CSCE 475/875
Handout 10: The
Impossibility Theorems, Continued!
October 6, 2011
(Based on Shoham
and Leyton-Brown 2011)
The Gibbard-Satterthwaite
Impossibility Theorem, for Social Choice Functions …
Theorem
10.2.6 (Gibbard–Satterthwaite)
Consider any social choice function 
of 
and 
. If:
1.
(there are at least three outcomes);
2.
is onto;
that is, for every 
there is a
preference profile 
such that 
(this
property is sometimes also called citizen
sovereignty); and
3.
is
dominant-strategy truthful,
then is
dictatorial.
Theorem 10.2.6 is reminiscent of the
Muller–Satterthwaite theorem (Theorem 9.4.8) in
Handout 8.
A Way to Get Around the Impossibility Theorem? Quasilinear Preferences
If we are to design a
dominant-strategy truthful mechanism that is not dictatorial, we are going to have to relax some of the
conditions of the Gibbard–Satterthwaite
theorem.
·
First, we relax the requirement that agents be
able to express any preferences and replace it with the requirement that agents
be able to express any preferences in a limited
set.
·
Second, we relax the condition that the
mechanism be onto.
(Note: There exists some outcomes that nobody prefers)
Definition
10.3.1 (Quasilinear utility function) Agents
have quasilinear utility
functions (or quasilinear preferences) in
an 
-player
Bayesian game when the set of outcomes is 
for a
finite set 
, and the
utility of an agent 
given
joint type 
is given
by 
, where 
is an element of , 
is an arbitrary function and 
is a
strictly monotonically increasing function.
Intuitively, we split outcomes into
two pieces that are linearly related. First, represents
a finite set of nonmonetary outcomes, such as the allocation of an object to
one of the bidders in an auction or the selection of a candidate in an
election. Second, 
is the
(possibly negative) payment made by agent to the
mechanism, such as a payment to the auctioneer.
What does it mean to assume that
agents’ preferences are quasilinear?
·
First, it means that we are in a setting in
which the mechanism can choose to charge or reward the agents by an arbitrary monetary amount.
·
Second, and more restrictive, it means that an
agent’s degree of preference for the selection of any choice 
is
independent from his degree of preference for having to pay the mechanism some amount 
. Thus an agent’s utility for a choice cannot
depend on the total amount of money that he has (e.g., an agent cannot
value having a yacht more if he is rich than if he is poor).
·
Finally, it means that agents care only about
the choice selected and about their own payments: in particular, they do not care about the monetary
payments made or received by other agents.
We will often want the 
functions
to be nonlinear. The curvature of gives ’s risk attitude, which risk
attitude we can understand as the way that feels
about lotteries such as the one just described.
Risk attitudes include risk neutral, risk averse, and risk seeking,
corresponding to linear, sublinear, and superlinear curves for 
Mechanism
Design in the Quasilinear Setting
Basic Constraints:
Nice to have!
We assume that agents are risk neutral and have transferable utility. For convenience,
let
, meaning
that we can think of agents’ utilities for different choices as being expressed
in dollars. We concentrate on Bayesian games because most mechanism design is
performed in such domains. Also, we
point out that since quasilinear preferences split the outcome space into
two parts, we can modify our formal definition of a mechanism accordingly.
Definition 10.3.2 (Quasilinear
mechanism) A mechanism in the quasilinear setting
(for a Bayesian game setting (
) is a
triple 
where
• 
, where 
is the set
of actions available to agent 
,
• 
maps each action profile to a distribution
over choices, and
• 
maps each action profile to a payment
for each agent.
In effect, we have split the function 
into two
functions 
and 
, where is the choice rule and is the payment
rule. We will use the notation 
to denote
the payment function for agent .
Definition 10.3.3 (Direct quasilinear
mechanism) A direct quasilinear mechanism (for a
Bayesian game setting () is a pair 
. It
defines a standard mechanism in the quasilinear
setting, where for each , 
. (Note: Action = Preference!)
In many quasilinear
mechanism design settings it is helpful to make the assumption that agents’ utilities depend only on their own types, a
property that we call conditional
utility independence.
Definition 10.3.4 (Conditional utility
independence) A Bayesian game exhibits conditional
utility independence if for all agents , for all
outcomes and for all pairs of joint types 
and 
for which 
, it holds
that 
.
We
will assume conditional utility independence for the rest of this section. When we do so, we can write an agent ’s utility function as 
, since it
does not depend on the other agents’ types.
We
can also refer to an agent’s valuation for choice , written 
should be thought of as the maximum amount of money that would be willing to pay to
get the mechanism designer to implement choice x—in fact, having to pay this much would exactly make indifferent about whether
he was offered this deal or not.
Let 
denote the set of all possible valuations for agent
. We will use the notation 
to denote the valuation that agent declares to such a direct
mechanism, which may be different from his true valuation . We denote the vector of all agents’ declared valuations as 
and the set of all possible
valuation vectors as 
Finally, denote the vector of declared valuations from all agents
other than as 
.
Definition 10.3.5 (Truthfulness) A quasilinear mechanism is truthful if it is direct and 
, agent ’s
equilibrium strategy is to adopt the strategy 
.
Definition 10.3.6 (Efficiency) A quasilinear mechanism is strictly Pareto efficient, or just efficient,
if in equilibrium it selects a choice 
such that 
Definition 10.3.7 (Budget balance) A quasilinear mechanism is budget
balanced when 
, where 
is the
equilibrium strategy profile.
In other words, regardless of the
agents’ types, the mechanism collects and disburses the same amount of money
from and to the agents, meaning that it makes neither a profit nor a loss.
(Note: The system earns nothing and loses nothing.)
Sometimes we relax this condition and
require only weak budget balance,
meaning that 
.
Finally, we can require that either
strict or weak budget balance hold ex ante, which means that 
. (Note: That
is, the mechanism is required to break even or make a profit only on
expectation.)
Definition 10.3.8 (Ex interim individual
rationality) A quasilinear
mechanism is ex interim individually rational when 
where is the
equilibrium strategy profile.
This
condition requires that no agent loses by participating in the mechanism. We call it ex interim because
it holds for every possible valuation for agent , but
averages over the possible valuations of the other agents. This approach makes
sense because it requires that, based on the information that an agent has when
he chooses to participate in a mechanism, no agent would be better off choosing
not to participate. Of course, we can also strengthen the condition to say that
no agent ever loses by participation.
Definition 10.3.9 (Ex post individual
rationality) A quasilinear
mechanism is ex post individually rational when 
, where is the equilibrium
strategy profile.
Definition 10.3.10 (Tractability) A quasilinear
mechanism is tractable when 
and 
can be computed in polynomial time.
Optimality
Properties: Even better!
Finally, in some domains there will be
many possible mechanisms that
satisfy the constraints we choose, meaning that we need to have some way of
choosing among them. The usual approach is to define an
optimization problem that identifies the optimal outcome in the feasible set.
For example, although we have defined efficiency as a constraint, it is also
possible to soften the constraint and
require the mechanism to achieve as much social welfare as possible. Here
we define some other quantities that a mechanism designer can seek to optimize.
Definition 10.3.11 (Revenue maximization) A quasilinear mechanism is revenue maximizing when, among the
set of functions and
that
satisfy the other constraints, the mechanism selects the and that
maximize where 
denotes the agents’ equilibrium strategy
profile.
Definition 10.3.12 (Revenue minimization) A quasilinear mechanism is revenue minimizing when, among the
set of functions and that
satisfy the other constraints, the mechanism selects the and that
minimize
in equilibrium, where denotes the agents’ equilibrium strategy
profile.
The mechanism designer might be
concerned with selecting a fair outcome.
Here we define so-called maxmin fairness,
which says that the fairest outcome is
the one that makes the least-happy agent the happiest. We also take an
expected value over different valuation vectors, but we could instead have
required a mechanism that does the best in the worst case.
Definition 10.3.13 (Maxmin
fairness) A quasilinear mechanism is maxmin fair when, among the set of functions and that
satisfy the other constraints, the mechanism selects the and that
maximize 
where 
denotes
the agents’ equilibrium strategy profile.
Finally,
the mechanism designer might not be able to implement a social-welfare
maximizing mechanism (e.g., in order to satisfy a tractability constraint) but
may want to get as close as possible. Thus, the
goal could be minimizing the price of anarchy, the worst-case ratio
between optimal social welfare and the social welfare achieved by the given
mechanism. Here we also consider the
worst case across agent valuations.
Definition 10.3.14 (Price-of-anarchy minimization)
A quasilinear mechanism minimizes
the price of anarchy when, among the set of functions and that
satisfy the other constraints, the mechanism selects the and that minimize 
, where denotes the agents’ equilibrium strategy
profile in the worst equilibrium of the mechanism—that is, the
one in which 
is the smallest.