Robust Implementation:
The Case of Direct Mechanisms
Dirk Bergemanny
Stephen Morrisz
First Version: August 2005
This Version: October 2005
Abstract
A social choice function is robustly implementable if there is a mechanism under which the
process of iteratively eliminating strictly dominated messages leads to a unique outcome and
agrees with the social choice at every true type pro le. In an interdependent value environment,
we identify a strict contraction property on the preferences which together with strict ex post
incentive compatibility and the strict single crossing property is su cient to guarantee robust
implementation in the direct mechanism.
The contraction property essentially requires that the interdependence is not too large. In
the important case of a linear model of signals, the contraction property is shown to be equivalent
for the matrix of interdependence to have an eigenvalue smaller than one. Finally, a slightly
weaker contraction property is shown to be necessary for robust implementation.
Keywords: Mechanism Design, Implementation, Robustness, Common Knowledge, Interim
Equilibrium, Iterative Deletion, Direct Mechanism.
Jel Classification: C79, D82
This research is supported by NSF Grant #SES-0095321 and #SES-0518929. We would like to thank Sandeep
Baliga, Federico Echenique, Donald Goldfarb, Tom Palfrey and Daniel Spielman for helpful discussions. Some of the
results reported in this paper appeared in early versions of Bergemann and Morris (2004); earlier versions of this
paper contained results on ex post implementation which now appear in Bergemann and Morris (2005).
y
Department of Economics, Yale University, 28 Hillhouse Avenue, New Haven, CT
06511,
dirk.bergemann@yale.edu.
z
Department of Economics, Princeton University, Prospect Street, Princeton, NJ 08544, smorris@princeton.edu.
1
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
2
Contents
1 Introduction
3
2 Setup
6
2.1
Payo Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Robust Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3 A Public Good Example
8
4 Robust Implementation
11
5 The Linear Model
14
5.1
Contraction Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2
Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3
Nonlinear Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Necessity of Contraction Property
23
7 Single Unit Auction
28
8 Conclusion
30
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
1
3
Introduction
The mechanism design literature provides a powerful characterization of which social choice functions can be achieved when the designer has incomplete information about agents' types.
If we
assume a commonly known common prior over the possible types of agents, the revelation principle
establishes that if the social choice function can arise as an equilibrium in some mechanism, then
it will arise in a truth-telling equilibrium of the direct mechanism (where each agent truthfully
reports his type and the designer chooses an outcome assuming they are telling the truth). Thus
the Bayesian incentive compatibility constraints characterize whether a social choice function is
implementable in this sense.
There are two important limitations of Bayesian incentive compatibility analysis. First, the
analysis typically assumes a commonly known common prior over the agents' types. This assumption may be too stringent in practise. In the spirit of the \Wilson doctrine" (Wilson (1987)), we
would like implementation results that are robust to di erent assumptions about what players do
or do not know about other agents' types. Second, the revelation principle only establishes that the
direct mechanism has an equilibrium that achieves the social choice function.
In general, there
may be other equilibria that deliver undesirable outcomes. We would like to achieve full implementation, i.e., show the existence of a mechanism all of whose equilibria deliver the social choice
function.
We studied the rst \robustness" problem in an earlier work, Bergemann and Morris
(2005c). The second \full implementation" problem has been the subject of a large literature. In
the incomplete information context, key full implementation references are Postlewaite and Schmeidler (1986), Palfrey and Srivastava (1989) and Jackson (1991). In this paper, we study \robust
implementation" where we require robustness and full implementation simultaneously. Requiring
both simultaneously adds extra structure to the problem and enables us to derive distinctive new
economic results.
Interim implementation on all type spaces is possible if and only if it is possible to implement
the social choice function using an iterative deletion procedure. We refer to the resulting notion as
robust implementation. We x a mechanism and iteratively delete messages for each payo type
that are strictly dominated by another message for each payo type pro le and message pro le that
has survived the procedure. This observation about iterative deletion illustrates a general point
well-known from the literature on epistemic foundations of game theory (e.g., Brandenburger and
Dekel (1987), Battigalli and Siniscalchi (2003)): equilibrium solution concepts only have bite if we
make strong assumptions about type spaces, i.e., we assume small type spaces where the common
prior assumption holds.
We obtain necessary and su cient conditions for robust implementation using the iterative
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
4
procedure in a wide class of environments. The iterative characterization comes with the additional
bene t that tight implementation results can be proved via a xed point of a contraction mapping.
In particular, we consider a general class of interdependent preferences in which the private types of
the agents can be linearly aggregated. In this environment we show that the social choice function
can be robustly implemented if and only if the interdependence is not too large. If
is the weight
of the type of agent j (relative to the type of agent i) for the utility of agent i, then the robust
implementation condition can simply be stated as:
Surprisingly, the converse result for
In other words, we also show that if
> 1= (I
> 1= (I
< 1= (I
1), where I is the number of agents.
1) even extends to a robust version of virtual utility.
1), then not only robust implementation, but even
robust virtual implementation fails. We further illustrate the strength of the contraction mapping
idea in the implementation context with two examples, one of a public good and one of a private
good (single unit auction) allocation problem with quasilinear utility. An important paper of Chung
and Ely (2001) analyzed the single (and multi-unit) auction with interdependent valuations with
dominance solvability (elimination of weakly rather than strictly dominated actions). In a linear
and symmetric setting, they reported su cient conditions for direct implementation that coincide
with the ones derived here. We show that in the environment with linear aggregation, under strict
incentive compatibility, the basic insight extends from the single unit auction model to general
allocations models, with elimination of strictly dominated actions only (thus Chung and Ely (2001)
require deletion of weakly dominated strategies only because incentive constraints are weak). We
also prove a converse result: if there is too much interdependence, then neither the direct nor any
augmented mechanism can robustly implement the social choice function (this result will also hold
with deletion of weakly dominated strategies).
In the implementation literature, it is a standard practice to obtain the su ciency results
with augmented mechanisms. By augmenting the direct mechanism with additional messages, the
designer may elicit additional information about undesirable equilibrium play by the agents. Yet, in
many environments common to applied mechanism designs, such a single crossing or supermodular
preferences, the structure of the preferences may already permit direct implementation (see a
companion paper, Bergemann and Morris (2005a), for direct implementation results in ex post
equilibrium). We thus provide necessary and su cient conditions for robust implementation in the
direct mechanism. In the direct mechanism, the agents can alert the designer only by a report of
their type. In consequence, the incentive compatibility conditions for the rewards are identical to
the truth-telling constraints, and the necessary and su cient conditions for robust implementation
coincide.
The results in this paper concern full implementation. An earlier paper of ours, Bergemann and
Morris (2005c), addresses the analogous questions of robustness to rich type spaces, but looking at
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
5
the question of truthtelling in the direct mechanism. In the literature, this is frequently referred to as
partial implementation. The notion of partial implementation asks whether there exist a mechanism
such that some equilibrium under that mechanism implements the social choice function. By
the revelation principle, it is then su cient to look at truthtelling in the direct mechanism. In
Bergemann and Morris (2005c), we showed that a social choice function robustly satis es the
interim incentive constraints, i.e. satis es the interim incentive constraints for any type space, if
and only if the ex post incentive constraints are satis ed.
In this sequel, we extend the robustness analysis from partial implementation (truthtelling in the
direct mechanism) to full implementation. The notion of full implementation requires that every
equilibrium under a given mechanism leads to the realization of the social choice function. By
considering robust implementation, we analyze, when a social choice function can be implemented
for any arbitrary type space. By using the equivalence between the iterative solution procedure and
the set of all interim equilibria under some type space, we can formulate the problem exclusively
in terms of the payo types and the messages without reference to type space, and associated prior
or posterior beliefs.
The analysis will deliver necessary and su cient conditions for robust implementation. We
derive the necessary conditions for general, in particular indirect and augmented mechanisms.
However, the main insight of our analysis is that the necessary conditions are essentially (with
one minor caveat) su cient to guarantee robust implementation in the direct mechanism. In
consequence, we will be able to pursue all su ciency arguments in the direct mechanism.
In this paper we consider full rather than partial implementation. Yet, with the existing robustness result from partial implementation, it is natural to ask whether the notion of ex post
equilibrium implementation plays the same role in the robust analysis of full implementation as it
did in the partial implementation set-up. In a companion paper, Bergemann and Morris (2005a),
we therefore investigate the notion of ex post implementation. The necessary and su cient conditions straddle the Nash and Bayesian implementation conditions as an ex post equilibrium is a
Nash equilibrium at every incomplete information (Bayesian) type pro le. However in contrast to
the iterative argument pursued here, the basic reasoning in Bergemann and Morris (2005a) invokes
more traditional equilibrium arguments. The limitation of the implementation result in ex post
equilibrium is that for a given type space, there might still exist interim equilibria which do not
coincide with the social choice function. Thus, in contrast to the partial implementation analysis,
in the context of full implementation, the notion of ex post equilibrium does not necessarily lead to
robust equilibria. In fact, by comparing the conditions for ex post and robust implementation, it
is apparent that robust implementation typically imposes additional constraints on the allocation
problem. In this paper, we show that robust implementation imposes a strict bound on the inter-
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
6
dependence of the preferences, which is not required by the ex post truthtelling conditions. The
contraction mapping behind the iterative argument directly points to the source of the restriction
of the interaction term.
The ex post incentive constraints in the direct mechanism remain necessary conditions for robust
implementation. Recently, Jehiel and Moldovanu (2001) and Jehiel, Meyer-Ter-Vehn, Moldovanu,
and Zame (2005) have shown that in an environment with multi-dimensional signals, the ex post
incentive constraints may be di culty to satisfy. Clearly, to the extent that an ex post equilibrium
does not exists in the direct mechanism, we cannot hope to deliver any positive implementation
results. While this provides a natural limit for our analysis, there are many interesting applications
for which ex post equilibria do exist, among them one-dimensional signal models (Dasgupta and
Maskin (2000), Perry and Reny (2002), Bergemann and V•alim•aki (2002)), models without allocative externalities (Bikhchandani (2004)) and many other models (see the recent survey Jehiel and
Moldovanu (2005)) for many positive and negative results) to which our analysis applies.
The remainder of the paper is organized as follows. Section 2 describes the formal environment
and solution concepts. Section 3 considers a public good example that illustrates the main ideas
and results of the paper. Section 4 establishes necessary conditions for robust implementation in
the direct mechanism. Section 5 considers the preference environment with a linear aggregation
of the types and obtains sharp implementation results. Section 6 reports su cient conditions for
robust implementation. Section 7 considers a single unit auction with interdependent values as a
second example of robust implementation. Section 8 concludes.
2
Setup
2.1
Payo
Environment
We consider a nite set of agents, 1; 2; :::; I. Agent i's payo type is
set. We write
2
=
1
I.
2
i,
where
i
is a compact
Let Y be a compact set of outcomes. Let agent i's utility if
outcome y is chosen and agents' type pro le is
f:
i
be ui (y; ). A social choice function is a mapping
!Y.
The main structural assumption of this paper is the existence of a monotonic aggregator hi ( )
for each i, which allows us to rewrite the utility function of every agent i as:
ui (y; )
where hi :
1
vi (y; hi ( ))
! R is continuous, strictly increasing in
i
and vi : Y
(1)
R ! R is continuous.1
We brie y discussed the impossibility results of ex post incentive compatibility with multi-dimensional signals,
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
2.2
7
Mechanisms
A planner must choose a game form or mechanism for the agents to play in order the determine
the social outcome. Let Mi be a compact set of messages available to agent i. Let g (m) be the
outcome chosen if action pro le m is chosen. Thus a mechanism is a collection:
M = (M1 ; :::; MI ; g ( )) ;
where g : M ! Y . The direct mechanism has the property that Mi =
2.3
i
for all i and g ( ) = f ( ).
Robust Implementation
In a xed mechanism M, we call a correspondence S = (S1 ; ::::; SI ), with each Si :
message pro le of the agents. We write S for the collection of message pro les.
In the direct mechanism a message pro le S may be thought of as a deception
i
=(
! 2Mi , a
1 ; :::;
I ),
or
i
A deception
i ( i)
:
i
!2
?; for all i.
i
is a set of possible reports of agent i to the principal regarding his true type. We
shall assume without loss of generality that
deception, speci cally truthtelling, with
i
i
2
i ( i)
for all i and all
( i ) = f i g for all i and
i.
Let
be the minimal
i.
Next we de ne the process of iterative elimination of never best responses. To this end, we
denote the belief of agent i over message and payo type pro les of the remaining agents by:
i
2
(M
i
i) :
see Jehiel and Moldovanu (2001) and Jehiel, Meyer-Ter-Vehn, Moldovanu, and Zame (2005) in the introduction. In
this context it is worthwhile to notice that the impossibility results are obtained in a setting where the utility function
is de ned separately for every allocation y. The utility function of every agent i at every allocation y, ui;y ( ), is
then assumed to be twice di erentiable in the signal. Yet, no continuity or monotonicity assumption are made across
allocations. In our setting, the aggregation of private types acts independently of the particular allocation. Yet,
provided the existence of an aggregating function hi ( ), we could allow the signal space of each agent i to be multidimensional without any further modi cation. Our analysis uses the single-crossing condition and hence a systematic
interaction between the set of allocations and singals and provided that aggregations is possible, the dimensionality
of the signal per se is not an issue.
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
8
We initiate Si0 = Mi and de ne inductively:
8
k
>
(1)
>
i (m i ;
i ) > 0 ) mj 2 Sj ( j ) ; 8 j 6= i;
>
>
>
>
>
<
P
k+1
Si ( i ) = mi 2 Mi 9 i s.th.:
i (m i ;
i ) ui (g (mi ; m i ) ; ( i ;
i ))
>
>
m i; i
>
>
(2)
P
>
0
0
>
>
i (m i ;
i ) ui (g (mi ; m i ) ; ( i ;
i )) ; 8mi 2 Mi :
:
m
i;
i
We observe that Sik is (weakly) decreasing in k. We denote the limit set by S M ( ), or
S M ( ) , lim S k ( ) , for all
k!1
2
.
By compactness of the message sets, we have
SiM ( i ) , \ Sik ( i ) .
k 1
For brevity and for lack of a better expression, we refer to the messages mi 2 SiM ( i ) as rationaliz-
able messages. We call a social choice function f robust implementable if there exists a mechanism
M under which the social choice can be recovered through a process of iterative elimination of
never best responses.
De nition 1 (Robust Implementation)
Social choice function f is robustly implemented by mechanism M if m 2 S M ( ) ) g (m) = f ( ).
The robustness notion is motivated by an equivalence between the set of reports surviving
iterated elimination of never best responses and the set of Bayesian equilibria which arise under
the mechanism M for some type space of the agents. The equivalence is formally established in
Proposition 1 of our extended working paper, Bergemann and Morris (2005b). The basic logic of
the argument follows the well-known argument of Brandenburger and Dekel (1987) for complete
information games.
3
A Public Good Example
We precede the formal results with an example which is meant to illustrate the main insights
of the paper. At the same time, the example facilitates a brief review of the key results in the
implementation literature. The example involves the provision of a public good with quasilinear
utility. The utility of each agent is given by:
0
ui ( ; y) = @
i
+
X
j6=i
1
j A y0
+ yi ;
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
:
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
9
where y0 is the level of public good provided and yi is the monetary transfer to agent i. The utility
of agent i depends on his own type
i
2 [0; 1] and the type pro le of other agents, with
0. The
utility function of agent i has the aggregation property with
X
hi ( ) = i +
j;
j6=i
but we notice the aggregator function hi ( ) depends on the agent i. In particular, a given type
pro le
is leads to a di erent aggregation result for i and j, provided that
The cost of establishing the public good is given by c (y0 ) =
(y0 ; y1 ; :::; yI ) 2 R+
1 2
2 y0 .
i
6=
j.
The planner must choose
RI to maximize social welfare, i.e., the sum of gross utilities minus the cost
of the public good:
(1 + (I
1))
I
X
i=1
i
!
1 2
y .
2 0
y0
The socially optimal level of the public good is therefore equal to
f0 ( ) = (1 + (I
1))
I
X
i.
i=1
We choose the generalized Vickrey-Groves-Clark transfers, essentially unique up to a constant, that
give rise to ex post incentive compatibility:
fi ( ) =
0 0
X
1)) @ @
(1 + (I
j6=i
1
jA i
+
1
2
1
2A
.
i
(2)
It is useful to observe that the generalized VCG transfers given by (2) guarantee ex post incentive
compatibility for any
2 R. Hence, ex post incentive compatibility per se does not impose any
constraint on the interdependence parameter .
Now we shall argue that if
<
1
I 1,
the social choice function f is robustly implementable in
the direct mechanism where each agent reports his payo type
i
and the planner chooses outcomes
according to f on the assumption that agents are telling the truth. Consider an iterative deletion
procedure. Let
0
( i ) = [0; 1] and, for each k = 1; 2; :::, let
k
( i ) be the set of reports that agent
i might send, for some conjecture over his opponents' types and reports, with the only restriction
on his conjecture being that each type
Suppose that agent i has payo
j
of agent j sends a message in
type
i,
i
0
i .and
0
and report their types to be
expected payo is a constant (1 + (I 1)) times:
0
10
1 0 0
X
X
X
0
0A
@ i+
@ @
jA @ i +
j
j6=i
( j ).
but reports himself to be type
conjecture that other agents have type pro le
j6=i
k 1
j6=i
1
1
0A 0
j
i+
2
1
0 2A
.
i
has a point
i.
Then his
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
The rst order condition with respect to
0
i
X
j
0
i
i
+
i
j6=i
is then
0
X
@
1
0A
j
j6=i
so he would wish to set
=
X
+
0
i
0
j
j
10
= 0,
.
(3)
j6=i
In other words, his best response to a misreport
0
i
by the other agents is to report a type so that
the aggregate type from his point of view is exactly identical to the true aggregate type generated
by the true type pro le . Note that the above calculation also veri es the strict ex post incentive
compatibility of f . The quadratic payo / linear best response nature of this problem means that
we can characterize
k
( i ) restricting attention to such point conjectures. In particular, we will
have
k
where
Analogously,
8
<
k
( i ) = min 1;
:
8
<
= min 1;
:
k
and
k
0
i
6=
i
i
+
)
0
i
2
=
k
f(
+
8
<
k
( i ) = max 0;
:
Thus
Thus
i
( i) =
0
h
i
( i) ;
k
i
( i) ;
max
k
0
( j)
j2
i ):
i;
max
i
k
X
j
for all j6=ig
k 1
( j)
j6=i
max
n
( i ) = min 1;
n
( i ) = max 0;
i
X
k 1
9
=
;
( j)
+ ( (I
i
( (I
o
1))k ;
o
1))k .
( i ) for su ciently large k, provided that
<
j
j6=i
.
j
j6=i
i
X
9
=
;
0
j
9
=
;
.
1
I 1.
At the end of the paper we shall present an additional example, namely a single unit auction with
symmetric bidders. The generalized Vickrey-Groves-Clark mechanism for the single unit auction
only satis es weak rather than strict incentive compatibility constraints. We therefore propose an
"-e cient allocation rule with strict ex post incentive constraints. The " e cient allocation rule can
also be interpreted as the virtual implementation of the e cient rule. In either case, we show that
the rule itself can be either robustly implemented or robust virtually implemented, respectively, if
there is not too much interdependence among the payo types.
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
4
11
Robust Implementation
We begin with the standard condition for truthful implementation.
De nition 2 (Ex Post Incentive Compatibility)
The social choice function f satis es ex post incentive compatibility if
ui (f ( i ;
for all i,
and
i) ; ( i;
i ))
0
i;
ui f
i
; ( i;
i)
0
i.
In the subsequent analysis we use the strict version of the incentive constraints.
De nition 3 (Strict Ex Post Incentive Compatibility)
The social choice function f satis es strict ex post incentive compatibility if
ui (f ( i ;
for all i,
and
0
i
6=
i) ; ( i;
i ))
0
i;
> ui f
i
; ( i;
i)
i.
We shall also assume the standard single crossing condition in its strict version.
De nition 4 (Strict Single Crossing)
The environment satis es strict single crossing (SSC) if vi (y; ) > vi (y 0 ; ) and vi y;
vi
y0;
0
implies vi y;
00
y0;
< vi
00
if either
<
0
<
00
or
>
0
>
00
= ( 1 ; :::;
I)
=
.
The single crossing condition is stated in way that we only require the aggregation
pro le
0
of the type
to be ordered on the real line, but it does not require that the allocation y is
one-dimensional or can be ordered on the real line.
The key property for our analysis is the following contraction property.
De nition 5 (Strict Contraction Property)
The aggregator functions h satisfy the strict contraction property if, for all
and
0
i
2
i ( i)
with
0
i
6=
i,
such that
sign
for all
i
and
0
i
2
i(
i ).
i
0
i
= sign hi ( i ;
i)
hi
0 0
i;
i
;
6=
, there exists i
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
12
The strict contraction property essentially says that for some agent i the direct impact of his
private signal
i
on the aggregator hi ( ) is always su ciently strong such that the di erence in the
aggregated value between the true type pro le and the reported type pro le always has the same
sign as the di erence between the true and reported type of agent i by itself. A slightly weaker
version of the contraction property will emerge as a necessary condition for robust implementation.
De nition 6 (Contraction Property)
The aggregator functions h satisfy the contraction property if, for all
0
i
2
i ( i)
0
i
with
6=
i,
6=
such that either
hi ( i ;
i)
0 0
i;
i
= hi
, there exists i and
;
or
sign
for all
i
and
0
i
2
i(
0
i
i
= sign hi ( i ;
i)
hi
0 0
i;
i
;
i ).
The strict contraction property is a very slight strengthening of the contraction property. We
shall henceforth assume that
i ( i)
a compact set, for all i,
i
and
i ( i)
continuous in
i.
Theorem 1 (Robust Implementation)
If strict EPIC and the strict contraction property are satis ed, then there is robust implementation
in the direct mechanism.
= S M and suppose that
Proof. We argue by contradiction. Let
contraction property, there exists i and
sign
for all
i
and
0
i
2
i(
i ).
,
where
0
i
i
0
i
2
i ( i)
such that
= sign hi ( i ;
i)
hi
0 0
i;
i
i)
hi
0 0
i;
i
6=
. By the strict
.
Let
i;
0
min
i2
i(
i)
hi ( i ;
,
> 0 by the strict contraction property. Suppose (without loss of generality) that
i
>
0
i.
Let
(") , max
hi
0
0
i
+ ";
0
i
hi
0 0
i;
i
.
i
As hi ( ) is strictly increasing in
i,
(") ! 0 as " ! 0.
i,
we know that (") is increasing in " and by continuity of hi in
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
13
Thus we have
hi ( i ;
for all
i
0
and
i
2
i(
i );
0
i.
0 0
i;
i
hi
;
(4)
and
0 0
i;
i
hi
for all
i)
0
i
hi
+ ";
0
(") ;
i
(5)
By strict EPIC,
0 0
i;
i
vi f
; hi
0 0
i;
i
; hi
0
i
0
i
> vi f
0
i
; hi
0 0
i;
i
0 0
i;
i
; hi
0
i
+ ";
;
(6)
for all " > 0 and
0
i
vi f
0
+ ";
i
+ ";
0
> vi f
i
for all " > 0. Now continuity of ui with respect to
+ ";
0
i
,
implies that for each " > 0 and
(7)
0
i,
there
exists
0
";
0
i
hi
i
+ ";
0
;
i
(8)
such that
vi f
0 0
i;
i
;
";
0
i
= vi f
0
i
+ ";
0
i
;
< vi f
0
i
+ ";
0
i
;
";
0
i
;
and SSC implies that
0 0
i;
i
vi f
for all
>
";
0
;
;
. Now x any " with
i
(") < .
0
Now for all
i
2
i(
(9)
i ),
hi ( i ;
i)
hi
0 0
i;
i
+ , by (4)
hi
0
i
+ ";
0
i
> hi
0
i
+ ";
0
i
";
0
i
(") + , by (5)
, by (9)
, by (8).
So
for every
i
and
vi f
0
i
0
i(
i
2
+ ";
0
i ).
i
; hi ( i ;
i)
> vi f
0 0
i;
i
; hi ( i ;
This contradicts our assumption that
i)
;
= SM.
The surprising element in this result is that we do not need to impose any conditions on how
the social choice function varies with the type pro le. In particular, it does not have to respond
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
to the reported pro le
14
in a manner similar to the response of any of the aggregators hi . Merely,
the strong single crossing condition is su cient to make full use of the contraction property. In
contrast to the classic results in Nash and Bayesian Nash implementation we do not have to impose
a condition on the number of agents, such as I > 2:
The argument is centered around the true type pro le
0
0 0
i;
i
=
= ( i;
. Without loss of generality we may assume that
i
>
i)
0
i:
property to establish a positive lower bound on the di erence h ( i ;
0
and
i
2
i(
i ).
0 0
i;
i
<
< hi
aggregator value
0
i
marginally upwards in the direction of
0
i
+ ";
0
i
i,
would be indi erent between the social allocations f
i)
>
us to assert that an agent with a true preference pro le
0
i
+ ";
0
i
0 0
i;
i
h
for all
i
rather than f
in other words to report
for the aggregator, with
, such that agent i with the utility pro le corresponding to the
By choosing " su ciently small, we know that h ( i ;
f
i)
+ ". This is achieved by showing that there is an intermediate value
hi
We use the contraction
With this positive lower bound, we then show that agent i is strictly better
o to move his misreport
0
i
and a reported pro le
0 0
i;
i
0 0
i;
i
0
i
and f
+ ";
0
i
.
and strict single crossing then allows
= ( i;
i)
would also prefer to obtain
. But this yields the contradiction to
0
i
2
i ( i)
being part
of the xed point of the iterative elimination. Consequently we show that the misreport
0
i;
which
established the same sign on the di erence between private type pro les and aggregated public
pro les can be eliminated as a best response to the set of misreports of the remaining agents.
5
The Linear Model
In this section, we consider the special case where the preference aggregator hi ( ) is linear for each
i and given by:
hi ( ) =
I
X
ij j ;
j=1
with
ij
2 R for all i, j and
ii
> 0 for all i. Without loss of generality, we may set
i, so we have
hi ( ) =
i
+
X
ii
= 1 for all
ij j .
j6=i
The parameters
ij
the exception of
ii
represent the in uence of the signal of agent j on the value of agent i. With
> 0 for all i, we do not impose any further a priori sign restrictions on
We denote the square matrix generated by the absolute values of
ij ;
namely
ij
ij .
; for all i; j with
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
i 6= j and zero entries on the diagonal by :
2
0
j 12 j
6
6 j j
0
6 21
,6 .
6 ..
4
j I1 j
We refer to the matrix
j
..
1I j
.
0
15
3
7
7
7
7.
7
5
as the interdependence matrix. The matrix
= 0 then constitutes the
case of pure private values.
5.1
Contraction Property
We shall rst give necessary and su cient conditions for the matrix
to satisfy the (strict) ex-
traction property. We then use duality theory to give a dual characterization of the contraction
property, which is very useful to nally obtain necessary and su cient conditions for the contraction
property in terms of the eigenvalue of the matrix .
Lemma 1 (Linear Aggregator)
1. Linear aggregator functions h satisfy the strict contraction property if and only if, for all
c 2 RI+ with c 6= 0, there exists i such that
ci >
X
ij
cj :
(10)
j6=i
2. Linear aggregator functions h satisfy the contraction property if and only if, for all c 2 RI+
with c 6= 0, there exists i such that
ci
X
ij
cj .
(11)
j6=i
Proof. (1) ()) We proof the contrapositive. Thus suppose there exists c 2 RI+ with c 6= 0,
such that for all i:
X
ci
ij
cj .
j6=i
We now show that this implies that the strict contraction property fails. Choose " > 0 such that
2ci " <
i
i
for all i. Now consider deceptions of the form:
i ( i)
=[
i
"ci ;
i
+ "ci ] \
i,
(12)
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
for all i. Then for all i and all j 6= i, let
and
0
j
ij
j
=
+ "ci if
j
0
j
="
ij
ij
0
j
< 0. By (12), we have
0
j
j
ij
=
i
+ "ci ,
0
i
i
+
j
X
="
j6=i
0
i
j
2
cj . Thus
X
Now if
1
2
=
j
ij
and let
j
0
j
16
=
j
"ci if
0
ij
( j ) for each j 6= i. Also observe that
cj
"ci .
j6=i
is strictly negative but
0
i
i
+
X
0
j
j
ij
;
j6=i
is non-negative. A symmetric argument works if
says that for all
sign
for all
i
i
and
6=
0
i
0
, there exists i and
= sign hi ( i ;
i
2
0
i
i(
i)
i)
2
>
i
i ( i)
So the strict contraction property, which
with 0i 6=
0
= sign @
0 0
i;
i
hi
0
i.
i,
0
i
i
such that
+
X
j
ij
1
A;
0
j
j6=i
(13)
fails. This proves the necessity of condition (10) of Lemma 1.1.
(() To show su ciency, suppose that condition (10) of the lemma holds. Fix any deception .
For all j, let:
cj =
0
j
max
0
j2 j( j)
By hypothesis, there exists i such that ci >
X
j
.
cj . Let
ij
j6=i
0
i
i
and suppose without loss of generality that
ij
j
0
j
ij
i
>
= ci ;
0
i.
Observe that for all
i
cj and thus
X
ij
j
0
j
j6=i
X
ij
and
0
i
2
i(
i ),
cj ;
j6=i
so
i
0
i
X
ij
j
0
j
= ci
j6=i
X
ij
j
0
j
j6=i
ci
X
ij
cj > 0,
j6=i
and hence the strict contraction property, or (13), is satis ed.
(2.) The proof for the contraction property is identical to the one of the strict contraction
property after replacing the strict with the weak inequality.
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
The absolute values of the matrix
17
are required to guarantee that the linear inequality implies
the strict contraction property. We observe that the condition (10) is only required to hold for a
single agent i. In fact, for c
private values, or
0, the condition (10) could hold for all i only in the case of pure
= 0.
The proof of the contraction property is constructive. We identify for each player i an initial
deception of the form
i ( i)
=[
i
ci ";
i
+ ci "] for some " > 0; common across all agents. The
size of ci is therefore proportional to the size of the set of candidate reports by agent i. It can
be thought of as the set of rationalizable strategies at an arbitrary stage k. The inequality of
the contraction property then says that for any arbitrary set of deceptions, characterized by the
vector c, there is always an agent i whose set of deceptions is too large (in the sense of being
rationalizable) relative to the set of deceptions by the remaining agents. It then follows that the set
of deceptions for this agent can be chosen smaller than ci , allowing us to reduce the set of possible
reports for a given agent i with a given type
i.
The inequality (10) asserts that for any given set
of deceptions, there is always at least one agent i whose deception
i
represents a set too large to
be rationalizable. Moreover, if the set of deceptions by i is too large, then there is an \overhang"
which can be \nipped and tucked". We will shortly see that a dual interpretation of the condition
(10) leads us from the idea of the overhang directly to the contraction property.
5.2
Duality
The following lemma gives a dual representation of the strict contraction condition, for the linear
case.
Lemma 2 (Duality)
The following two properties of
are equivalent:
1. for all c 2 RI+ with c 6= 0, there exists i such that:
X
ci >
ij cj ;
(14)
j6=i
2. there exists d 2 RI+ such that:
di >
X
ji
dj ;
(15)
j6=i
for all i.
Proof. Consider the following contrapositive restatement of condition (14): there does not
exist c 2 RI+ such that
I
X
i=1
ci > 0;
(a)
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
and
X
ij
cj
ci
18
0 for each i:
(b)
j6=i
Writing
for the multiplier of constraint (a) and di for the i multiplier of constraint (b), Farkas'
lemma states that such a c does not exist if and only if there exist d 2 RI+ and
di +
X
ji
2 R+ such that
dj = 0 for all i;
(a')
j6=i
and
> 0:
(b')
But this is true if and only if condition (15) of the lemma holds.
An analogous exercise leads to the duality result for the contraction property, where the strict
inequalities in (14) and (15) are simply replaced by weak inequalities. The dual variable d has the
usual interpretation of the Lagrange multiplier as a price. The multiplier di is the value of marginal
decrease in the misreport ci of agent i. With this interpretation of di , the dual conditions have
natural interpretation as price system. For the sets of reports to converge to the true pro le, it
must be that there is a system of prices such that the direct value, di , of decreasing the misreport
of agent i is larger than the indirect value which is generated by relaxing the constraints of all other
agents j 6= i, weighted by the impact
ji ;
i has on the preferences of the remaining agents.
The condition (14) represents a natural generalization of the notion of diagonal dominant matrix
due to Hadamard. A matrix
is said to be diagonally dominant if
ii
>
X
ij
, for all i = 1; :::; I.
j6=i
The relationship between the diagonal dominant and generalized diagonal dominant matrix is easily
established. Let D be a diagonal matrix whose diagonal elements dii are the above di 's, then the
matrix D A has a dominant diagonal in the Hadamard sense.2
We can use the duality result to obtain an alternative characterization of the contraction property in terms of eigenvector and eigenvalue of the matrix
some d 2
RI+ ,
then it follows that there exists some 0
still holds:
di >
X
ji
. If the dual condition (15) holds for
< 1 such that the modi ed condition
dj :
(16)
j6=i
2
See Berman and Plemmons (1994) or Minc (1988) for an introduction into the theory of nonnegative matrices
where the notion of diagonally dominant is usefully employed.
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
The parameter
19
now has a natural interpretation as the contraction constant. Jointly with the du-
ality condition, we then obtain a exact characterization of contractibility in terms of the eigenvalue
of the interdependence matrix .
Theorem 2 (Contraction Property via Eigenvalue)
The matrix
has the contraction property if and only if its eigenvalue
< 1.
Proof. If we try solve the strict inequalities (15), or for all i:
X
di >
ji dj ;
(17)
j6=i
with the assistance of a contraction constant , or
X
di =
ji
dj ;
j6=i
then by the Froebenius-Perron Theorem for nonnegative matrices (see Minc (1988), Theorem 1.4.2),
there exists a positive right and a left eigenvector, both with the same positive eigenvalue . The
associated eigenvector is positive as well. We can use the above dual property to establish that
clearly a ( ; d) solution exists for:
di =
X
ji
dj ,
j6=i
but from the duality relationship (15), we know that for every d > 0,
X
di >
ji dj ,
j6=i
so it follows that
< 1.
The dual condition directly illustrates the contraction property in the linear setting. By Theorem 2, we know that the matrix
has the contraction property if and only if we can nd a vector
d, which happens to be the eigenvector of the matrix
T
such that
d < d.
An additional insight, possible due to the nonnegativity of the interdependence matrix, is that the
primal condition can be strengthened for some c. Namely, there exists an eigenvector c such that
for all i
ic
< ci ;
and c is the vector which describes the set of deceptions for which the iterative procedure shrinks
at the fastest rate. By linking the contraction property to the eigenvalue of the matrix
, we
can immediately obtain necessary and su cient condition for robust implementation for di erent
classes of preference environments.
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
20
Symmetric Preferences In the symmetric model, the parameters for interdependent values are
given by
ij
=
(
1; if
;
j = i;
if
j 6= i:
The eigenvalue of the resulting matrix can be computed to be:
1+
= 1 + (I
1) ;
and hence from Theorem 2, we immediately obtain a necessary and su cient condition by requiring
that
<
1
I
1
.
Cyclic Preferences A weaker form of symmetry is incorporated in the following model of cyclic
preferences. Here, the interdependence matrix is determined by the distance between i and j
(modulo I), or
ij
=
(i j)mod I .
In this case, the positive eigenvalue is given by:
1+
=1+
X
(i j) ,
j6=i
and consequently a necessary and su cient condition for robust implementation is given by:
X
(i j)
< 1.
j6=i
Two Bidders With two bidders, the matrix of interdependence, , is given by
"
#
1
12
=
:
1
21
The eigenvalue of the matrix
can again be immediately computed by requiring that
1+
=1+
p
12 21
< 1.
12 21 ,
or
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
21
Central Bidder Finally, we may consider a model in which each bidder only cares about his
own type and the type of bidder 1, the central or informed bidder. The matrix of interdependence
is then given by
ij
=
In this case, the eigenvalue is given by:
8
>
>
< 1 if
j = i;
if
j = 1;
>
>
: 0 if otherwise.
1+
= 1 + 0,
and hence robust monotonicity holds vacuously for all , independent of I. The intuition in this
case is that bidder 1 has a private value utility model. In conjunction with the strict ex post
incentive constraints, this essentially means that agent 1 will always have a strict incentive to tell
the truth. But as the utility of all the other agents depends only the their own utility and the
utility of agent 1, and agent 1 is known to tell the truth, all other agents will also want to report
truthfully.
5.3
Nonlinear Conditions
The linear model has the obvious advantage that the local conditions for contraction agree with
the global conditions for contraction as the derivatives of the mapping hi ( ) are constant and
independent of . Conversely, with a nonlinear model, we can present weak local conditions for every
and stronger global conditions. With this we can extend the idea behind the linear aggregator
function to a general nonlinear and di erentiable aggregator function hi ( ) as follows.
De nition 7 (Local and Global Contraction Property)
1. The aggregator function hi satis es the local contraction property if for all c 2 RI+ and
int ( ), there exists i such that
ci
@hi ( )
@hi ( ) X
>
cj
.
@ i
@ j
2
(18)
j6=i
2. The aggregator function hi satis es the global contraction property if for all c 2 RI+ , there
exists i such that,
ci
@hi ( ) X
@hi ( )
>
cj
@ i
@ j
j6=i
for all .
(19)
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
22
Proposition 1 (Local and Global Contraction Property)
1. If hi satis es the strict contraction property, then it satis es the local contraction property.
2. If hi satis es the global contraction property, then it satis es the strict contraction property.
Proof. (1.) The proof is by contradiction. The strict contraction property requires that if, for
all
6=
0
i
, there exists i and
sign
for all
and
i
0
2
i
of the form
i(
i ).
2
i ( i)
0
i
i
ci
0
j
=
j
0
i
2
i ( i)
6=
i,
such that
i)
0 0
i;
i
hi
;
Fix any c 2 RI+ and choose small " > 0. Now consider deceptions
2 int ( ),
for all i, then if
0
i
= sign hi ( i ;
i ( i)
If for some
with
and (wlog)
=[
"ci ;
i
>
+ "ci ] \
X
cj
then
i
@hi ( )
@ i
0
i
i
j6=i
i,
i.
@hi ( )
.
@ j
0
i
is negative. Now choose
"cj . Now
hi ( i ;
i)
0 0
i;
i
hi
"
!
ci
@hi ( ) X @hi ( )
+
cj
@ i
@ j
j6=i
as " ! 0. This contradicts the strict contraction property.
(2.) Fix any deception. Let
cj =
There exists i
ci
max
0
j2 j( j)
0
j
j
.
@hi ( ) X
@hi ( )
>
cj
@ i
@ j
j6=i
for all . Let
i
0
i
= ci
0
0
i
such that
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
and suppose wlog that
hi ( i ;
i)
0 0
i;
i
hi
0
i.
>
i
0
Now x any
as:
Z1 X
I
@hi t + (1
@ j
t)
0
0
j
j
i
2
i(
i ),
23
we can then write the di erence
dt
t=0 j=1
Z1
=
t=0
Z1
t=0
@hi t + (1
@ i
@hi t + (1
@ i
t)
0
i
0
i
Z1 X
@hi t + (1
dt +
@ j
t)
0
j
0
j
dt
t=0 j6=i
t)
0
ci dt
Z1
X @hi t + (1
@ j
t)
0
cj dt
t=0 j6=i
> 0;
where the last inequality comes from the hypothesis of the global contraction property. This
establishes the claim.
6
Necessity of Contraction Property
The contraction property appears to be a natural condition in the context of robust implementation.
In fact, we now show that the weka contraction property constitutes a necessary condition for robust
implementation. In particular, the necessity of the contraction property allows us to give a sharp
impossibility result in the context of the linear model just discussed. The idea behind the necessity
argument is to show that the hypothesis of robust implementation leads inevitably to a con ict
with a deception pro le
which fails to satisfy the contraction property.
Theorem 3 (Necessity)
If f is robustly implementable, then f satis es strict EPIC and the contraction property.
Proof. The restriction to compact mechanisms ensures that S M is non-empty. It follows that
if mechanism M robustly implements f , then, for each i, there exists mi :
i
! Mi such that
g (m ( )) = f ( ) and m ( ) 2 S M ( ) ,
we can simply let mi ( i ) be any element of SiM ( i ).
We rst establish strict EPIC. Suppose strict EPIC fails. Then there exists i,
that f
0
i;
i
6= f ( i ;
i)
and
ui f
0
i;
i
;
ui (f ( ) ; ) .
and
0
i
such
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
24
Now, for any message mi with
mi 2 arg max ui g m0i ; m
i(
m0i
since m
i(
0
for all
i.
i)
2 Si1 (
i ),
Thus let
0 0
i;
i
0
i
0
i
= g mi
; ( i;
i)
,
we must have mi 2 Si1 ( i ) and thus g mi ; m
mi
and f
i)
2 arg max ui g m0i ; m
i(
m0i
;m
0
i
=f
i
0
i;
i)
0
for all
i
; ( i;
i,
i)
i
0
=f
i
i;
0
i
;
a contradiction.
Now we establish the contraction property. The proof is by contradiction and we show that if
f is robustly implementable, then every , with
this end, suppose
0
i
6=
i,
there exists
6=
i
>
0
i,
, must satisfy the contraction property. To
does not satisfy the contraction property. Then for all i,
i
and
sign
Thus for
6=
0
i
2
0
i
i
there exists
i(
i
hi
i)
=
and
0
0
i
i;
such that:
sign hi ( i ;
i
2
i(
> hi
i)
0 0
i;
i
hi
i)
such that
0 0
i;
i
> hi ( i ;
0
i
2
i ( i)
with
.
i) .
Now, by single crossing, if
vi (y; hi ( i ;
i ))
> vi f
0 0
i;
i
; hi ( i ;
vi f
0 0
i;
i
; hi
0 0
i;
i
< vi f
0 0
i;
i
; hi
i;
i)
(20)
and
vi y; hi
0 0
i;
i
vi y; hi
i;
,
then
0
i
0
.
i
(21)
In other words, there does not exist y such that:
vi y; hi
0 0
i;
i
vi f
0 0
i;
i
; hi
0 0
i;
i
> vi f
0 0
i;
i
; hi
i;
(22)
and
vi y; hi
for all e
i
such that e
and (21) apply.
i
2
i(
i;
e
i ),
i
as by hypothesis,
i;
0
i
2
e
;
i
i(
(23)
i)
and hence both (20)
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
25
We now show that (20) and (21) are in con ict with the hypothesis of robust implementation.
Consider an arbitrary deception
6=
. Let b
k be the largest k such that for every i, i and
0
i
2
i ( i ):
Si1
Sik ( i ) .
0
i
We know that such a b
k exists because Si0 ( i )\Si1
f , we must have
Si1 ( i )
\
Si1
0
i
0
i
0
i
;and since M robustly implements
0
i
.
= Si1
= ?.
0
i
Now we know that there exists i and
2
i ( i)
b
Sik+1 ( i ) \ Si1
such that
0
i
6= Si1
Let
b
m
b i 2 Sik ( i ) \ Si1
and
Since message m
b i gets deleted for
b
m
bi 2
= Sik+1 ( i ) \ Si1
i (m i ;
i)
0
i
.
i
2
(M
b
i ))
>
m
i
X
i (m i ;
i;
i
Let
m
b j 2 Sj1
b i; m i) ; ( i;
i ) ui (g (m
probability 1 on m
b i . Thus for every
i)
i
2
(
i)
0
i
2
that
i(
i ),
X
i
such that
i ))
>
X
b i; m
b i) ; ( i;
i ) ui (g (m
i(
i
, so (since M robustly implements f ), g (m
b i; m
b i) = f
then m
b
i(
i
2
b
Sk
i(
putting
b
i
0
i
>0)m
b j 2 Sjk ( j ) for all j 6= i,
there exists mi such that
X
b i) ; ( i;
i ) ui (g (mi ; m
i(
But m
b 2 S1
i )) .
0
j
for all j 6= i. Now the above claim remains true if we restrict attention to distributions
i(
i)
i
> 0 ) mj 2 Sjk ( j ) for all j 6= i,
there exists mi such that
X
i (m i ;
i ) ui (g (mi ; m i ) ; ( i ;
i;
,
at round b
k + 1, we know that for every
i
such that
m
0
i
i ).
Thus for every
b i) ; ( i;
i ) ui (g (mi ; m
i ))
>
X
i
i
2
i(
i ) ui
1
i
0
f
0
0
i
i )) .
. Also observe that if
, there exists mi such
; ( i;
i)
.
(24)
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
26
We also observe that the message pro le mi (and the associated allocation g (mi ; m
b i )) which
eliminates bi must be strictly dominated by the social choice function at the type pro les ei ; 0
i
for all ei 2
i,
or
ui f ei ;
0
i
; ei ;
0
i
> ui g (mi ; m
b i ) ; ei ;
0
, 8ei 2
i
i.
(25)
The argument here is by contradiction. Suppose the ex post incentive inequalities, (25), are not
satis ed strictly, and hence:
for some ei 2
i.
ui g (mi ; m
b i ) ; ei ;
0
ui f ei ;
i
Now, for any
mi 2 arg max ui g m0i ; m
b
m0i
since m
b
2 Si1
i
0
0
; ei ;
i
; ei ;
i
0
0
;
i
,
i
(26)
, we must have mi 2 Si1 ei and thus g (mi ; m
b i) = f
i
i;
0
i
. Thus from
(26) we also know that mi achieves the maximum:
mi 2 arg max ui g m0i ; m
b
m0i
and, for all ei , if
ui g (mi ; m
b i ) ; ei ;
then g (mi ; m
b i ) = f ei ;
and
0
i
i
; ei ;
ui f ei ;
i
.
0
i
0
i
; ei ;
0
g (mi ; m
b i ), we have established that for each
Now setting y
1
i
0
0
i
0
, there exists y such that
ui f ei ;
X
i
i(
0
i
; ei ;
i ) ui (y; ( i ;
0
> ui y; ei ;
i
i ))
>
X
i(
0
i ) ui
i
f
,
i
i
2
, 8ei 2
i,
0
i)
; ( i;
i(
i)
and
i
2
(27)
.
(28)
i
But now we arrive at the desired contradiction as the inequalities (27)-(28), coming from robust
implementation, and the inequalities (22)-(23), coming from the failure of the contraction property
cannot be true simultaneously. A symmetric argument works if
i
<
0
i.
We brie y sketch the idea of the necessity part of the proof. The proof is by contradiction.
We start with the hypothesis of robust implementation and consider a deception
for which the
contraction property fails to be satis ed. The failure of the contraction property in the single
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
0
i
crossing environment is then shown to imply that for all i,
0
i
2
i ( i)
we cannot
2
i ( i)
27
and some
nd an allocation y such that it weakly dominated if
0
i;
0
i
with
is the true type
pro le, or
ui f
0
but is preferred over f
i
2
i(
0
;
i;
i ).
e
,
(29)
i:
0
> ui f
i
0
ui y; f
if agent i is of type
ui y;
for all e
0
;
i;
e
i
,
(30)
Yet, if we start the iterative process, then for f to be robustly implementable, it must be that
there is a rst time, denoted by stage b
k, such that for some i and some i :
0
i
but then in stage b
k + 1:
0
i
b
2 Sik ( i )
b
2
= Sik+1 ( i ) .
0
i
The question then arises what is a necessary condition for
to be eliminated if all misreports
( ) are still possible candidate strategies at stage b
k. The answer is simply that there
feasible by
must be some allocation y, induced by a report
ui y;
but of course f
report of
0
i
i;
0
i
i;
e
i
of agent i, or y = f
0
> ui f
i
;
i;
e
i
i;
0
i
such that
,
would still have to satisfy incentive compatibility relative to a truthful
(if and when the true type is indeed
ui f
0
;
0
0
i ),
or
ui y; f
But now we observe that the necessary condition for
0
i
0
:
to be eliminated from Sik ( i ) in stage b
k + 1,
is precisely the condition which fails to hold if the contraction property fails to hold, leading to the
desired conclusion.
The above inequalities, (29) and (30) in fact describe a condition, termed robust monotonicity, in
Bergemann and Morris (2005b). There it shown that robust monotonicity is a necessary and almost
su cient condition if we want to guarantee Bayesian equilibrium implementation for all possible
priors. In Bergemann and Morris (2005b), the notion of Bayesian equilibrium implementation
for all possible priors allows the use of complicated augmented mechanism. In contrast, here we
focus on robust implementation in the direct mechanism, yet as the argument shows the robust
monotonicity condition emerges again as necessary condition for implementation.
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
28
For the linear model discussed in the previous section, with
hi ( ) =
X
ij j ;
j
we have an impossibility result as an immediate consequence of Theorem 3.
Corollary 1 (Impossibility of Iterative Implementation)
If the contraction property fails, i.e. there exists c 2 R+ n f0g such that for all i:
ci <
X
ij
cj ,
j6=i
then robust implementation fails.
7
Single Unit Auction
We conclude our analysis with a second example, namely a single unit auction with symmetric
bidders. The model has I agents and agent i's payo type is
agent i's valuation of the object is
i
+
X
i
2 [0; 1]. If the type pro le is ,
j,
(31)
j6=i
where 0
1.
An allocation rule in this context is a function q :
! [0; 1]I , where qi ( ) is the probability
P
qi ( ) 1. The symmetric e cient allocation rule is given
that agent i gets the object and so
i
by:
qi ( ) =
(
1
#fj:
j
k
for all kg ,
0;
if
i
if
k
for all k;
otherwise.
Maskin (1992) has shown that the e cient allocation can be truthfully implemented in a generalized
Vickrey-Clark-Groves mechanism, according to which the monetary transfer of the winning agent
i is given by
yi ( ) = max
j6=i
8
<
:
j
+
X
j6=i
j
9
=
;
.
We observe that the winning probability qi ( ) and the monetary transfer are piecewise constant.
The generalized VCG mechanism therefore does not satisfy the strict ex post incentive compatibility
conditions which we assumed as part of our analysis. We therefore modify the generalized VCG
mechanism to a symmetric "-e cient allocation rule given by:
qi ( ) = "
i
I
+ (1
") qi ( ) :
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
29
" 3
2.
We then argue that
Under this allocation rule, the object is not allocated with probability
the symmetric " e cient allocation rule can be robustly implemented if
<
1
I 1.
Alternatively,
we can say that the generalized VCG mechanism itself is virtually and robustly implementable if
<
1
I 1.
It is easy to verify that the resulting generalized VCG transfers satisfy strict ex post incentive
compatibility and show that this "-e cient allocation is robustly implementable. The corresponding
essentially unique ex post transfer rule is:
0
1
X
" @
"
( i )2 +
yi ( ) =
jA
2I
I
+ (1
i
j6=i
0
") @max
j6=i
8
<
:
j
+
X
j6=i
91
=
j A qi ( ) :
;
The rst two components of the transfers guarantee incentive compatibility with the respect to the
linear probability assignment and the third component with respect to the e cient allocation rule.
0
The best response of agent i for misreport
i
of the remaining agents at a true type pro le
is
given as the public good example by:
0
i
=
i
+
X
0
j
j
.
j6=i
We can therefore exactly repeat our earlier argument in the context of the public good and get
robust implementation in the direct mechanism if
<
1
I 1.
The implementation conditions here are substantially di erent from the generalized single crossing conditions recently obtained by Birulin and Izmalkov (2003) and the dominant e ect property
identi ed by Echenique and Manelli (2004). The single crossing condition obtained in Birulin and
Izmalkov (2003) provide necessary and su cient conditions for the existence of an e cient equilibrium in an English auction for a single object.4 In our linear and symmetric environment, see
(31) their necessary and su cient condition reduces to the condition of
< 1, independent of
the number of agents, I. Echenique and Manelli (2004) present a dominant e ect property which
guarantees the existence of an e cient ex post equilibrium without further continuity and di erentiability conditions. However in our linear and symmetric environment, their condition again
reduces to
3
< 1, independent of the number of agents, I
At the cost of some additional algebra, we could modify the allocation rule qi ( ) to be:
"P
i
+ (1
") qi ( ) ;
j
j
which in turn allocates the object with probability 1.
4
Their conditions generalize earlier results by Maskin (1992) for two bidders. The novel issue with many bidders
is that a marginal change in the signal pro le should favor one of the currently winning bidders over a currently
loosing bidder. With many bidders, a pairwise condition comparing the e ect of signal
anymore.
i
on i and j is not su cient
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
8
30
Conclusion
Interdependent Valuations In this paper we considered implementation in an environment
with interdependent valuations. We provided conditions for full implementation which did not
depend on the prior or posterior belief of the agents. More precisely, we provided conditions
under which the social choice function can be implemented in the direct mechanism by iteratively
eliminating strictly dominated reports. The idea of robust implementation appeals to the same
epistemic insight as iterative elimination of strategies of strictly dominated strategies in complete
information games (see Brandenburger and Dekel (1987)). In particular, robust implementation is
equivalent to Bayesian equilibrium implementation for all possible priors, which we called robust
monotonicity in a companion paper, Bergemann and Morris (2005b). In contrast to much of the
recent literature on implementation which relies heavily on complicated augmented mechanisms
to achieve full implementation, here we pursued implementation in the direct mechanism without
relying on augmented mechanisms. The resulting su cient and almost necessary condition for
robust implementation, the contraction property, was shown to essentially require that there is not
too much interdependence in the valuation of each agent across signals received by the agents. In
the important case of the linear model in signals, the contraction property was shown to reduce to
a single condition on the eigenvalue of the interdependence matrix.
The nature of the contraction property also highlighted that robust implementation is considerably more demanding than ex post truthful implementation. Finally, the example of the e cient
single unit auction suggested that the dividing line between positive and negative robust implementation results might also be the exact dividing line for the more permissive notion of virtual
implementation.
Contraction Property The robust implementation argument rested essentially on the single
crossing property and the contraction property. The single crossing is essentially symmetric in
allocation and type. It therefore would have been possible to impose the contraction property on
the outcome function rather than on the preference aggregator. In fact, given that the misreports
can only alter the outcome function, but certainly not the preferences, one might have thought it
would be more natural to impose the contraction property on the outcome function rather than
on the preference aggregator.5 The advantage of using the contraction property on the aggregator
function arises from the single crossing condition. The true type
and the misreported types
0
can potentially be very far from each other. Consequently, the preferences at the type pro les
5
In fact, the contraction property has been employed successfully in games with complete information and linear
best responses to prove the uniqueness of the Nash equilibrium, see Luenberger (1978) and Gabay and Moulin (1980).
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
and
0
over a pair allocations, in particular f
0 0
i;
i
and f
0
i
+ ";
i
31
, can be very di erent.
With the contraction property on the preference aggregator, it su ces to compare the allocations,
f
0 0
i;
i
and f
be valid for
0
i
+ ";
i
for a type pro le near
0
=
0 0
i;
i
and then extend the ranking to
through the existence of an aggregator hi and the single crossing property. Without
the aggregator hi but a contraction property on the social choice function, we would be forced to
rank the allocations f
= ( i;
i ).
0 0
i;
i
and f
0
i
+ ";
i
for some preferences near the true type pro le
In particular, in order to be able to use the single crossing condition fruitfully, it
would have to be the case that the allocations f
the equilibrium allocation for some reports
i; i
0 0
i;
i
and f
0
i
+ ";
i
would also arise as
of agent i given the truthful report
i
of the
remaining agents. But such a \full support" requirement is rather strong. In particular, it will
rarely be satis ed in models with quasilinear utilities, where each agent has preferences over a
two-dimensional object, the allocation and the monetary transfer.
Robust Implementation: The Case of Direct Mechanisms November 7, 2005
32
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