CSCE 475/875
Handout
11: Efficient Groves Mechanisms and Clarke Tax
October 13, 2011
(Based on Shoham and Leyton-Brown 2011)
Groves Mechanisms
Efficiency (Definition 10.3.6) is
often considered to be one of the most important properties for a mechanism to
satisfy in the quasilinear setting. Consequently, a great deal of research has
considered the design of mechanisms that are guaranteed to select efficient choices when agents follow dominant
or equilibrium strategies.
The most important family of efficient
mechanisms are the Groves mechanisms.
Definition
10.4.1 (Groves mechanisms) Groves mechanisms are direct
quasilinearmechanisms 
, for
which
In other words, Groves mechanisms are
direct mechanisms in which agents can declare any valuation function 
:
·
The mechanism then optimizes its choice
assuming that the agents disclosed their true utility function.
·
An agent is made to pay an arbitrary amount 
which does not depend on its own declaration
and is paid the sum of every other agent’s declared valuation for the
mechanism’s choice.
The fact that the mechanism designer
has the freedom to choose the
functions
explains why we refer to the family of Groves mechanisms rather than to
a single mechanism.
The
remarkable property of Groves mechanisms is that they provide a dominant
strategy truthful implementation of a social-welfare-maximizing social choice
function.[LS4]
Theorem
10.4.2 Truth telling is a dominant strategy under any Groves mechanism.
Intuitively, the reason that Groves mechanisms
are dominant-strategy truthful is that agents’ externalities are internalized.[LS5]
INTUITION!! Imagine a mechanism in which agents declared
their valuations for the different choices 
and the mechanism selected the efficient
choice, but in which the mechanism did not impose any payments on agents. Clearly, agents would be able to change the
mechanism’s choice to another that they preferred by overstating their
valuation. Under Groves mechanisms, however, an agent’s utility does not depend
only on the selected choice, because payments are imposed. Since agents
are paid the (reported) utility of all the other agents under the chosen
allocation, each agent becomes just as interested in maximizing the other
agents’ utilities as in maximizing his own. Thus, once payments are taken into
account, all agents have the same interests.[LS6]
The Vickrey-Clarke-Groves
(VCG)Mechanism (aka Pivot Mechanism)
So far, we have said nothing about how
to set the function in a
Groves mechanism’s payment function.
Here we will discuss the most popular answer, which is called the Clarke
tax.
Definition
10.4.4 (Clarke tax) The Clarke tax sets theterm in a
Groves mechanism as 
, where
is the Groves mechanism allocation function.
Definition
10.4.5 (Vickrey–Clarke–Groves (VCG) mechanism) The VCG
mechanism is a direct quasilinearmechanism , where
First, note that because the Clarke
tax does not depend on an agent 
’s own
declaration 
,our
previous arguments that Groves mechanisms are dominant-strategy truthful and
efficient carry over immediately to the VCG mechanism[LS9] .
Now, we try to provide some intuition
about the VCG payment rule. Assume that all agents follow their dominant
strategies and declare their valuations truthfully. The second sum in the VCG
payment rule pays each agent the sum of
every other agent 
’s utility
for the mechanism’s choice. The first sum charges each agent the sum of
every other agent’s utility for the choice that would have been made had
not
participated in the mechanism. Thus, each agent is made to pay his socialcost—the
aggregate impact that his participation has on other agents’ utilities.[LS10]
What can we say about the amounts of
different agents’ payments to the mechanism? If some agent does not
change the mechanism’s choice by his participation—that is, if 
then the
two sums in the VCG payment function will cancel out. The social cost of ’s
participation is zero, and so he has to pay nothing.
In order for an agent to be made
to pay a nonzero amount, he must be pivotal in the sense that the
mechanism’s choice 
is different from its choice without , 
This is
why VCG is sometimes called the pivot mechanism—only pivotal agents are made to
pay.
Of
course, it is possible that some agents will improve other agents’
utilities by participating; such agents will be made to pay a negative amount,
or in other words will be paid by the mechanism.
Drawbacks
of VCG
The VCG mechanism is one of the most
powerful positive results in mechanism design: it gives us a general way of
constructing dominant-strategy truthful mechanisms to implement
social-welfare-maximizing social choice functions in quasilinear settings. We
have seen that no fundamentally different mechanism could do the same job. And
VCG gives us even more: under the right conditions it further guarantees ex post individual rationality
and weak budget balance.
Thus, it is not surprising that this
mechanism has been enormously influential and continues to be widely studied.
However, despite these attractive
properties, VCG also has some undesirable characteristics.
·
Agents must fully disclose private information.
·
Susceptibility to collusion.
·
VCG is not frugal.
·
Dropping bidders can increase revenue. (Note:
If we have agents that are not pivotal, then they don’t have to pay …)
·
Cannot return all revenue to the agents
·
Computational intractability. (Note: Evaluating
the 
can
require solving an NP-hard problem in many practical domains.)