 3.1 For an IC mechanism M, the expected discounted sum of payments by
agent i is equal to
"1
#
"1
#
X
X
⇥
⇤
t
t
E
pi,t = E
↵i (si,0 )vi,t (at , sti ) + i,t (at , si,0 )
E UiM,T (si,0 = 0)
t=0
t=0
by⇥ taking expectations
over s i,0 (see Eq. (7)). Since the mechanism satisfies IR,
⇤
M,T
E Ui
(si,0 = 0)
0 and, therefore,
E
"
1
X
t=0
t
#
pi,t  E
"
1
X
t=0
t
↵i (si,0 )vi,t (at , sti ) +
i,t (a
t
, si,0 )
#
.
The profit of M is given by the sum of payments minus the cost of actions (see Eq. (4)),
"1
!#
n ⇣
⌘
X
X
t
ProfitM  E
↵i (si,0 )vi,t (at , sti ) + i,t (at , si,0 ) ct (at )
.
t=1
i=1
The bound above is valid for all IC and IR mechanisms. By maximizing over the set of all possible
allocation rules (payment rules do not enter the equation above), we obtain the desired result.
Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
4
Proof of Lemma 3.5
type z is
The expected utility of agent i under reporting strategy x0 ! z and initial
0
0
0
Uix !z (z) = Vix !z (z)
Pix !z (z),
(30)
0
Pix !z (z)
where
is the expected discounted sum of payments of agent i under reporting strategy
0
0
0
x ! z and initial type z (see similar definitions of Uix !z (z) and Vix !z (z) in Eqs. (9) and (10)).
Under the same reporting strategy x0 ! z, but under initial type z 0 , the utility of agent i is
0
0
Uix !z (z 0 ) = Vix !z (z 0 )
0
Pix !z (z 0 ).
(31)
0
The payments are functions only of reported types, not true types, and therefore, Pix !z (z) =
0
Pix !z (z 0 ). Therefore, for any z 6= z 0 , combining Eqs. (30) and (31) yields
0
Uix !z (z)
z
0
0
0
0
Periodic ex-post IC guarantees that Uix !z (z 0 )
0
Uix !z (z)
z
0
Uix !z (z 0 ) Vix !z (z)
=
z0
z
0
Vix !z (z 0 )
.
z0
0
Uix !z (z 0 ). Therefore, for any z > z 0 ,
0
0
0
Uix !z (z 0 ) Uix !z (z)

z0
z
Uix !z (z 0 )
.
z0
z!z
Since the partial derivative @V @x (x) exists for all x (see Lemma A.1), we can take the limit as
0
z 0 " z and obtain that the left-hand side derivative of Uix !z (z) for any constant x0 satisfies
0
0
d Uix !z (z) @Vix !z (s)

dz
@s
s=z
.
0
Using the same argument for z 0 > z, we obtain that the right-hand side derivative of Uix !z (z)
satisfies
0
0
d+ Uix !z (z) @Vix !z (s)
.
dz
@s
s=z
0
@V x !z (s)
Since | i @s
| is bounded by 1V̄ by Lemma A.1, we get that the absolute value of both the left0
0
hand and right-hand side derivatives of Uix !z (z) are also bounded by 1V̄ . The function Uix !z (z)
is, therefore, 1V̄ -Lipschitz-continuous and, thus, di↵erentiable almost everywhere. At all points
where the derivative exists,
0
Uix !x (x)
0
dUix !z (z)
dz
0
0
=
0
@Vix !z (s)
@s
Uix !x (x0 ) =
Z
x
x0
s=z
0
. Therefore, the envelope condition follows:
dUix !z (z)
dz =
dz
Z
x
x0
0
@Vix !z (s)
@s
s=z
dz.
Proof of Lemma 3.6 The envelope condition from the relaxed environment (see Lemma 3.2)
also applies to this setting since a deviation that is feasible in the relaxed environment (that is,
using reporting strategy z ! z for an initial type z 0 ) is also feasible in the dynamic environment.
Therefore, if the mechanism is incentive compatible, then it satisfies Eq. (29), which is identical to
Eq. (15).
To see that IC implies the dynamic monotonicity condition in Eq. (16), simply note that IC
is equivalent to Eq. (11) and Eqs. (15) and (13) are respectively equal to the left-hand and the
right-hand side of Eq. (11). We thus obtain that IC implies Eq. (16).
We now show that if both Eqs. (15) and (16) hold, then the mechanism is IC. If both equations
hold, then for all x and x0 ,
Z x
Z x
0
0
0
0
@Viz!z (s)
@Vix !z (s)
x!x
x0 !x0
0
Ui (x) Ui
(x ) =
dz
dz = Uix !x (x) Uix !x (x0 ),
@s
@s
s=z
s=z
x0
x0
Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
5
where the last equality follows from Lemma 3.5. The equation above is equivalent to IC (see Eq.
(11)), when the mechanism is periodic ex-post IC for t 1.
Appendix B: Proofs for Section 4
Lemma B.1 Suppose Assumptions 4.1 and 4.2 hold. Then ↵i is strictly increasing for multiplicatively separable functions and i,t is strictly increasing for additively separable functions.
Proof:
For simplicity of nation, let s = si,0 . Also, let ⌘i (s) denote the hazard rate, i.e.,
⌘i (s) =
fi (s)
.
1 Fi (s)
In the additive case,
@
i,t (a
t
, s)
@s
=
⌘i0 (s) 0
A (s)Ci,t (at )
⌘i2 (s) i
1
A00 (s)Ci,t (at )
⌘i (s) i
where (·)0 denotes a partial derivative with respect to s. By the assumptions that Ai is concave and
strictly increasing, and the hazard rate is positive and strictly increasing, we have that the above
has the same sign as Ci,t . In the multiplicative case, first note that ↵i (s) = 1 ⌘i1(s) (log Ai (s))0 .
Therefore,
⌘ 0 (s) A0i (s)
1
↵i0 (s) = i 2
(log Ai (s))00
⌘i Ai (s) ⌘i (s)
which is positive by the assumption.
Proof of Lemma 4.1 If agents are truthful, by Eq. (24), the expected payment of each agent i
given si,0 is equal to max{p?i (si,0 ), 0}, where 0 occurs if agent i is excluded from the system (i 2
/ ai ).
Namely,
Z ŝi,0
@Viz!z (si,0 , ŝ i,0 )
?
t
pi (ŝ0 ) = V (si )
dz
(32)
@si,0
si,0 =z
0
where
@Viz!z (si,0 , ŝ
@si,0
i,0 )
si,0 =z
=E
"
1
X
t=1
?t
t @vi,t (q , si,0 , si,1 , . . . si,t )
|si,0 =z si,0 = z, s
@si,0
i,0
= ŝ
i,0
#
For notational convenience, we write:
@vi,t (at , si,0 , si,1 , ..., si,t )
@si,0
si,0 =si,0
=
@vi,t (at , sti )
@si,0
where the sti implicitly depends on the first signal. The expected payment of agent i is equal to:
Z
1
max{p?i (s, s0, i ), 0}fi (s)ds
0
"1
Z 1
X
t
=
E
vi,t (q ? t , sti ) si,0 = s, s
0
t=1
i,0
#
Z
s
E
0
"
1
X
t=1
t @vi,t (q
?t
@si,0
, sti )
si,0 = z, s
i,0
#
!
dz fi (s)ds,
where we can drop the max with zero since the agent obtains value zero at all periods when she is
excluded from the system. By changing the order of integration, we have
Z 1
max{p?i (s, s0, i ), 0}fi (s)ds
0
Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
6
=
=
Z
Z
1
E
0
1
E
0
=E
"
"
✓
1
X
t
" t=1
1
X
1
?t
vi,t (q , sti )
Fi (s) @vi,t (q ? t , sti )
fi (s)
@si,0
?t
t
↵i (si,0 )vi,t (q , sti ) +
i,t (q
?t
t
↵i (si,0 )vi,t (q ? t , sti ) +
i,t (q
?t
si,0 = s, s
, si,0 ) si,0 = s, s
t=1
1
X
◆
, si,0 ) s
t=1
i,0
i,0
#
i,0
#!
#!
fi (s)ds
fi (s)ds
(33)
Therefore, the profit of the mechanism matches the upper-bound provided in Corollary 3.1.
Hence, to prove the optimality, it suffices to show that the mechanism is individually rational. By
construction, we have the utility of agent i equal to 0 if si,0 = 0 for any s i,0 . Therefore,
Ui (s0 ) =
Z
si,0
E
0
@v
(q ?t ,s
ui,t = vi,t (a? t , sti )
pi,t
"
1
X
t @vi,t (q
?t
t=1
,s
,...s
, si,0 , si,1 , . . . si,t )
|si,0 =z si,0 = z, s
@si,0
i,0
#
dz.
)
i,0 i,1
i,t
By Assumption 4.2, i,t
is non-negative. Hence, the mechanism is individually ratio@si,0
nal. Precisely, periodic ex-post IR at time 0.
Proof of Lemma 4.2 Define ui,t to be the instantaneous utility of agent i at time t. We get
= vi,t (a? t , sti )
vi,t (a? t , ŝti ) +
= vi,t (a? t , sti ) +
1
+
↵
ˆi
X⇣
mi,t
↵
ˆi
ˆi (a? t )
↵
ˆi
?t
j,t
↵
ˆ j vj,t (a , ŝ )
j6=i
⌘
?t
ct (a )
(↵,
ˆ ˆ)
W i (a? t 1 , ŝt ) +
E
h
(↵,
ˆ ˆ)
W i (a? t i , ŝt+1 )
i
!
(↵,
ˆ ˆ)
The last equality follows from Eq. (21). We dropped the conditioning of W i (a? t , ŝt+1 ) on st = ŝt ,
a? t , and a? i,t , as it is clear from the context. For ease of notation, let s = s0 . Because all agents
except i are truthful, we have
✓ n
⌘
1 X⇣
uit =
↵
ˆ j vj,t (a? t , stj ) + ˆj (a? t ) ct (a? t )
↵
ˆ i j=1
h
i◆
(↵(s), (s))
(↵(s), (s))
W i
(a? t 1 , st ) + E W i
(a? t i , st+1 )
If agent i is truthful and other agents are truthful, we have
1
X
t0 =t
t
uit0 =
1 ⇣ (↵(s),
W
↵
ˆi
(s))
(a? t 1 , st )
W
(↵(s), (s))
(a? t 1 , st )
i
⌘
Hence, the allocation rule is aligned with the incentive of agent i. She can maximize her utility
by reporting truthfully.
P1
0
Observe that agents with ↵
ˆ i  0 would have been excluded. Hence, we have
0.
t0 =t uit
Therefore, the mechanism is periodic ex-post IR.
Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
7
Proof of Lemma 4.3 Observe that Eq. (15) is followed from Lemma 4.1 and Eq. (33). To establish Eq. (16), we show that the inequality holds point-wise, i.e., if x x0 , then
@Vixi !xi (s)
@s
x0i !xi
@Vi
By Eq. (14), this is equivalent to
"1
#
X @vi,t (a? t , st )
i
t
Exi !xi
s = xi
si,0 =s i,0
@s
i,0
t=0
(s)
@s
s=xi
Ex0i !xi
"
(34)
s=xi
1
X
t @vi,t (a
0t
, sti )
si,0 =s
@si,0
t=0
si,0 = xi
#
(35)
where Exi !xi is the expectation under the stochastic process determined by agent i reporting
according to xi ! xi (while other agents are truthful) and a?t represents the allocation at time t
in this case. Similarly, for reporting strategy x0i ! xi , we use the notation Ex0i !xi and represent the
0
allocation at time t by a t .
Recall that we have:
1
vi,t (at , sti )
Fi (si,0 ) @vi,t (at , sti )
= ↵i (si,0 )vi,t (at , sti ) +
fi (si,0 )
@si,0
i,t (a
t
, si,0 )
Hence, we get
@vi,t (at , sti )
fi (si,0 )
=
(1
@si,0
1 Fi (si,0 )
↵i (si,0 ))vi,t (at , sti )
i,t (a
t
, sti )
(36)
Therefore, by Eq. (36), the inequality below is equivalent to the desired equation, Eq. (34):
"1
#
X
t
t
Exi !xi
(37)
((1 ↵i (xi ))vi,t (at , sti )
i,t (a , xi ))
t=1
Ex0i !xi
"
1
⇣
X
t
(1
0t
↵i (xi ))vi,t (a , sti )
i,t (a
0t
t=1
, xi )
⌘
#
In the following we prove the inequality above. For k 6= i, define xk and x0k to be equal sk,0 .
Because a? and a0 are optimal allocation rules with respect to (↵(x), (x)) and (↵(x0 ), (x0 )), we
have:
"1
!#
n
X
X
t
t
t
t
Exi !xi
↵j (xj )vj,t (a? , stj ) + j,t (a? , xj ) ct (a? )
Ex0i !xi
"
Exi !xi
"
t=1
j=1
1
X
n
X
t
t=1
t
↵j (xj )vj,t (a0 , stj ) +
j,t (a
0t
, xj )
ct (a0t )
j=1
!#
and similarly
 Ex0i !xi
"
1
X
t
t=1
1
X
t=1
n
X
↵j (x0j )vj,t (a? t , stj ) +
j,t (a
?t
, xj )
?t
ct (a )
j=1
t
n
X
j=1
t
↵j (x0j )vj,t (a0 , stj ) +
j,t (a
0t
, x0j )
0t
ct (a )
!#
!#
Kakade, Lobel, Nazerzadeh: Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism
Article submitted to Operations Research; manuscript no. OPRE-2012-02-064
8
Subtracting these inequalities we get:
"1
n ✓
X X
t
Exi !xi
(↵j (xj ) ↵j (x0j ))vj,t (a? t , stj ) + (
Ex0i !xi
"
t=1
, xj )
j,t (a
?t
, x0j ))
j=1
1
n ✓
X
X
t
t=1
(↵j (xj )
0t
↵j (x0j ))vj,t (a , stj ) + (
j,t (a
0t
, xj )
j,t (a
0t
, x0j ))
j=1
Because for k 6= i, agents are truthful and x0k = xk , we have
"1 ✓
X
t
Exi !xi
(↵i (xi ) ↵i (x0i ))vi,t (a? t , sti ) + (
Ex0i !xi
j,t (a
?t
" t=1
1
X
t=1
t
✓
(↵i (xi )
0t
↵i (x0i ))vi,t (a , sti ) + (
i,t (a
i,t (a
?t
0t
, xi )
i,t (a
, xi )
i,t (a
?t
0t
, x0i ))
, x0i ))
◆#
◆#
◆#
(38)
◆#
Now suppose vi is multiplicative separable (i.e., i,t (·, ·) = 0) and Assumption 4.2 holds — we
consider the additive valuations later. Because x x0 , by Assumption 4.2 and Lemma B.1, we have
↵i (xi ) > ↵i (x0i ); moreover ↵i (xi ) is less than 1 for x 2 [0, 1). Multiplying both sides of the inequality
1 ↵i (xi )
above by ↵i (x
0 , yields the following:
i ) ↵i (x )
i
Exi !xi
"
1
X
t
(1
↵i (xi ))vi,t (a? t , sti )
t=1
#
Ex0i !xi
"
1
X
t
(1
0t
↵i (xi ))vi,t (a , sti )
t=1
#
which is equivalent to Eq. (37) for multiplicative-separable valuations.
Now consider the case of additive-separable value functions. We have ↵i (x) = ↵i (x0 ) = 1. Plugging
into Eq. (38) we get
"1
#
"1
#
X
X
t
t
t
?t
0
t
0
0
Exi !xi
( i,t (a? t , xi )
Ex0i !xi
( i,t (a0 , xi )
i,t (a , xi ))
i,t (a , xi ))
t=1
Recall that
i,t (a
Lemma B.1, we have
above by
t=1
t
1 Fi (xi ) 0
Ai (xi )Ci,t (at ). Because x
x0 , by
fi (xi )
0
1 Fi (xi ) 0
1 Fi (xi ) 0
Ai (x0i ). By multiplying both
Ai (xi ) >
fi (xi )
fi (x0 )
, xi ) =
1 Fi (xi ) 0
Ai (xi )
fi (xi )
1 Fi (x0 ) 0 0
1 Fi (xi ) 0
i A (x )
Ai (xi )+
i i
fi (xi )
fi (x0 )
i
Exi !xi
"
1
X
t=1
t
i
Assumption 4.2 and
sides of the inequality
, we get:
i,t (a
?t
, xi )
#
Ex0i !xi
"
1
X
t=1
t
i,t (a
0t
, x0i )
#
which produces Eq. (37) and, thus, completes the proof.
Appendix C: Proof for the Single Agent Case
Proof of Corollary 5.1 Simply note that under the Virtual-Pivot Mechanism, if the agent is
allocated the item at any time t, the price she pays, under the Virtual-Pivot Mechanism, is not
a function of her report at time t (or any report after t = 0). Furthermore, the prices that the agent
is charged at t 1 are identical to that in the Virtual-Pivot Mechanism (see Eq. (22)). Also,
the prices charged at t = 0 is identical to that in the Virtual-Pivot Mechanism by construction.