GAMES
AND
ECONOMIC
BEHAVIOR
4, 463-483 (1992)
Continuous Implementation in Economies with
Incomplete information*
DAVID WET-I-STEIN
University of Western Ontario, London, Canada N6A 5C2
Received November 20, 1990
We study the problem of implementing a given Social Choice Correspondence,
via a continuous outcome function, in economic environments with incomplete
information and assume a continuous informational structure. We impose the
assumption of Non-Exclusive Information and construct a mechanism with an
“almost continuous” outcome function implementing a given Social Choice Correspondence, provided it satisfies monotonicity and some further technical conditions. Journal of Economic Literature Classification Numbers: 025, 026. o 1992
Academic Press. Inc.
1.
INTRODUCTION
The theory of mechanism design in general and implementation theory
in particular deals with ways by which a society can achieve desired
outcomes.
Hurwicz (1960) presented several mechanisms analyzing their performance according to various criteria, thus embarking-as
he put it-on a
line of research in which the mechanism is the unknown of the problem.
In this approach one does not take the prevailing set of rules and institutions as given. The theory of design should look at various criteria according to which mechanisms should be judged or ranked. It should determine what goals are attainable and at which price.
* The research reported in this paper is based in part on my Ph.D. Thesis submitted to the
University of Minnesota. 1 thank my Thesis adviser Professor Leonid Hurwicz for helpful
discussions. Helpful conversations with James Jordan, Andrew Postlewaite, and David
Schmeidler are gratefully acknowledged. I also thank an anonymous referee for several
helpful remarks.
463
0899-8256/92 $5.00
Copyright 0 1992 by Academic Press, Inc.
All rights of reproduction in any form reserved.
464
DAVID
WETTSTEIN
We deal with economic environments; the society or the environment
consists of a set of individuals characterized by preferences and initial
endowments. Thus we restrict our attention to a pure exchange setting.
An outcome is a reallocation of the initial aggregate endowment vector to
the various individuals.
A Social Choice Correspondence (XC) maps environments into sets of
allocations, the set corresponding to a given environment can be thought
of as the set of desirable or acceptable allocations given the environment.
This set depends on the characteristics of the individuals comprising the
environment: their preferences and initial endowments.
The basic question leading to the issue of implementability is the following: How can an outside designer guide the economy toward a desired
outcome? In much of the literature the notion of implementation
has
been interpreted as follows: Given a family of environments and an SCC,
implementing the SCC amounts to designing a mechanism whose set of
equilibrium outcomes for each member of that family coincides with the
set of outcomes prescribed for that member by the SCC.
As is well known, the problems start when the designer does not possess
all the information needed to carry out the implementation and has to rely
on the individuals to supply it.
The basic result on implementation with complete information was formulated by Maskin (1977). He employed the Nash equilibrium (NE) concept and showed that any SCC satisfying monotonicity and no veto power
can be implemented provided the number of individuals is greater than or
equal to three. Since in economic environments with private goods and
more than two persons the no veto power assumption is usually satisfied,
it follows that a monotonic SCC can be implemented. The proof was
carried out by explicitly constructing the desired mechanism.
Clearly the results one gets crucially depend on the equilibrium (solution) notion used. We restrict our attention to Nash and/or Bayesian-Nash
equilibria. Several works dealt with notions of subgame perfect equilibria
(Moore and Repullo, 1988) and equilibria in undominated strategies (Palfrey and Srivastava, 1991). Their results indicate that using those more
sophisticated notions of equilibria considerably enlarges the set of implementable SCCs.
The next stage in the development of the field was to define and investigate implementation
in the presence of incomplete information. PostleWaite and Schmeidler (1983, 1986) defined economies with differential
information using the following model: There is a space Y of possible
states of nature, a y in Y completely determines all the characteristics of
the economy. Each individual has a partition on Y and knows only what
member of his partition occurred.
They have shown that if (a) the number of individuals is greater than or
CONTINUOUS
IMPLEMENTATION
465
equal to three and (b) Non-Exclusivity
of Information (NEI) is satisfied,
then any SCC closed under common knowledge concatenation and satisfying a strong version of the monotonicity condition can be implemented.
The solution concept is no longer the NE but the Bayesian-Nash Equilibrium (BNE) introduced by Harsanyi (1967) for games with incomplete
information. NE1 is the assumption that any N - 1 individuals, where N
is the number of individuals, can tell what the true state of nature is if they
pool together their information. Blume and Easley (1990) demonstrate
that for some SCCs the violation of the NE1 assumption implies that
implementation is impossible. The common knowledge concatenation assumption is extensively discussed and motivated in Jackson (1991).
Palfrey and Srivastava (1989) allowed for general information structures,
removing the restrictive NE1 assumption. They have shown that if the
number of individuals is greater than or equal to three, any SCC closed
under common knowledge concatenation,
satisfying a suitably defined
version of the monotonicity condition and a-self selection (E-SS) can be
implemented. The usual self selection (SS) condition (which is slightly
weaker than E-SS) and monotonicity were shown to be necessary for the
implementability
of a given SCC.
Jackson (1991) demonstrates that one could dispense with the E-SS. The
definition of a state used in that work was different than the one used by
Palfrey and Srivastava (1989) and Postlewaite and Schmeidler (1986). The
information sets of the individuals were taken as the primitive concept, and
a state was defined as any collection of such information sets regardless of
whether or not they were consistent. Thus an allocation rule was defined
for a strictly larger set than the set of possible events. This gave rise to a
somewhat stronger monotonicity condition. However, the conditions of
closure under common knowledge concatenation, SS, and monotonicity
were shown to be necessary and sufficient (provided some technical conditions were satisfied) for implementability,
thus fully characterizing the set
of implementable SCCs.
The analysis in the above mentionedworks
was restricted to a discrete
informational structure, the set of possible states of nature was taken to
be finite, and it was assumed that each event in an individual partition
member has a positive probability of occurring.
Two drawbacks of the above results are (i) the inherent discontinuity of
the outcome functions employed in the mechanisms constructed to establish implementability
and (ii) a discrete informational structure. Admittedly the discreteness of the informational
structure makes continuity
with regard to this component of the strategy space trivial. However, the
outcome functions used so far were discontinuous with regard to all the
components of the strategy space.
Implementation
theory among other things tries to come up with reason-
466
DAVID
WETTSTEIN
able mechanisms (institutions) for the individuals to take part in. Discontinuous outcome functions suffer from the fault that a slight change in one’s
strategy which might be interpreted as a slight mistake is liable to cause
a considerable change in the outcome reached. We feel continuity of the
outcome function is an important feature to look for in any proposed
mechanism. Furthermore our models at best approximate reality. In order
to make our conclusions and suggestions robust with regard to slight
misspecifications, we should be employing a mechanism with a continuous
outcome function. See also Postlewaite and Wettstein (1989) for further
discussion of the continuity issue. The informational structure we use does
allow for a continuum of possible states of the world. It subsumes the
previous models with a discrete informational structure as a special case.
The extension to a continuum of states will make it possible, in the future,
to study sequences of information structures and thus test the robustness
of our results with respect to assumptions similar to the NE1 assumption.]
It is the objective of this work to avoid these drawbacks. Thus we build
a continuous version of the above models. The informational structure is
generalized by regarding R” with its Bore1 sets and a probability measure
p defined on them as a probability space and letting the set of possible
states of nature be given by some measurable subset of R”; p is common
knowledge as in Aumann (1976). However, each individual observes only
one coordinate of the vector determining the state of nature. The individual’s preferences depend on the prevailing state of nature and the consumption undertaken. The outcome function of the mechanism constructed,
would be “almost” continuous with regard to the natural topology, the
strategy space is endowed with (i.e., if part of the strategy space calls for
sending in elements of R’ the outcome function should be continuous with
respect to the usual topology on R’).2
This formulation of the environment is general enough to include economies with complete information as in Maskin (1977), in which case in our
model we interpret the support of p as “diagonal”:
any single coordinate
uniquely determines all the others. In the Jackson, Palfrey and Srivastava,
’ I am grateful to an anonymous referee for pointing out this line of research.
2 Naturally we cannot expect the outcome function to be continuous in regions where the
Social Choice Correspondence itself exhibits discontinuous behavior. Our goal is to construct
an outcome function that would not impose any further discontinuities beyond those implied
by the structure of the economy or the Social Choice Correspondence. This should not be
taken as indicating that discontinuities in the SCC must be inherited by any implementing
mechanism. It is well known there are continuous mechanisms implementing the Walrasian
correspondence which is discontinuous. However, the outcome functions of the above
mechanisms do not use the Walrasian correspondence. In contrast, the outcome function of
our mechanism, due to the general nature of the problem, makes use of the actual correspondence being implemented.
CONTINUOUS
IMPLEMENTATION
467
and Postlewaite and Schmeidler framework, the random variable assumes
only a finite number of values.
We assume that our informational
structure does satisfy the NE1 assumption. Palfrey and Srivastava (1989) and Jackson (1991) allowed for
more general information structures and used in a very essential way the
SS assumption. Their outcome function associated the zero allocation (the
“worst outcome”) with any detectable deviation from truth telling. The
SS assumption is then used to generate truth telling as a Bayesian Nash
Equilibrium. Associating a certain “worst outcome” with any deviation,
no matter how minute, is not a continuous operation. Thus the use of a
continuous outcome function rules out these constructions and our analysis is conducted in environments satisfying the NE1 assumption.
We note however that were we to consider the other extreme of environments with completely exclusive information, i.e., when it is impossible
to detect any false reports on part of the individuals, a suitably modified
version of our mechanism would continuously implement any SCC satisfying a monotonicity condition and SS condition as in Palfrey and Srivastava. Furthermore by assuming an appropriately defined, continuous version of the SS condition, we could modify the mechanism constructed to
continuously handle intermediate informational structures. However, this
would require excessive amounts of notation which would not be justified
given that the main difficulties and techniques associated with continuous
implementation
are aptly covered in our present, admittedly less general
setting.
At this stage it is advisable to sum up what are the informational assumptions in our model (“Who knows what?“). We assume the designer knows
the initial endowments which are fixed over the states of nature. The
designer and the individuals know the probability space, how the individual characteristics depend on the state of nature and the SCC.
The designer has no information on the state of nature while the ith
individual observes the ith component of the state. The designer has to
construct a mechanism implementing the SCC.
Our result can be described as follows: A monotonicity condition, basically the one used by Palfrey and Srivastava (1987), is necessary if the
implementing mechanism satisfies certain properties. Then we show that
if NE1 is satisfied any SCC satisfying monotonicity can be implemented if
the number of individuals is greater than or equal to three and some
technical conditions (continuity and strong monotonicity of preferences)
are satisfied. The proof is carried out by constructing a mechanism, with
an “almost” continuous outcome function, implementing the SCC. Some
convexity demands regarding the support of p and the SCC will guarantee
continuity of the outcome function.
Unlike Postlewaite and Schmeidler (1983, 1986) and Jackson (1991),
468
DAVID
WETTSTEIN
we do not assume that the SCC is closed under common knowledge
concatenation. Instead, we assume the game can consist of several stages.
The first stage is played before any private information is observed, then
the private information is observed and the second stage takes place. The
assumption that there is an initial stage in which everyone possesses
symmetric information restricts the applicability of our results. However,
this assumption can be replaced by the common knowledge concatenation
assumption at the cost of further complicating the mechanism. The second
section introduces several notations and elaborates on various definitions.
In the third section we prove an implementability
theorem by constructing
a mechanism, with an “almost continuous” outcome function, implementing a given XC. The final section discusses possible applications and
further lines of research.
2.
NOTATION
AND
DEFINITIONS
(R”, B”, p)-The n-dimensional Euclidean space viewed as a probability
space with B” denoting the set of Bore1 sets and p a probability measure
defined on B”. A generic element of this space is denoted by
&
=
(El,.
.
. )
e,)whereeiisinRforalli
= 1,. . . ,n.
J-The support of p. It is interpreted as the set of possible states of the
world. J is assumed to be compact.
Ji, >* . “i, -The projection of J onto the (i, , . . . , i,) axis.
The economy E containing n individuals and k commodities consists of
the following:
1. An n-tuple of von Neumann-Morgenstern
utility functions
Ui(x, E) denotes the satisfaction individual i derives from consuming bundle
x, when the state of the world is given by E. The Ui’s are assumed to be
measurable3 in their last n coordinates.
2. An n-tuple of initial endowments (w,, . . . , wJ, where Wi E Rk, for
all i = 1, . . . , fz. W = Gin_, wi. So we assume endowments are certain
and do not vary with E. This assumption could be relaxed if one were
willing to consider implementation
in terms of net trades and allow for
nonfeasible outcomes outside of equilibrium. This issue was treated extensively in Hurwicz, Maskin, and Postlewaite (1982).
3 Unless otherwise specified, measurability
is taken to be in the Bore1 sense.
CONTINUOUS
469
IMPLEMENTATION
We assume that the economy and the probability space are common
knowledge as in Aumann (1976). Individual i observes q and this constitutes private information.
X-An
allocation.
X E R;k, X = (x1, . . . , x,), where Xi denotes the commodity bundle
allocated to individual i.
A-The
set of feasible allocations for E.
A=
T-The
(x,,.
. .,x*)ER:k
&+f:w<,
i=l
forj=l,.
. .,k
i=l
set of net trades.
n
(z,, . . . ,z,) E Rnk 2 z+
0
forj=l,.
. .,k
i=l
f-An
allocation rule defined as a measurable function
feasible allocation with each E in J.
associating
a
hf-A
trade rule from the allocation rule f, defined as a measurable
function associating a feasible trade with each E in J.
hf: J + T and furthermore for all E in J, hf(.z) satisfies h{(e) + A(E) 2 0
for all i = 1, . . . , 12.
h<-A trade rule for individual i from allocation rule f, defined as a
measurable function associating a feasible trade for individual i with each
E in J.
h:: J--f Rk and furthermore for all E in J, h{(F) satisfies:
h{(e) + J;(E) 2 0
and
h;(E) + A(E) 5 W.
F-The
set of all allocation rules.
Eif-The
set of all trade rules from the allocation rulef.
@-The
set of all trade rules from allocation ruleffor
individual
F-A Social Choice Correspondence (SCC).
FCF
f *-An allocation fule in E designated as the default rule.
II-A
measurable function from R” into R” with the following
properties:
(i) II(F) is in J for all E in R”;
(ii) II(.?) = 2 for all 2 in J.
i.
two
470
DAVID
WETTSTEIN
The way we define an XC deserves some comment. One could define
an SCC as a mapping associating a set of good allocations (not allocation
rules) with each state of the world. Our definition, which coincides with
the definition of Postlewaite and Schmeidler (1983, 1986), is more specialized. An allocation rule may embody the idea that there is a certain
property we want the good allocation to satisfy over all states of the world.
It is true that any SCC defined as a mapping of the type mentioned above
can be imitated by a set of allocation rules. The allocation rules will be all
the possible selections from the sets of good allocations.
Another argument in favour of using allocation rules rather than allocations has to do with the structure of timing and uncertainty in these models.
The individual has to take an action after observing his private information.
This information is usually not enough to determine the precise state of
the world. Hence the action he takes may have different consequences,
perhaps even in terms of the commodity bundles he gets, for different
states among which he cannot distinguish when taking the action. This
will lead individuals to consider the desirability of alternative allocation
rules.
A monotonicity condition plays a central role in our subsequent discussion. The definition of monotonicity is broken into two parts. First we
define the acceptability of a mapping which is then used in the monotonicity definition.
In the definition of acceptability we use the notion of a strong null set
which is defined as follows:
L, a subset of J, is called a strong null set i$
Foralli=
1,. . ., n,P(Ljq)
= OforallEiinJi.
Or in words L is a strong null set if every individual assigns a zero posterior
probability to L for any ci he might have observed.
A measurable mapping o = (u, , . . . , (T,) from J into R” will be called
acceptable for an allocation rulef, if the following holds:
(i) pi: Ji * Ji ;
(ii) U(E) is in J for all E’S in J/L, where L is a strong null set;
(iii) Define &((E) by s(e) = II 0 V(E).
Foralli
= 1,. . . , n if an allocation rule a satisfies for an Ei in Ji,
then we have for all ci in v,~‘(v~(E~))
E(Uj(~(&((E))y
E)
1 EJ
2
E(Ui(ui(h((E))T
&)
I &i);
CONTINUOUS
IMPLEMENTATION
Remarks.
1. The conditional expectations are defined through the
of a conditional probability measure on J given that the realization of
ith coordinate equals ai
in the first pair of expectations and &i in
second pair.
2. We assume all the expectations taken are finite.
3. The use of II is necessary to ensure thatfand
a are well defined
all points of J, even those which c maps into the complement of J.
471
use
the
the
for
An XC F will be called monotonic if the following holds:
For any f in F and for any measurable mapping o acceptable for f, the
allocation f defined by
J‘(E)= f(W))
coincides with some allocation
set.
rule in F, except perhaps on a strong null
Remarks.
1. The basic ingredient in the definition of acceptability is an
allocation rule. This follows directly from the very notion of implementation we use, in which the basic object of discussion is an allocation rule.
2. Naturally monotonicity
should be a property of an SSC and not
be dependent on any specific mechanism being used. However, we can
interpret u as having individuals behave as if they were observing a(~)
instead of E. Condition (ii) serves to make this behavior pattern consistent,
except, perhaps on a strong null set.
3. The identity mapping is clearly acceptable and that of course imposes
no restrictions on the structure of the SCC. However, there might be other
acceptable mappings, for instance where all the E’S are mapped into a
certain a*. These will indeed impose restrictions on the structure of the
see.
4. This definition of monotonicity is basically the one used by Palfrey
and Srivastava (1987) with a finite set of states of the world. It reduces to
the usual concept of monotonicity introduced in Maskin (1977) when one
deals with the case of complete information.
A mechanism G will consist of the following:
1. An n-tuple of measurable strategy sets (S, , . . . , S,)
s; = Bj x D;
Bi denotes acts that have to be taken before the observation of any private
information, Di denotes acts taken after the observation of the private
information.
472
DAVID
s = fpi;
WETTSTEIN
B = fiBi;
i=l
i=l
D = fiDi.
i=l
2. A measurable outcome function g: B x D * A, g = (g,, . . . ,
g,), where gdenotes the bundle received by individual i. A strategy for
individual i wotrld%e a choice of bi in Bi and a measurable function di:
Ji ---, Di. In defining the BNE of such a mechanism we use the following
notation:
s = (s, ) . . . ) SJ,
for s^in Si define (sBi, .?) = (~1, . . . , Si-1, i, Si+i, . . . , sJ. A BNE of
the mechanism is an n-tuple of strategies,
s = (S,) . . . , S,)
where Si = (bi , ai).
that satisfies for all i = 1, . . . , n the following:
(i) E(Cri(gi(~),-- E)) 2 E(Ui(gi(s_i, s^),--E)) for all s^in Si;
(ii) E(C;l,(gi(b,d)7 El ( &i) 2 E(Ui(gi(b2d-i,
617 E) 1 &J
for all 6 in Di , for all Ei in Ji .
Note that (ii) is actually implied by (i) almost everywhere (in Ji). The
strategies chosen by the individuals yield an allocation rule a,
a(E) = &TM&)).
N&)-The
set of allocation rules corresponding to BNE of the mechanism for E.
G implements F if
(0 FC~&%
(ii) Any allocation rule in N,(E) coincides, except, perhaps on a
strong null set with an allocation rule in F.
3.
NECESSARY
AND SUFFICIENT
CONDITIONS
FOR IMPLEMENTABILITY
Describing a set of conditions which are both necessary and sufficient
for implementability
(in the framework we have outlined) is still an open
question. We shall start by stating a theorem, regarding monotonicity as
a necessary condition. The proof except for some purely technical points
is identical to the one given by Postlewaite and Schmeidler (1986).
CONTINUOUS
IMPLEMENTATION
THEOREM
1. If an SCC F is implementable by a mechanism
satisfies the following condition:
For every f in F there exists a BNE s = (6, d) satisfying:
(cl) For all-8 in D, for all i = 1, . . .’ n
g(6_i) 6, 6) = g~6-~,6, G)for
all b and 6 in B;.
(~2) g(6, ak)) = f(~)for all E in J.
Then F must satisfy monotonicity.
Proof.
473
G which
See the Appendix.
In order to prove that a monotonic
that J satisfies NE1 (Non-Exclusivity
will be formulated as follows:
F can be implemented we assume
of Information).
This assumption
(NEI) for all E in J, 2 ci = 0.
i=l
Admittedly this is not the most general way of formulating the NE1 assumption. We use this convenient form to avoid additional notation. In
the proof we only use the fact that any ci can be uniquely expressed as
a function of all the other coordinates. We also need three technical
assumptions; that the Ui’s are continuous and strictly increasing in their
first k arguments, that no one ever gets the zero bundle allocated to him
and an integrability assumption on the Ui’S. These assumptions are needed
amongst other things, to guarantee that various fines imposed by the
mechanism are indeed effective.
2. Given an economy E and an SCC F, if the following
assumptions are satisfied:
(Al) n 2 3
(A2) For all i = 1, . . ., n, Ui is continuous and strictly increasing
in its first k arguments.
(x 2 y, x # y implies for all E in J
UJx, E) > Ui(Y, &)I.
(A3)Foralli
= 1,. . . , n, Ui(W, E) is integrable in its last n
arguments.
(A4) F satisJes monotonicity.
(A5) J satisfies NEI.
(A6) For all f in F and all E in J
./x4 2 0, .I%&) # 0
foralli=
1,. . . ,n,
where J; denotes the bundle allocated to individual i.
then F can be implemented by an (almost) continuous mechanism.
THEOREM
474
DAVID
WETTSTEIN
Proof.
We shall construct an explicit mechanism implementing
strategy space for individual i will be:
F. The
si = Bj x Di,
where
Bi = F
Di = Ji
X
Hi(b)
X
N
X
M.
N denotes the set of all positive natural numbers and M the set of all strictly
positive numbers, Hi(b) denotes a set of “trade rules” for individual i.
A generic element of the strategy space will be denoted by
The first component belongs to Bi and the last four to Di . The strategy of
individual i can be given the following interpretation:
Y-An allocation rule individual i would like to have. This has to be
decided upon before the observation of any private information.
rrThe
q he “observed.”
+-The
trade rule individual i would like to have.
nrAn
“indication”
as to how much weight should be assigned to the
trade rule demanded.
m,-A number affecting the “fines” imposed on individual i for any
detected lies (declared profiles outside J) and deviations from allocations
in F.
The outcome function is defined as follows:
Stage 1. Construct a weighted
f;f(.s), the allocation corresponding
Define
where ‘f denotes the allocation
average of the ‘f’s and denote it by
to an E in J, is constructed as follows:
rule announced
Now define (Y = 2 aj
j=l
ifa>O
ifa = 0
by
t.
CONTINUOUS
IMPLEMENTATION
475
and finally
If f is in F it is called f and the mechanism moves to the next stage.4
Otherwise the fin F which minimizes f If(e) - f(e)12 &(E),
where I(E)
denotes the distribution function of E, is chosen. If it has no solution then
the default allocation rule f* is chosen as jY In any case the mechanism
provides an f in F. This f determines the set Hi(b) as @.
Stage 2. In this stage n-profiles in J are constructed. The ith profile is
denoted by (?, , . . . , 7:). The profiles are constructed based on the r,
messages sent by the individuals. The details of the construction are
identical to Wettstein (1990) and can be found in the Appendix. The
construction satisfies the following two properties.
(i) If (r, , . . . , r,) is in J then all the n-profiles are identical and equal
to r.
(ii) A change in the strategy of individual i will not change the ith
profile.
Stage 3. Construct out of the announced trade rules hi (belonging to
@) and the various ni’s a trade rule h in Eif.
Before proceeding with the technical details, we would like to point out
the following difficulty. An individual may announce different trade rules
for different observations. Taking for example individual 1, for each 6 he
observes he chooses some trade rule hb,(.s,, . . . , E,). Even if all the
h,, are measurable, as we assume they are, the “diagonal function”
+, , . . . , E,,)= he,h , . . . , E,)
may not be measurable. However, this would imply that the individual at
the uninformed stage of the game cannot evaluate the utility associated
with the strategic choices he plans to make after having observed his
private information. Hence we must restrict our analysis to the case where
individual strategies lead to measurable “diagonal functions.” This assumption guarantees that the trade rule constructed by the mechanism is
“diagonally measurable.”
The first step consists of transforming the individually announced trade
rules into a trade rule which is feasible in the aggregate.
Define n(c) by:
4 We note that F(E) is a continuous function of the ‘f’s even though the p,‘s are not.
476
DAVID
WETTSTEIN
n
de) = minbini,l,.
,,(ni)-‘;
minj,,,
-1
.
,k Wj C n,(h{(~) + f{(e))
(
( i=l
1)
We recall that Wj denotes the aggregate endowment of commodity j
whereas h-jandfi respectively denote the net trade in commodity j requested by individual i and the amount of commodity j allocated to him
by f.
The trade rule constructed by the mechanism Z(E) will be
&I = e&L . . . , h,(E)),
where
hi(E) = It(&)ni ’ (hi(E) + I)
- I.
Remarks.
The trade rule constructed is measurable
might have occurred, as well as diagonally measurable.
for each E that
Stage 4. Define an allocation rule T by T = h + f, where h and fare
respectively the trade rule and allocation rule constructed in the previous
stages.
Now components of a set of new allocation rules are constructed (one
component for each individual). For all i = 1, . . . , n define pi by
~i(&l
) .
. . ) ri,
. .
. ) E,)
=
6i.
Ti(&l)
. . . , ri,
. . f ) E,),
where 0 < ai I 1 is the largest number for which
E(Ui(~(E), E) 1rj)~ E(Ui(pi(&)y E) 1Ti).
Now define
where 0 < y ~5 1 is the largest such number for which
Stage 5. Similar to our discussion in the Appendix, we now project on
J, using a measurable projection function constructed prior to the game
CONTINUOUS
477
IMPLEMENTATION
and assumed to be common knowledge among the individuals.
(rl, . . . , r,) on J and get a point E* in J and let
&b,
3 - .
. ) s,)
=
Project
tj * ‘I)i(Ti),
where
First we prove that F C N,(E).
BNE:
‘$=f
forallalli=
Givenfin
F we construct
the following
1,. . .,n
rj = q; hi = 0; ni = mi = 1
foralli
= 1,. . . ,nandallEinJ.
At Stage 1 we obtainf, which no single individual can change by deviating and declaring some other allocation rule. At Stage 2 we get the profile
(q,.
. *, E,) for all the individuals. At Stage 3 we get the zero trade rule.
At Stage 4 we get T = fand p, where defined, equals 7. Finally since the
profile is the same for all individuals, 7) equals p and hencef. At Stage 5
wegetforalli=
1,. . .,n
g,(b, d(c)) = A(E) for all E in J.
since tj = 1.
So this n-tuple of strategies does yield the allocation rulef. It remains
to be shown that it indeed forms a BNE.
Individual i cannot affect the allocation rule f, or the profile (P’) by
changing his strategy. Since all the other individuals report truthfully Y’ is
the true profile. He can change the trade rule constructed by the mechanism, but this will never yield him an allocation rule which is strictly
preferred tof, given the true E; observed by him.
In order to show that any allocation rule, arising as a BNE, coincides
with an allocation rule in F up to a strong null set we first show that all
BNE must have
h=O
(rI(q),
. . . 1 r,(4)
E J,
for all E in J, except, perhaps on a strong null set.
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DAVID
WETTSTEIN
Assume by way of contradiction that one of these conditions is violated
on a set L that satisfies for some i and some ai in Ji, P(L 1q) > 0. By Stage
5 we get that in the case where individual i does observe ai, no choice of
mi would yield an equilibrium;
mi must be strictly positive but the ith
individual would like to make it as small as possible.
Hence those conditions must hold up to a strong null set in any equilibrium.
Now we show that in a BNE the mapping T(E) defined by
Y(E)= G-,(q), . . * , Y,(E,))
is acceptable for the allocation rule f constructed in that BNE.
By assumption, the individuals choose measurable strategies and thus
the mapping Y is measurable. r; maps Ji into Ji, and as shown above, T(E)
is in Jfor all E in J/L, where L is a strong null set. So the first two properties
of acceptability are satisfied. Now suppose by way of contradiction that
there exists an individual i, an pi in Ji, and an allocation rule a for which
whereas
E(Ui(J;(~E)),
where P(s) = 7~0 I-(E)
Since r is acceptable,
set,
E) 1Ei) < E(Ui(ai(i(E)),
the following
E) I Ei),
equality holds, up to a strong null
where (b, d(e)) denotes the BNE that the Y(E) reporting strategy is part of.
Thus
E(Ui(gi(b, ~E)),E) (EJ < E(Ui(ai(4&)), ~11EJ*
If that is the case, when observing Ei, individual i can improve his
position by deviating from his equilibrium strategy.
By declaring an Izi large enough, an mi small enough, and the trade rule
hi(E)= ai - fitE),
the ith individual
can get arbitrarily
close to the allocation
rule
CONTINUOUS
479
IMPLEMENTATION
Ui(F-i, Yip).
So the ith individual
can get a sequence of allocations
tZl(E-i, rj(Ei))zai(E-jy
Since Ui is continuous,
rj(Fj)).
this sequence of allocations
satisfies.
E(Ui(gi(b,d(e)),~11Fi)< E i k% (Ui(al(f(e)),E))1Zi
since r is acceptable, interchanging r and i does not change the value of
the expectation.
By the integrability assumption we made on the Ui’s, we can take the
limit out of the expectation operator and obtain
E(Ui(gi(b,
4~11, ~1 I ai) < !:zE(U~(~‘(~(E)),
~11EJ*
So the ith individual can, by getting close enough to the ai allocation
(by a suitable change of his strategy), improve his position contradicting
the fact that we were at a BNE.
Thus r(E) satisfies the third property of acceptability as well, and is
acceptable for-f. However, the allocation rule yielded by the BNE is up
to a strong null set
and since r is acceptable forf, F satisfies monotonicity, andfis in F, we
have thatf(i(s)) coincides with some allocation rule a in F, except perhaps
on a strong null set. This implies the BNE allocation rule also coincides
with this a in F up to a strong null set.
Hence we conclude the mechanism constructed does implement F.
Q.E.D.
Remark.
If we assume that J and F are closed and convex sets then
all the projections we make are continuous and we have a truly continuous
mechanism.
4.
CONCLUSION
We have succeeded in continuously implementing a given SSC, in a
framework which encompasses the previous setting of a discrete informational structure. One undesirable feature of our construction is the resort
480
DAVID
WETTSTEIN
to noncompact stategy spaces. This led to strategy choices by (n - 1)
individuals to which the remaining individual could not find a best response. The problem whether continuous implementation could be carried
out using strategy spaces for which there always exists a best response
strategy deserves further investigation.
In Wettstein (1987) we considered another version of the mechanism
which allowed for an explicit stage of signaling. The main motivation was
the study of market related correspondences. You do not get more SCCs
being implementable, but the signaling will change the definition of a strong
null set and might eliminate some of the previously strong null sets.
A natural question is what SCC’s of economic interest are implementable. This was pursued in Palfrey and Srivastava (1987). Furthermore
when dealing with a particular SCC, one should look for simpler or more
appealing mechanisms than the general one suggested here. Wettstein
(1990) addresses this issue for the SCC induced by constrained rational
expectations equilibria.
Our entire framework dealt with pure exchange environments. One
could use similar tools to examine the problem of resource allocation
within large enterprises.
The designer would be the manager of the whole enterprise, whereas
the heads of the various divisions would be the individuals taking part in
the designed mechanism. Work in this vein would try to analyze the
question of incentives in organizations and teams as did Groves (1973) and
Groves and Loeb (1979). The general equilibrium framework in which
both exchange and production take place, under conditions of incomplete
information, remains to be thoroughly analyzed from the point of view of
design.
APPENDIX
Proof of Theorem I.
The method of proof is identical to the one used in Postlewaite and
Schmeidler (1986) and we provide it for the sake of completeness.
Suppose that F is implementable by a mechanism G satisfying the above
properties and take an f in F and a mapping u which is acceptable for f.
In the mechanism G we must have a BNE satisfying (Cl) and (C2) with
respect tofi Denote that BNE by (6, d); thus we have
(9 g@, 4~)) = fW;
(ii) A change in the Bi strategies on the part of a single individual will
have no effect whatsoever on the outcome.
CONTINUOUS
We shall show that the following
well.
481
IMPLEMENTATION
n-tuple of strategies forms a BNE as
-(b, d(E) = (b, 4dE)h
where 4d~))
= (~((T~(E)),. . . , 4(~,(~,,))).
By the measurability of o and the di’s these strategies are indeed measurable.
By (ii) above it is enough to show that no individual can gain by a
deviation from his di strategy, at the second stage of the mechanism.
Suppose by way of contradiction that there exists an individual i and an
Ei in Ji for which there exists a 8 in Di that satisfies
(A)
--
--
E(Ui(gi(6, d(c)), E) 1Ei) < E(Ui(gi(b, d-i(E), 8), E) I Ei)*
Now define an allocation
rule a in the following
way:
4s) = t-(E)
for .5i # ai
u(E) = db, d-i(E)7 8,
if Ei = ci(Ei).
Once more note that the fact that the individuals choose measurable
strategies implies this allocation rule is indeed measurable.
Since CTis acceptable, the following two equalities hold, except perhaps
on a strong null set:
-g@, &)) = f(e))
-db, d-i(E), 8>= a(+((E))
ifEi = Ei.
This implies we can substitute the terms on the right hand side into the
(A) inequality without changing the value of the expectation taken. Thus
we get
Hence for this i the allocation rule a (since c was acceptable) must satisfy
By the definition
of a and since the dj’s together with b yieldf,
we have
482
DAVID
WETTSTEIN
But that contradicts the fact that the dls are a part of a BNE. Individual
i, at the second stage of the game, after observing (+i(Ei), is better off doing
8 and not following his equilibrium strategy when the others follow their
equilibrium -- strategies.
--Hence (6, d(a)) does constitute a BNE and the allocation rule ?((E) =
g(b, d(c)) is induced by a BNE. Since G implemented F,~(‘(E)must coincide,
except perhaps on a strong null set with an allocation rule in F. Since
f(a) = ~(G(E)) up to a strong null set, we obtain thatf(&(e)) also coincides
with an allocation rule in F, except perhaps on a strong null set. This in
turn shows that F indeed satisfies monotonicity.
Theorem 2: The Construction
of Stage 2
There are n closed sets (in R”-‘).l,,z ,,,,,i-,,i+ ,,,,,,n for i = 1, . . . , n on
which we need to project tuples of numbers announced by the individuals.
Prior to starting the operation of the mechanism, n measurable functions
carrying out this projection are constructed. When the sets in question are
convex the projection operation yields a continuous and thus certainly
measurable function. If the sets are not convex the projection operation
turns out to be a correspondence. The problem is resolved by Lemma 1 in
Hildenbrand (1974, p. 55), which shows it is possible to select a measurable
function out of such a correspondence. The n-functions are assumed to be
common knowledge among the individuals.
Those functions are used by the mechanism to construct n-profiles in J.
The ith profile denoted by (7;) . . . , F’,) is constructed in the following
manner. The projection (by the above mentioned function) of (r, , . . . ,
ri- I , ri+ l , . . . , r,) on Ji ,,,,,i- I,;+ I,...,n is denoted by
(F’,, . . . ,7j_,&,
. . . (7’) and rj is defined by 7: = - c Fj .
j#i
This way we indeed get a profile in J given by
r’ = (Ff) . . . (Yk).
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