Uniqueness of the VCG Mechanism
Professors Greenwald and Oyakawa
2017-03-15
We prove that VCG mechanism is the unique direct, DSIC, welfaremaximizing auction. We also analyze the consequences of this theorem
and the conditions under which it holds.
1
Introduction
Recall the multi-parameter setting, defined as follows:
• n bidders;
• a finite set Ω of possible outcomes;
• for each bidder i, a private valuation vector vi ∈ Ti ⊂ R|Ω | across
all outcomes.
Previously, we showed that the VCG mechanism, defined by the
allocation rule
x (b ) = arg max ∑ bi (ω )
ω ∈Ω
i∈ N
and payment rule
"
∗
∗
p i ( ω ) = bi ( ω ) −
∑ b j (ω
j∈ N
!
∗
) − max
∑
ω ∈Ω i 6= j∈ N
!#
b j (ω )
.
The theorem we will prove is the following:
Theorem 1.1. If the set of possible valuations Ti is connected for every
bidder i, then the solution found by the VCG mechanism is the unique DSIC
direct-revelation mechanism, up to an additive constant.
Before proving the theorem, we analyze two keywords that we
omitted from the informal statement of the theorem at the top of this
document.
1.1 “Up to an additive constant”
Ultimately, the theorem will show that for any two outcomes a, b ∈
Ω, the difference in the resulting payments is a fixed value. This
means that we set a single payment for a particular outcome, the rest
of the payments for all outcomes are fixed. However, any such initial
assignment will result in a DSIC mechanism, which is why we add
this clause. However, typically only one such payment scheme will
satisfy IR—specifically, the one in which the outcome corresponding
to no allocations results in a 0 price vector.
uniqueness of the vcg mechanism
1.2 “If Ti is connected”
Informally, we say that Ti ⊆ R|Ω | is connected if it geometrically consists of no more than one solid object in Rn . We give a more formal
treatment of this condition in the appendix.
If Ti is not connected, then DSIC payments may not be unique.
Consider, for example, a single-item auction in which Ti = Z+ for
each bidder i. Define x to be the allocation rule in which the highest
bidder wins the item. If p is the second-price payment rule, then
we know that ( x, p ) is a DSIC mechanism, namely the second-price
auction. However, consider the payment rule p0 , defined as follows:
• If there is a unique highest bidder, she will pay 1/2 more than the
second highest bid.
• In the event that multiple bidders tie for first place, a winner is
arbitrarily selected and she will pay her bid.
We argue that ( x, p0 ) is also a DSIC mechanism. To see this let b∗ ∈
Z+ be the highest bid. Then the price for winning the auction is
b∗ + 1/2. If bidder i’s valuation is less than b∗ , then she should lose
the item to maximize utility. This will be achieved by bidding her
valuation. Similarly, if her valuation is greater than b∗ + 1/2, she
should win the item, which will also be achieved by bidding her
valuation. Now consider the interval [b∗ , b∗ + 1/2]. Since the type
space is restricted to integers, the only possible valuation on this
interval is b∗ , in which case the most utility she can achieve is 0,
which will occur if she bids truthfully.
To maximize utility, bidder i should win the auction if and only if
her valuation is at least b∗ + 1/2.
2
Proof
In this section, we prove Theorem 1.1.
Let x be the welfare-maximizing allocation. We begin with some
arbitrary payment rule p such that ( x, p ) is DSIC; then, we show that
our choice of p was unique, i.e. we could not have chosen anything
else, up to an additive constant.
Fix some bidder i and v −i . We write x (·) and p (·) in place of
xi (·, v −i ) and pi (·, v −i ), respectively. This will mean that x and p are
restricted to the space of outcomes and payments than can possibly
occur as the result of some bid by bidder i.
Since ( x, p ) is DSIC, we have that for any two valuations v and v0
for bidder i, x (v) = x (v0 ) implies p (v) = p (v0 ).1 Therefore, for some
outcome a, we can let p a be bidder i’s unique associated payment.
(Formally, p a is such that for all v satisfying x (v) = a, p a = p (v).)
If p (v) < p (v0 ) and i’s true valuation
is v0 , she would have incentive to
instead report v.
1
2
uniqueness of the vcg mechanism
3
Now, we define the notion of closeness.
Definition 2.1. Two outcomes a, b ∈ Ω are close if for every e > 0,
there exist valuations v and v0 such that
x (v ) = a, x (v0 ) = b, and kv0 − v k∞ < e,
where k · k∞ is defined as follows2 :
Recall that v, v0 are vectors of degree
|Ω |. v ( a) then refers to the valuation of
the outcome a according to v.
2
kv k∞ = max |v ( a)|.
a
Consider the following graph:
Ω
d
Figure 1: Outcomes as a function of a
one-dimensional v.
c
b
a
v
Outcomes a and d are close because it is possible to pick two arbitrarily close valuations on either side of the dotted line. Outcomes
a and b are not close because for some small value of e, we cannot
find any two valuations within e of each other that result in the two
different outcomes.
The proof of Theorem 1.1 follows from the following two lemmas:
Lemma 2.2. If a, b ∈ Ω are close, then the difference in payments p a − pb
is a fixed value.
The proof of Lemma 2.2 is given as a homework question.
Remark 2.3. Since we argued that p a is unique for any outcome a ∈ Ω,
why does this not automatically prove Lemma 2.2? This only shows
that within one payment scheme, say, p1 , that | pb − p a | has a fixed
value. But it is conceivable that there is another payment scheme p2 ,
in which | pb − p a | has a different value. You will show that | pb − p a |
has only one value in any DSIC payment scheme.
Lemma 2.4. If Ti is connected, then for any two outcomes a, b ∈ Ω, there
exists a chain of outcomes
a = a1 , a2 , a3 , . . . , a m = b
such that every pair ak , ak+1 is close.
uniqueness of the vcg mechanism
Proof. A rigorous proof requires some analysis, so we’ve deferred
it to the Appendix. Here, we give a simple proof for the case where
Ti ⊆ R.
Let a, b ∈ Ω and let x (v ) = a and x (v0 ) = b for some v, v0 ∈ Ti ⊆
R. WLOG let v < v0 . Since Ti is connected, [v, v0 ] ⊆ Ti , and so there is
some corresponding allocation for every v0 ∈ [v, v0 ]. Since Ω is finite,
these allocations form a sequence beginning with a and ending with
b.
Example 2.5. In Figure 1, the outcomes a and b can be joined by the
chain
a, d, c, b,
where a and d are close, d and c are close, and b and c are close.
Given these two lemmas, we have that for any pair of outcomes
a, b ∈ Ω, there is only one possible value for pb − p a . This means
that all DSIC payment functions for x (·) differ only by an additive
constant.
3
Consequences of Theorem 1.1
Given a welfare-maximizing allocation x, charging VCG payments
is equivalent to charging each bidder their externality on the other
bidders and the auctioneer. Therefore, if we run a direct mechanism in which (1) we know we arrive at the welfare maximizing
outcome and (2) we charge each bidder their externality, then we
know that the result is a DSIC (welfare-maximizing) mechanism.
Theorem 1.1 gives us the converse: that if a mechanism is DSIC and
welfare-maximizing, then it must be that bidders are charged their
externalities.
What about ascending auctions? We have the following corollary
of Theorem 1.1:
Corollary 3.1. If a mechanism is EPIC and welfare-maximizing, then
bidders must be charged the VCG payments (their externalities).
To see why this is true, if we apply the Revelation Principle to
the EPIC, welfare-maximizing auction, we obtain a direct, DSIC,
welfare-maximizing auction, in which the payments must be VCG
by Theorem 1.1. Since the Revelation Principle has no impact on
payments, it must be that the original EPIC mechanism must charge
VCG payments as well.
What about the converse? Is it true that any ascending auction
that arrives at the welfare-maximizing allocation and charges VCG
payments is EPIC? We will soon find out!
4
uniqueness of the vcg mechanism
4
Appendix: Formal treatment of connectedness
Definition 4.1. We say that a set S is connected if S cannot be partitioned into two nonempty subsets that are open in the relative
topology induced by S. Equivalently, S is connected if it cannot be
partitioned into two nonempty closed subsets.
In light of this definition, we prove Lemma 2.4 as follows:
Proof. We argue if Ti is connected, for any pair of outcomes a, b ∈ Ti ,
there exists a sequence a = a0 , a1 , . . . , am = b such that every adjacent
pair is close. Another way of expressing this is the following: let A
be the set of points reachable from a by a sequence of pairwise-close
points. Then Ti is connected if and only if A = Ti .
Suppose A 6= Ti . Then the following sets partition Ti and are both
nonempty:
X = {v ∈ Ti : x (v ) ∈ A}
Y = {v ∈ Ti : x (v ) 6∈ A}.
Further, the set closures of X and Y are disjoint; if they overlapped,
containing some point v0 , then x (v0 ) would be close to some point in
X and some point in Y, in which case the outcomes corresponding to
valuations in Y are in fact reachable by a. Therefore, cl( X ) and cl(Y )
partition Ti , a contradiction since we assumed that Ti is connected.
Therefore, we conclude that A = Ti and so for any pair of outcomes
a, b ∈ Ti , there exists a chain of pairwise-close outcomes connecting
the two.
5