On Combining Implementable Social Choice Rules
Jean Pierre Benoît
Efe A. Ok
Department of Economics
Department of Economics
New York University
New York University
Remzi Sanver
Department of Economics
Bilgi University
June, 2004
Abstract
We study if (and when) the intersections and unions of social choice rules that are
implementable with respect to a certain equilibrium concept are themselves implementable with respect to that equilibrium concept. While the results for dominant
strategy equilibrium are mostly of negative nature, the situation is di¤erent in the case
of Nash implementation. We …nd that the union of any set of Nash implementable
social choice rules is Nash implementable (for societies of at least three constituents),
while the intersection of …nitely many such correspondences is “almost” Nash implementable (even in the case of two-person societies). The former observation allows
us to formulate the notion of the largest Nash implementable subcorrespondence of a
social choice rule.
Keywords: Nash implementation, dominant strategy implementation, Maskin monotonicity
We thank Matt Jackson and Tar¬k Kara for helpful discussions on the intersections of Nash implementable
social choice rules, and William Thomson for providing us with his lecture notes on implementation theory
in which a question related to the ones that we study here is discussed.
1
1. Introduction
Despite the fact that implementation theory is a relatively mature …eld, the literature does
not provide a systematic analysis of the basic structure of the class of social choice correspondences (SCCs) that are implementable with respect to a given equilibrium concept. In
particular, it appears that little is known about the conditions under which this class is closed
under intersections or unions. This is unfortunate because an analysis of this issue may have
implications for applications of the theory. Indeed, many interesting SCCs can be expressed
as the intersection or union of two or more SCCs.1 Since, depending on the environment,
the latter SCCs may be either known to be implementable or relatively easy to implement,
it may be bene…cial to …rst study the implementability of the components of these SCCs,
and then apply a suitable closure result in order to settle the issue of implementability of
the target SCC.2
In this paper we examine this problem with respect to the dominant strategy and Nash
equilibrium concepts. Our main results regarding dominant strategy implementation are
negative. We present in Section 3 two examples to show that neither the union nor the
intersection of two dominant strategy implementable SCCs is necessarily dominant strategy
implementable.
The situation is somewhat more satisfactory in the case of Nash implementation. In
particular, when there are at least three constituents of the society, the union of an arbitrary
class of Nash implementable SCCs is Nash implementable (Section 4). We apply this result to
study a new implementation notion which is of independent interest. The idea stems from the
fact that an SCC F which is not (fully) implementable, may still be weakly implementable;
that is, there may exist an implementable subcorrespondence of F: Among its implementable,
if any, subcorrespondences, the one of most interest would naturally be the largest, as this
one would be the closest to the original correspondence F . But does there exist such a
largest subcorrespondence? The answer is yes in the case of Nash implementation, as the
largest Nash implementable subcorrespondence (lis) is nothing more than the union of all the
Nash implementable subcorrespondences – and our aforementioned result establishes that
this union is itself Nash implementable. Motivated by this observation, in Section 6 the lis
operator is studied in some detail.3
1
For instance, in an economic domain, the “fair” SCC is the intersection of the “envy-free” and “Pareto
optimal” SCCs, and when there are two agents, the “core” SCC equals the intersection of the “individually
rational” and “Pareto optimal” SCCs.
2
A concrete illustration of this point is given in Example 5 below.
3
The concept of the largest Nash implementable subcorrespondence can be thought of as the ‡ip-side of
2
While it is closed under unions, the class of Nash implementable SCCs is not closed under
intersections. This fact is already known from an example due to Kara and Sönmez (1997).
However, that example –which shows that the intersection of the Pareto and individually rational rules need not be Nash implementable in the context of the college admissions problem
– is somewhat involved. In the hope of clarifying the potential di¢ culty of implementing
the intersection of two Nash implementable SCCs, we provide in Section 5 a very simple
illustration involving only three agents and two states of nature.
Given that the intersection of Nash implementable SCCs may not itself be Nash implementable, next we inquire into how “close” one may come to (fully) Nash implementing it.
Recall that there are two aspects to implementation. On the one hand, any element of the
target SCC must arise as some equilibrium of the mechanism. On the other hand, only
the elements of the SCC must be equilibrium outcomes. Informally speaking, (i) all “good”
alternatives must be equilibrium outcomes and (ii) no “bad”alternatives can be equilibrium
outcomes. We de…ne a concept of almost implementation that satis…es condition (i), but fails
with respect to condition (ii); however, this failure is slight in two respects: the probability
of a “bad”outcome being chosen is arbitrarily small and there is some control over how bad
a chosen “bad” outcome can be. More precisely, provided that the SCCs at hand satisfy a
weak Pareto consistency condition, we prove that, for any 0 <
< 1; there is a stochastic
mechanism such that (i) any alternative chosen by the intersection of …nitely many Nash
implementable SCCs at a given state obtains as a Nash equilibrium of the game induced by
the mechanism at that state, (ii) the probability that a Nash equilibrium at a given state
results in an alternative that would not be chosen by the intersection at that state is at most
1
, and (iii) all Nash equilibrium outcomes at any state are chosen by at least one of the
SCCs.4;5
2. De…nitions
De…ne an environment to be a triple (N; X; R), where N := f1; :::; ng is interpreted as the
set of all individuals in the society, X as the set of all (mutually exclusive) social outcomes,
and R as the set of all admissible preference pro…les. Throughout, we assume that n
2,
the notion of the minimal monotonic extension (mme) of an SCC (cf. Sen, 1995, and Thomson, 1999). The
relation between these two concepts is also discussed in Section 6.
4
Note that almost implementation di¤ers from virtual implementation due to (i ) and (iii ). (More on this
in Section 7.)
5
This result does not extend to dominant strategy implementation: neither the intersection nor the union
of two dominany strategy implementable SCCs need be almost dominant strategy implementable (see Section
8).
3
and that X is an arbitrary set with jXj
2: Generic members of X are denoted as x; y; z
etc... Let PX denote the set of all complete preorders on X; and write LX for the set of all
n
linear orders on X: Then R is, by de…nition, a nonempty subset of PX
:= PX
PX
(n times). We think of R as the set of all possible preference pro…les in the society, or
equivalently, the set of all states of nature relevant to the problem. A generic element of R
is denoted as %; the ith component of which is denoted as %i , i 2 N: The asymmetric part
of any %i is denoted as
i
:
Fix an environment (N; X; R): By a social choice correspondence (SCC) on R; we
mean a correspondence F : R
X with F (%) 6= ; for all % 2 R: There is a natural order
on the class of all SCCs on R which is de…ned pointwise: F w G i¤ F (%)
G(%) for all
% 2 R: When F w G holds, we say that G is a subcorrespondence of F; and F is a
supercorrespondence of G:
We say that an SCC F on R is Pareto-comprehensive if, for all (y; %) 2 X
R; we
have y 2 F (%) if there exists an x 2 F (%) with y %i x for all i 2 N: We say that F is
Maskin monotonic if, for all %; %0 2 R with x 2 F (%), we have x 2 F (%0 ) whenever there
is no i 2 N and y 2 X with x %i y
for all (x; %) 2 X
0
i
x: Finally, we say that F satis…es no-veto power if,
R, jfi 2 N : x %i y for all y 2 Xgj
n
1 implies x 2 F (%):
If F and G are SCCs on R; then we de…ne the correspondences F _ G and F ^ G on R by
(F _G)(%) := F (%)[G(%) and (F ^G)(%) := F (%)\G(%); respectively. (It is obvious that
F _ G is an SCC on R; and F ^ G is an SCC on R, provided that F (%) \ G(%) 6= ; for all
% 2 R:) More generally, for any nonempty set F of SCCs on R, we de…ne the correspondences
W
V
F and F on R by
W
S
( F) (%) := fF (%) : F 2 Fg
and
V
T
( F) (%) := fF (%) : F 2 Fg:
A (normal-form) mechanism is a list (fMi gi2N ; h); where Mi is a nonempty set, i 2 N;
and h is a map from M into X; where M := M1
Mn : As usual, we refer to Mi as
the message space of person i; to a member of M as a message pro…le, and to h as an
outcome function. To simplify the notation, we will denote the mechanism (fMi gi2N ; h)
simply as (M; h) throughout this paper.
Given the environment (N; X; R); a mechanism (M; h) induces a class of normal-form
games fhM; h j %i : % 2 Rg; where hM; h j %i stands for the game in which the set of
players is N; the action space of player i is Mi ; and the preference relation R(%i ) of player i
is de…ned on M as m R(%i ) m0 i¤ h(m) %i h(m0 ): If
4
is an equilibrium concept de…ned for
normal-form games, we denote the set of equilibria of hM; h j %i according to
j %i: The dominant strategy and Nash equilibrium concepts are denoted as
as hM; h
DS
and
N;
respectively.
A mechanism (M; h) is said to weakly -implement the SCC F on R if
;=
6 h( hM; h j %i)
F (%)
for all % 2 R:
If there is such an (M; h); then we say that F is weakly -implementable. On the other
hand, (M; h) is said to -implement F on R if
h( hM; h j %i) = F (%)
for all % 2 R:
If there is such a mechanism, then we say that F is said to be -implementable. Obviously,
F is weakly -implementable if, and only if, there is a -implementable subcorrespondence
of F . We use the terms “
“(weakly)
DS -implementable”and
N -implementable”and
“dominant strategy implementable,”and
“(weakly) Nash implementable”interchangeably through-
out the exposition.
We recall that a classic result of implementation theory is that Nash implementability
implies Maskin monotonicity, but not conversely. However, every Maskin monotonic SCC
on R that satis…es no-veto power is Nash implementable; this is Maskin’s Theorem. (See
Maskin, 1999).6
We let F (R) stand for the class of all
-implementable SCCs on R; and denote by
F P (R) the class of all Pareto-comprehensive members of F (R): For any nonempty subset
K of F (R); we say that K is closed under (…nite) unions if, for any nonempty (…nite)
W
subset F of K; we have F 2 K: The closure of K under arbitrary unions is, of course,
W
equivalent to the partially ordered set (K; w) being a complete -semilattice. In turn, K
is closed under (…nite) intersections if, for any nonempty (…nite) subset F of K with
V
V
( F) (%) 6= ; for all % 2 R; we have F 2 K:
3. Combining Dominant Strategy Implementable SCCs
Our …rst set of results about dominant strategy implementation is of a negative nature; we
…nd that the class of dominant strategy implementable SCCs is, in general, not closed under
…nite unions or …nite intersections.
6
For detailed overviews of Nash implementation theory, see the surveys of Jackson (2001) and Maskin
and Sjöström (2002). Benoît and Ok (2004b) have recently obtained a generalization of Maskin’s Theorem
by replacing no-veto power with a property called limited veto power. A corollary of this generalization which will be used in Section 6 - is that the (weak) core is Nash implementable in any environment in which
it is well-de…ned as an SCC.
5
(Counter-)Example 1. Consider the environment (N; X; R); where X := fx; yg, N :=
f1; 2g and R := f(%1 ; %2 )g, with x
1
y and y
2
x: De…ne the SCCs F and G on R as
F (%) := fxg and G(%) := fyg: It is obvious that both F and G are
However, F _ G is not
DS hM; h
h(
DS -implementable.
DS -implementable.
Indeed, if (M; h) is a mechanism such that
j %i) = fx; yg; then there must exist m; m0 2
DS hM; h
j %i such that h(m) = x
and h(m0 ) = y: Since m1 and m01 are dominant strategies for player 1 in the game hM; h
j %i; we must have h(m1 ; ) = h(m01 ; ): So
h(m1 ; m02 ) = h(m0 ) = y
2
x = h(m1 ; m2 );
contradicting that m2 is a dominant strategy for player 2.7
(Counter-)Example 2. Let X := fx; yg, and let us write
xRy and not yRx;
y
x
x
y
for the preorder R 2 PX with
is understood similarly. Let xy stand for the preorder R 2 PX with
xRyRx: Finally, let c(R) := fa 2 X : aRb for all b 2 Xg for any R 2 PX :
Now consider the environment (N; X; R); where N := f1; 2g and R :=
2
PX
n
n
o
x y
y x
( y ; x ); ( x ; y ) :
De…ne the SCCs F and G on R as F (%) := c(%1 ) and G(%) := c(%2 ): It can easily be
shown that both F and G are
DS -implementable.
implementable.
We now show that F ^ G is not
DS -
To see this, assume that (M; h) is a mechanism that
DS -implements F ^ G: Then player
1 should have a dominant strategy in the game hM; h j( xy ; xy )i; say m1 ( xy ); and also in the
game hM; h j( xy ; xy )i; say m1 ( xy ): De…ne m2 ( xy ) and m2 ( xy ) similarly. Now consider the game
hM; h j(xy; xy )i: Since in this game player 1 is indi¤erent between all of his actions in M1 ;
and m2 ( xy ) is a dominant strategy for player 2, we have
h m1 ( xy ); m2 ( xy )
2 h(
y
DS hM; hj(xy; x )i)
= (F ^ G)(xy; xy )
= fyg
so that h m1 ( xy ); m2 ( xy ) = y: Similarly, in the game hM; h j( xy ; xy)i; player 2 is indi¤erent
between all of his actions in M2 ; and m1 ( xy ) is a dominant strategy for player 1. Consequently,
h m1 ( xy ); m2 ( xy )
2 h(
x
DS hM; hj( y ; xy)i)
= (F ^ G)( xy ; xy)
= fxg;
7
This situation is not a consequence of the triviality of the preference domain we consider in this example.
The same conclusion holds if, for instance, R = LX
LX .
6
so that h m1 ( xy ); m2 ( xy ) = x: It follows that x = y; a contradiction.
Remark. Examples 1 and 2 demonstrate that neither a subcorrespondence nor a supercorrespondence of a dominant strategy implementable SCC need be dominant strategy implementable.
The major culprit behind Example 2 is the potential indi¤erence of the players between
some alternatives. As the next proposition establishes, the class of all SCCs (on a given
domain R) that are implementable in dominant strategy equilibrium is closed under inter-
sections if no one in the society is indi¤erent between any two distinct alternatives. (Example
1 shows that the same is not true for unions.)
Proposition 1. If (N; X; R) is an environment with R
intersections.8
LnX ; then F
DS
(R) is closed under
V
Proof. Take any nonempty subset F of SCCs on R such that ( F) (%) 6= ; for all % 2 R;
and assume that, for each F 2 F; there exists a normal-form mechanism (M F ; hF ) such that
hF (
DS hM
F
; hF j %i) = F (%)
for all % 2 R:
(1)
Fix an arbitrary G 2 F; and de…ne Mi := MiG ; i = 1; :::; n; M := M1
Mn ; and
T
h := hG : Now …x an arbitrary % 2 R; and note that fF (%) : F 2 Fg 6= ;; whereas the
antisymmetry of % entails that hF (
DS hM
F
; hF j %i) = 1 for all F 2 F: It follows from (1)
that there exists an x% 2 X with F (%) = fx% g for all F 2 F: Then, by (1),
h(
DS hM; h
j %i) = hG (
DS hM
G
; hG j %i) = G(%) = fx% g = (
as we sought.
V
F) (%);
4. Unions of Nash Implementable SCCs
We now turn to combining Nash implementable social choice rules by taking unions. The
following result shows that, in contrast to dominant strategy implementation, Nash implementation is well-behaved in this regard (at least when there are three or more individuals
in the society).9
Proposition 2. If (N; X; R) is an environment with n
W
…nite unions, that is, (F N (R); w) is a -semilattice.
8
9
3; then F N (R) is closed under
LnX := LX
LX (n times).
As Example 1 can easily be adopted to a three person society, this dissimilarity is not due to the number
agents.
7
Proof. Take any F; G 2 F N (R); and note that it is enough to establish that F _G 2 F N (R)
to prove the claim. Let (M H ; hH ) be a mechanism that
each i 2 N; de…ne Mi := MiF
MiG
the outcome function h : M ! X as:
G
F
h((mF1 ; mG
1 ; H1 ); :::; (mn ; mn ; Hn )) :=
We claim that
h(
N hM; h
N -implements
fF; Gg: Moreover, let M := M1
H 2 fF; Gg: For
Mn ; and de…ne
8
H
H
H
>
>
< h (m1 ; :::; mn ); if jfi 2 N : Hi = H
>
>
: hF (mF ; :::; mF );
1
n
j %i) = (F _ G) (%)
for some H 2 fF; Ggj
n
1
otherwise.
for all % 2 R:
F
G
To prove this, take any % 2 R; and let ((mF1 ; mG
1 ; H1 ); :::; (mn ; mn ; Hn )) 2
By de…nition of h; there must exist an H 2 fF; Gg such that
(2)
N hM; h
G
H
H
H
F
h((mF1 ; mG
1 ; H1 ); :::; (mn ; mn ; Hn )) = h (m1 ; :::; mn ):
j %i:
(3)
H
H
j %i to mHi , for each i: Then
It follows that, mH
i is a best response in the game hM ; h
H
(mH
1 ; :::; mn ) 2
N hM
H
; hH j %i, so (3) yields
F
G
H
h((mF1 ; mG
1 ; H1 ); :::; (mn ; mn ; Hn )) 2 h (
N hM
H
; hH j %i)
= H(%)
(F _ G) (%):
Conversely, let x 2 (F _ G) (%): Then x 2 H(%) for some H 2 fF; Gg: Take any (mFi ; mG
i ) 2
MiF
H
MiG ; i 2 N; with (mH
1 ; :::; mn ) 2
N hM
H
; hH j %i; and de…ne
F
G
m
^ := ((mF1 ; mG
1 ; H); :::; (mn ; mn ; H)) 2 M:
It is obvious that m
^ 2
proof of (2).
N hM; hj
H
%i and h(m)
^ = hH (mH
1 ; :::; mn ) = x: This completes the
It is worth noting that F N (R) is in fact closed under arbitrary unions (when n
3).
Indeed, the proof of Proposition 2 modi…es in a straightforward way in the case of an arbitrary
collection of SCCs on R; provided that we invoke the Axiom of Choice to well-de…ne the
cartesian product of the message spaces of all the members of this collection. Thus, we have
the following generalization of Proposition 2.
Proposition 3. If (N; X; R) is an environment with n
W
unions, that is, (F N (R); w) is a complete -semilattice.
8
3; then F N (R) is closed under
In Section 6, we shall present a potentially important application of this result.
5. Intersections of Nash Implementable SCCs
The following example shows that F N (R) need not be closed under …nite intersections.10
(Counter-)Example 3. Let N := f1; 2; 3g, X := fx; y; zg, and
0
1
0
x
y
B
C
B
1
2
B
C
B
% = @xyz; xyz; z A and % = @ x ;
y
z
R := f%1 ; %2 g; where
1
y x
C
x ; y C
A:
z z
(Here, again, we adopt the notation used in Example 2.) De…ne the SCCs F and G on R as
F (%1 ) := fx; zg;
F (%2 ) := fx; yg
and
G(%1 ) := fx; yg =: G(%2 ):
Both of these SCCs are implementable by a mechanism in which the message space of each
agent is X. To see this, de…ne the functions hF and hG on X 3 as follows:
8
>
(
>
< z; if m3 = z
y; if jfi 2 N : mi = ygj
hF (m) :=
y; if m1 = m2 = y; m3 6= z and hG (m) :=
>
x; otherwise
>
: x; otherwise
2
It is an easy exercise to show that (X 3 ; hF ) Nash implements F and (X 3 ; hG ) Nash implements G:
Here we have (F ^ G)(%1 ) := fxg and (F ^ G)(%2 ) := fx; yg: (Notice that F ^ G is none
other than the strong Pareto rule on R:) We claim that F ^ G is not Nash implementable.
Suppose (M; h) is a mechanism that Nash implements F ^ G: Then there must be an m 2
NhM; hj %2 i with h(m) = y: Clearly, this is possible only if h(m1 ; m2 ; M3 )
fy; zg: But if
z = h(m1 ; m2 ; m03 ) for some m03 2 M3 ; then (m1 ; m2 ; m03 ) is an equilibrium of hM; hj %1 i;
which contradicts (M; h) implementing F ^ G since this SCC does not choose z at state %1 .
On the other hand, if y = h(m1 ; m2 ; m03 ) for every m03 2 M3 ; then m is an equilibrium of
hM; hj %1 i; which again contradicts (M; h) implementing F ^ G since this SCC does not
choose y at state %1 . Thus: F ^ G is not Nash implementable.
We do not know at present if F N (R) is closed under …nite intersections when no individ-
ual in the society is indi¤erent between any two distinct alternatives (i.e. when R
10
LnX ):
As noted in Section 1, this fact is already known from an example due to Kara and Sönmez (1997).
Unfortunately, however, their example is quite involved, and speci…c to the college admissions problem,
which was their topic of interest. By contrast, the example produced here is very simple, and hence better
helps to see the source of the di¢ culty.
9
Put di¤erently, we do not know whether or not the analogue of Proposition 1 is valid for
Nash implementation.
Remark. As Example 3 shows, a subcorrespondence of a Nash implementable SCC need
not be Nash implementable. The same holds true for supercorrespondences of a Nash
implementable SCC as well.
For instance, consider the environment (N; X; R); where
N := f1; 2; 3g; X := fx; yg; and R := f%; %0 g with % := ( xy ; xy ; xy ) and %0 := (xy; xy ; xy ): De…ne the SCCs F and G on R as: F (%) := F (%0 ) := fxg; G(%) := fx; yg and G(%0 ) := fxg:
Then, obviously, F 2 F N (R) and F v G; but G is not even Maskin monotonic.
While F N (R) is, in general, not closed under …nite intersections, it turns out that there
is a sense in which this class is almost closed under …nite intersections. We take up this issue
in Section 7, after …rst presenting an application of our …ndings so far.
6. An Application: Largest Nash Implementable Subcorrespondences
Even if a social choice rule is not Nash implementable, it may still be weakly Nash implementable; that is, it may be possible to Nash implement a subcorrespondence of that rule.
In that case we would naturally be interested in Nash implementing the subcorrespondence
that di¤ers from the principal SCC in the minimum possible way. This leads to the notion
of the largest Nash implementable subcorrespondence (lis) of a social choice rule, which we
introduce next.
Let (N; X; R) be an environment, and denote by Fw- N (R) the class of all weakly Nash
implementable SCCs on R. For any F 2 Fw- N (R); we de…ne the largest Nash imple-
mentable subcorrespondence of F , denoted by lis(F ); as the largest SCC on R (with
respect to the partial order v) which is a Nash implementable subcorrespondence of F: For
any SCC F on R that lies outside Fw- N (R); we let lis(F ) := ? by convention. The following
result establishes that the lis of any weakly Nash implementable SCC is itself a well-de…ned
SCC.
Proposition 4. If (N; X; R) is an environment with n
have
lis(F ) =
Consequently,
W
3; then, for any SCC F on R; we
fG 2 F N (R) : G v F g :
lis(F ) 2 F N (R) for any F 2 Fw- N (R):
W
Proof. It is enough to show that fG 2 F N (R) : G v F g is Nash implementable for any
F 2 Fw- N (R). But this is an immediate consequence of Proposition 3.
10
The literature on Nash implementation provides a concept that has a similar spirit to
the notion of the largest Nash implementable subcorrespondence: the minimal monotonic
extension (mme). The minimal monotonic extension of an SCC F on R; denoted
henceforth as mme(F ); is de…ned to be the smallest Maskin monotonic supercorrespondence
of F; and equals the intersection of all monotonic supercorrespondences of F (cf. Sen (1995)
and Thomson (1999)). Their formal di¤erences aside, there are two important distinctions
between the lis and the mme of an SCC worth noting. First, while the mme of a given
SCC is always nonempty, it need not be Nash implementable, attenuating its usefulness in
solving Nash implementation problems. Second, the mme of a non-implementable SCC F
chooses outcomes in some states even though those outcomes are deemed as “undesirable”
by the target social choice rule F . Consequently, mme(F ) may well not be an SCC that
the planner (whose objectives are re‡ected in F ) would care to implement. The following
simple examples illustrate these two points.
Example 4. (a) Consider the environment given in Example 3, and let H be the strong
Pareto SCC on R: (That is, H := F ^ G; where F and G are as de…ned in Example 3.)
We have shown in Example 3 that H is not Nash implementable. Nonetheless, H is Maskin
monotonic, so that mme(H) = H, and the notion of an mme does not advance us. On
the other hand, lis(H)(%i ) = fxg for each i = 1; 2: This suggests that a “second best” to
implementing H is choosing x always. More precisely, if the planner wishes to implement
an SCC that stays within H, while coming as close as possible to H, the best (s)he can do
is to implement the SCC that chooses x in each state.
(b) Let N := f1; 2; 3g, X := fx; yg, and R := f%1 ; %2 g; where
!
!
x
y
y
y
y
%1 =
;
;
and %2 = xy;
;
:
y x x
x x
Consider the SCC F on R de…ned as F (%1 ) := fx; yg and F (%2 ) := fyg: (Again, F is the
strong Pareto rule on R.) Here we have mme(F )(%i ) = X for each i = 1; 2; and hence the
mme does not produce an interesting SCC to implement. On the other hand, lis is again
informative: lis(F )(%i ) = fyg for each i = 1; 2:11
Of particular interest are the largest Nash implementable subcorrespondences of various
concrete SCCs on speci…c domains. For instance, consider a coalitional game environment
11
These examples notwithstanding, we do not intend to argue that the notion of lis dominates that of
mme: In particular, lis(F ) = ; for any SCC F that is not weakly implementable, while mme(F ) may well
be a Nash implementable SCC for such an F:
11
in which the (weak) core, denoted as Fcore , is a well-de…ned SCC. It is well-known that the
Mas-Collel bargaining set, denoted as Fbarg , need not be Nash implementable in such an
environment (Serrano and Vohra, 2002). However, Fcore is a subcorrespondence of Fbarg , and
we know that the former is always Nash implementable (Benoît and Ok, 2004b). Therefore,
the Mas-Collel bargaining set (in any environment with nonempty core) is weakly Nash implementable. Thus, lis(Fbarg ) is a Nash implementable SCC, and we have Fcore
lis(Fbarg ):
In fact, lis(Fbarg ) is the largest monotonic subcorrespondence of the Mas-Collel bargaining
set.12 For another example, while the strong Pareto rule is not monotonic in an economic environment, and hence is not Nash implementable, its lis is a well-de…ned SCC, as the strong
Pareto rule is weakly Nash implementable in such an environment (because any dictatorial
rule is a Nash implementable).
Pursuing speci…c lis computations, however, would be too much of a digression for the
present paper. Consequently, we con…ne ourselves in the rest of this section to presenting
some basic properties of the largest Nash implementable subcorrespondences, and leave the
deeper analysis of the lis operator to future research. Our next result identi…es the relation
between an SCC and its largest Nash implementable subcorrespondence. The result follows
directly from the de…nition of lis.
Proposition 5. Let (N; X; R) be an environment with n
(a) lis(F ) v F;
3; and F an SCC on R: Then,
(b) lis(F ) = ; i¤ F is not weakly Nash implementable,
(c) lis(F ) = F i¤ F is Nash implementable.
The …nal result of this section identi…es the relation between the lis of the intersection
(and the union) of a family of SCCs and the lis of the members of the family. (The analogous
problem was studied by Thomson (1999) in the case of minimal monotonic extensions.)
Proposition 6. Let (N; X; R) be an environment with n
V
SCCs on R such that ( F) (%) 6= ; for all % 2 R: Then
and
12
V
V
lis ( F) v flis(F ) : F 2 Fg
W
W
lis ( F) w flis(F ) : F 2 Fg:
3; and F a nonempty class of
(4)
(5)
The largest monotonic subcorrespondence of the Mas-Collel bargaining set satis…es limited veto power,
and hence, it is Nash implementable. (See Benoit and Ok, 2004b).
12
Both of these relations may hold strictly.
Proof.
V
Take any % 2 R: By Proposition 4, if x 2 lis ( F) (%); then there exists a
G 2 F N (R) such that x 2 G(%) and G v F for all F 2 F: By de…nition of lis(F ); we have
G v lis(F ) for all F 2 F; so it follows that x 2 lis(F )(%) for all F 2 F: This establishes (4).
On the other hand, (5) is trivially true if lis(F ) = ; for all F 2 F: Moreover, if lis(F ) 6= ;
W
for some F 2 F; then, by Proposition 3, flis(F ) : F 2 Fg is a Nash implementable SCC
W
W
on R. Since lis(F ) v F for all F 2 F; we also have flis(F ) : F 2 Fg v F: The claim
(5) thus follows from the de…nition of the lis operator.
To prove that the relations (4) and (5) may hold strictly, consider the framework of
Example 3. Since both F and G are Nash implementable in this example, we have lis(F ) ^
lis(G) = F ^ G; whereas lis(F ^ G)(%i ) = fxg; i = 1; 2: Since (F ^ G)(%2 ) = fx; yg; we
thus have lis(F ) ^ lis(G) 6= lis(F ^ G): Now let U := F ^ G and let V be the SCC on R
that equals fzg everywhere. It is not di¢ cult to check that U _ V is Nash implementable,13
and hence lis(U _ V ) = U _ V; whereas lis(U ) equals fxg everywhere and lis(V ) = V: Since
(U _ V )(%2 ) = fx; y; zg; we thus have lis(U ) _ lis(V ) 6= lis(U _ V ):
Remark. The notion of mme suggests an alternative approach to the one pursued here,
namely, to consider the smallest Nash implementable supercorrespondence of a given SCC F .
Unfortunately, there may not be a smallest such supercorrespondence. This is a consequence
V
of the fact that (F N (R); w) is in general not a -semilattice. For example, consider the
environment (N; X; R) of Example 3, and let H be the strong Pareto rule on R, that is,
H(%1 ) := fxg and H(%2 ) := fx; yg: It is readily checked that there does not exist a smallest
implementable supercorrespondence of H:14
7. Almost Implementation
Before introducing the notion of almost implementation, we need to go through some basic
nomenclature of decision theory. For any nonempty set X; we denote the set of all simple
lotteries by P(X):15 As is usual, the degenerate lottery that puts unit mass on x 2 X is
13
An explicit implementing mechanism can easily be constructed. Alternatively, this SCC is Maskin
monotonic and satis…es the limited veto power condition of Benoît and Ok (2004b), and hence is implementable by Theorem 1 of that paper.
14
Proof. Let F and G be as de…ned in Example 3. F is a Nash implementable supercorrespondence of H:
Since the only nontrivial subcorrespondence of F which is larger than H is H; and H is not implementable,
the only candidate for the smallest implementable supercorrespondence of H is F: But while F v G is false,
G is a Nash implementable supercorrespondence of H.
15
Abusing the standard terminology of probability theory a bit, we de…ne the support of a probability
13
denoted as
x:
For convenience, we will sometimes identify x with
and x 2 X; write p(x) for p(fxg):
For any p; q 2 P(X) and
p with probability
2 [0; 1]; we denote by p
and q with probability 1
p
x;
and for any p 2 P(X)
q the compound lottery that gives
; that is,
q := p + (1
)q:
For any R 2 PX ; we say that R is an a¢ ne extension of R to P(X) if R 2 PP(X) extends
R (that is, xRy i¤
(p
r) R (q
xR
y)
and is a¢ ne (that is, for any p; q; r 2 P(X) and
2 (0; 1]; pR q i¤
r)). We denote the class of all a¢ ne extensions of R to P(X) by a¤(R): For
instance, if R is represented by a utility function u : X ! R and R 2 PP(X) is represented
P
by the map p 7! x2supp(p) p(x)u(x); then R 2 a¤(R):16
The following elementary lemma will be useful in what follows.
Lemma 1. Let R 2 PX and R 2 a¤(R), and denote the asymmetric parts of R and R
by P and P ; respectively. If xP y; then
(p
x)
P
(p
for all
y)
2 [0; 1) and p 2 P(X);
and
(
y)
x
Proof. Since xP y; we have
x
y
=p
x
and
x
1
2 (1=2; 1):
(
x
xP
y;
so the …rst claim follows readily from the a¢ neness
of R : On the other hand, where
y
=p
y)
for all
P
1
:= 2(1
y;
) 2 (0; 1); and p :=
x
1=2
y;
we have
so the second claim follows from the …rst one.
We turn now to implementation theory. By a stochastic (normal-form) mechanism,
we mean a list (fMi gi2N ; ); where Mi is a nonempty set, i 2 N; and
M := M1
: M ! P(X) where
Mn : Thus such a mechanism assigns to each message pro…le a simple lottery
in P(X): We denote the mechanism (fMi gi2N ; ) as (M; ).
measure p on X; denoted supp(p); as the set of all x 2 X with p(fxg) > 0: A simple lottery is a probability
measure on X whose support is …nite.
16
An a¢ ne extension of a preorder on X need not satisfy the Archimedean axiom of von NeumannMorgenstern utility theory, and hence it need not have an expected utility representation. More importantly,
when one is interested in the intersection of a small number of SCCs, the way we extend individual preferences
to the space of simple lotteries is not very restrictive. For instance, if jFj = 2 in Proposition 3 below, then
it is enough to work with the space of lotteries the supports of which contain at most two outcomes. Of
course, the independence axiom (which we refer to as a¢ neness here) is unexceptionable on such a space of
lotteries.
14
Now …x an environment (N; X; R); and an equilibrium concept
For any preference pro…le % 2
n
PX
;
for normal-form games.
we let
a¤(%) := f(%1 ; :::; %n ) : %i 2 a¤(%i ); i = 1; :::; ng;
and de…ne
a¤(R) :=
S
fa¤(%) : % 2 Rg:
Clearly, a stochastic mechanism (M; ) induces a class of normal-form games fhM;
j% i:
% 2 R g where hM; j % i stands for the game in which the set of players is N; the action
space of player i is Mi ; and the preference relation R(%i ) of player i is de…ned on M as z
R(%i ) z 0 i¤ (z) %i (z 0 ): The mechanism (M; ) is said to -implement the SCC F on R
with probability
2 (0; 1); if, for any % 2 R and any % 2 a¤(%);
(i) ( hM; j % i)
there exists an x 2 F (%) such that p(x)
(ii) ( hM;
j % i)
(that is, if p 2 ( hM; j % i); then
F (%) holds with probability
),
F (%) (that is, if x 2 F (%); then
F is said to be almost -implementable, if, for any
x
2 ( hM;
j % i)).
2 (0; 1); there exists a mechanism
that -implements F with probability : Finally, given any SCC V on R with F v V; we
say that F is almost -implementable within V; if it is almost -implementable and
supp( (m))
V (%)
for all % 2 a¤(%) and m 2 hM; j % i:
Because a stochastic mechanism may assign to a message pro…le a nondegenerate lottery,
we need to specify how the individuals compare such lotteries. The notion of almost implementation presupposes that the agents will do this by using an a¢ ne extension of their
original preferences (over certain outcomes) but does not specify which particular a¢ ne extension will be used for this purpose. Instead, it ensures that any equilibrium of any such
game at a given state % yields an outcome which is in F (%) with arbitrarily high probability,
and conversely, that any outcome in F (%) corresponds to an equilibrium in any such game
with probability one.
Almost implementation within an SCC V provides some control over the outcomes that
the mechanism may choose. Recall that if an SCC F is almost implementable and
then there is a mechanism (M; ) such that ( hM;
% 2 a¤(%)), but with probability 1
j % i)
2 (0; 1);
F (%) (for all % 2 R and any
a “bad” outcome that lies outside of F at state
% may be chosen. No matter how small is 1
(almost implementability makes sure that
we can take this positive number as small as possible), this is somewhat troublesome. If
the implementation is within V; however, then we are at least ensured that the choice of the
mechanism will never lead to an outcome that lies outside of V (%).
15
At one extreme, if V = F; then almost
-implementation within V coincides, essen-
tially, with the standard notion of -implementation. It is slightly weaker, in that, although
only outcomes within any V (%) may obtain with positive probability, the mechanism may
involve randomization in equilibrium. At the other extreme, if V = X, then almost
-
implementation within V (which is then equivalent to almost -implementation) is akin to
virtual implementation (cf. Abreu and Sen (1991) and Matsushima (1988)).17 It remains
stronger than virtual implementation, however, in that almost
-implementation requires
that all elements of the SCC arise as deterministic equilibria, whereas virtual implementation does not.18
In Section 9, we show that the intersection of a …nite number of Pareto comprehensive
implementable SCCs is always almost Nash implementable within the union of the intersections. First, however, we establish that neither the union nor the intersection of Pareto
comprehensive implementable SCCs need be almost dominant strategy implementable.
8. Combining Dominant Strategy Implementable SCCs, Again
We show here that the notion of almost implementation is not weak enough to overturn the
negative observations made in Section 3 about dominant strategy implementation. In fact,
we …nd that the union of two dominant strategy implementable (and Pareto comprehensive) SCCs need not be dominant strategy implementable with any probability, while their
intersection need not be dominant strategy implementable with probability greater than 1=2:
As jXj = 2 in both of the examples that follow, every preorder R in PX has a unique
a¢ ne extension R to P(X): (For instance, if xRy; then (
Consequently, it is not ambiguous to write hM;
examples.
x
y)
j %i for hM;
R (
x
y)
i¤
:)
j % i in either of these
(Counter-)Example 1. [Continued] Consider the environment (N; X; R) and the SCCs
F and G of Example 1. We claim that F _ G is not
17
DS -implementable
with any probability
Let X be …nite, and …x a set A in a¤(R) such that there is a bijection ! : R ! A: Given A; we say that
an SCC on R is virtually -implementable if, for any " > 0; there exists a mechanism (M; ) such that,
for every % 2 R; there exists a bijection
jXj
%
: F (%) ! ( hM;
j!(%)i) with d( x ;
% (x))
" (where d is
the Euclidean metric on R ):
18
This di¤erence has important consequences. In particular, Maskin monotonicity is necessary for almost
Nash implementability, but not for virtual Nash implementability. Moreover, depending on the choice of
V; the requirements of the notion of almost implementation within V is substantially more stringent than
virtual implementation. Indeed, virtual implementation does not impose any restriction on the structure of
the equilibrium outcomes that may obtain with positive probability (however small) outside the choice set
of the target SCC.
16
. Indeed, if (M; ) is a stochastic mechanism that
; then h(
(m) =
x
DS hM;
y
2
x
j %i) = fx; yg: Then there must exist m; m 2
0
and (m0 ) =
have (m1 ; )
1
DS -implements
y:
F _ G with probability
DS hM;
j %i such that
Since m1 and m01 are dominant strategies for player 1; we must
(m01 ; ); which holds i¤ (m1 ; ) = (m01 ; ): Then (m1 ; m02 ) = (m0 ) =
= (m1 ; m2 ); contradicting that m2 is a dominant strategy for player 2.
(Counter-)Example 2. [Continued] Consider the environment (N; X; R) and the SCCs F
and G of Example 2. We adopt here the same notation that we used in that example as well.
To derive a contradiction, assume that (M; ) is a mechanism that
DS -implements F ^G with
probability > 1=2: Player 1 has a dominant strategy in the game hM; j( xy ; xy )i; say m1 ( xy );
and also in the game hM; j( xy ; xy )i; say m1 ( xy ): De…ne m2 ( xy ) and m2 ( xy ) similarly. Let p :=
(m1 ( xy ); m2 ( xy )) 2 P(X): Notice that (m1 ( xy ); m2 ( xy )) 2 DS hM; j(xy; xy )i; and consequently,
we obtain p 2 ( DS hM; j(xy; xy )i: Then, since (M; ) DS -implements F ^G with probability
; and (F ^ G)(xy; xy ) = fyg; we must have p(y)
> 1=2 so that p(x) < 1=2: On the other
x
y
x
hand, we also have (m1 ( y ); m2 ( x )) 2 DS hM; j( y ; xy)i; so p 2 ( DS hM; j( xy ; xy)i: Then,
since (M; ) DS -implements F ^ G with probability ; and (F ^ G)( xy ; xy) = fxg; we …nd
p(x)
> 1=2; a contradiction.
9. Intersections of Nash Implementable SCCs Revisited
In Section 5 we have seen that F N (R) need not be closed under …nite intersections. This fact
also extends to the class F PN (R) of all Pareto comprehensive and Nash implementable SCCs
on R:19 However, it turns out that the intersection of …nitely many Pareto-comprehensive
and Nash implementable SCCs is almost Nash implementable within the union of these
SCCs. When the intersection consists of only two social choice rules, there is an obvious
sense in which this is a second best result one can obtain about the Nash implementability
of the intersection of Nash implementable SCCs.20
Proposition 7. If (N; X; R) is an environment and F is a nonempty …nite subset of
V
W
F PN (R); then F is almost Nash implementable within F.
Proof. Fix an arbitrary
2 (0; 1); and a nonempty …nite subset F of F N (R) with
V
( F) (%) 6= ; for all % 2 R: Now take any normal-form mechanism (M F ; hF ) that Nash
implements F; for each F 2 F: For each i 2 N; we de…ne Mi :=
19
20
F
F 2F Mi
F
Z+ for
Example 3 of Section 5, in fact, establishes this stronger result.
As is evident from the proof of Proposition 7, in this case, it would be enough to work with lotteries that
assign positive probability to at most two outcomes. The a¢ ne extension property (which is a prerequisite
behind the notion of almost implementation) is extremely weak on the class of such lotteries.
17
each i; denote a generic member of this set by (mi ; Fi ; ki ); and let M := M1
Mn :
For any m 2 M; let `(m) be the agent with the highest integer announcement. (If there
is more than one agent with the highest integer announcement, we choose ` to be the
one with the smallest subscript.) Let
0
:= maxf1=2; g: We de…ne
follows: For any m := ((m1 ; F1 ; k1 ); :::; (mn ; Fn ; kn )),
F
F
hF`(m) (m1 `(m) ; :::; mn`(m) )
0
with probability
0
: M ! P(X) as
(m) equals the lottery that gives
; and that gives hF (mF1 ; :::; mFn ) with probabil-
1
; F 2 FnfF`(m) g: We claim that the stochastic mechanism (M; ) almost
jFj 1 V
implements F with probability 0 :
ity
To prove this, take any % 2 R and any % 2 a¤(%). Now let m
^ 2
N hM;
N-
j % i and
(m)
^ = p: Applying the …rst claim of Lemma 1 we see that m
^ Fi must be a best response
in the game hM F ; hF j %i to m
^ F i , for each i 2 N and every F 2 F: It follows that
m
^ F := (m
^ F1 ; :::; m
^ Fn ) 2
N hM
F
^ ; hF`(m)
^ ) Nash implements
; hF j %i for all F 2 F: Since (M F`(m)
F`(m)
^ ; therefore, there must exist an x 2 F`(m)
^ (%) such that p(x)
0
: Now take any
F 2 FnfF`(m)
^ i ; Fi ; ki ) to be a best response to m
^ i ; it must be
^ g; and observe that for (m
the case that
^
^
^ F i)
x = hF`(m)
(m
^ F`(m)
) %i hF (mFi ; m
for all mFi 2 MiF ; i = 1; :::; n:
(We again use the fact that % is an a¢ ne extension of % here; recall the second claim of
Lemma 1.) Since F is Nash implemented by (M F ; hF ); it follows that
x %i hF (m
^ F ) 2 F (%)
for all i = 1; :::; n:
Since F is Pareto-comprehensive, therefore, we have x 2 F (%): But F is arbitrarily chosen
V
from FnfF`(m)
^ g; so we may conclude that x 2 ( F) (%); as we sought.
V
Conversely, let x 2 ( F) (%): For any F 2 F; since x 2 F (%) and (M F ; hF ) Nash
implements F; there must exist m
^ Fi 2 MiF (for each i 2 N ) such that hF (m
^ F1 ; :::; m
^ Fn ) = x:
Then we de…ne
m
^ := ((m
^ 1 ; F; 1); :::; (m
^ n ; F; 1)) 2 M;
where F 2 F is arbitrarily …xed. It is clear that m
^ 2
V
proves that ( N hM; j % i) ( F) (%):
Example 5. Let (N; X; R) be an environment where n
N hM;
j % i and (m)
^ =
x;
which
3 and jXj < 1, and let ! i : R !
X, i = 1; :::; n; be maps that satisfy fx 2 X : x %i ! i (%)g =
6 ; for all % 2 R: De…ne the
individually rational SCC F!;IR on R as F!-IR (%) := fx : x %i ! i (%)g:
Claim 1. F!-IR is Nash implementable.
18
Proof of Claim 1. Let Mi := R X N; i = 1; :::; n; and denote the generic member of Mi
as mi = (Di ; xi ; k i ); i = 1; :::; n: De…ne h : M ! X as follows: For any m = (m1 ; :::; mn ) 2 M ,
(i) if (Di ; xi ) = (D; x) for all i 2 N and x 2 F!-IR (D), then h(m) := x;
P
(ii) otherwise, h(m) := ! `(m) (D`(m) ) where `(m) := i2N k i (mod n).
It is not di¢ cult to verify that (M; h) Nash implements F!-IR : k
Let Fwpo be the weakly Pareto optimal SCC on R, that is, Fwpo (%) := fx 2 X : there
is no y 2 X such that y
i
x for all i 2 N g for any % 2 R: Consider the SCC F! on R that
chooses individually rational and Pareto optimal outcomes:
F! (%) := fx 2 Fwpo (%) : x %i ! i (%) for all i 2 N g:
It is not obvious how to approach the question of the Nash implementability of F! at this level
of generality. Note, however, that F! = F!-IR ^ Fwpo , and that F!-IR is Nash implementable
by Claim 1, and Fwpo is Nash implementable by Maskin’s Theorem. Furthermore, both
F! and Fwpo are Pareto-comprehensive. Therefore, by Proposition 7, F! is almost Nash
implementable within F!-IR _ Fwpo on any preference domain R
n 21
:
PX
Remark. Benoît and Ok (2004a) have recently shown that, given an environment (N; X; R)
with n
3; any weakly unanimous SCC on R is Nash implementable by means of a simple
stochastic mechanism (that does not use lotteries on the equilibrium path), provided that
R satis…es a relatively weak restriction (called the top-coincidence condition). In particular,
this result shows that if n
3 and R
LnX ; then the intersection of any number Nash
implementable (and Pareto-comprehensive) SCCs on R is Nash implementable by a simple
stochastic mechanism. While this result is superior to Proposition 7 on the grounds that it
is an exact Nash implementation result (as opposed to almost Nash implementation), the
results are not nested since Proposition 7 applies free of any domain restrictions and covers
the case of two agent environments.
10. Conclusion
In this paper we have studied the implementability of the unions and intersections of SCCs
that are separately implementable in dominant strategy or Nash equilibrium. The results
21
n
In fact, if R = PX
; then F! is Nash implementable. This follows from the following fact, which can be
established by an appropriate modi…cation of the proof of Proposition 7.
n
Proposition. For any n = 2; 3; ::: and any nonempty set X; F PN (PX
) is closed under …nite intersections.
19
in the case of dominant strategy equilibrium are negative: Neither unions nor intersections
of dominant strategy implementable SCCs need be dominant strategy implementable in
general. The situation is di¤erent in the case of Nash equilibrium. When there are at least
three agents in the society, the union of Nash implementable SCCs is Nash implementable.
An important corollary of this result is that every weakly Nash implementable SCC admits a
largest (fully) Nash implementable subcorrespondence. On the other hand, the intersection
of two Nash implementable (and Pareto-comprehensive) social choice rules need not be Nash
implementable. However, we have found that the intersection of such social choice rules is
almost Nash implementable.
Two related open problems remain. First, it is not known at present if the intersection of
two Nash implementable SCCs is also Nash implementable when preferences are strict ( as
we found to be the case for dominant strategy implementation). Second, we do not know if
the intersection of two Nash implementable SCCs (that need not be Pareto-comprehensive)
is almost Nash implementable; in other words, we do not know if in Proposition 7 F PN (R)
can be replaced with F N (R):
Concerning the broader research path, we note that very little is known about the al-
gebraic closure properties of classes of SCCs that are implementable with respect to other
interesting equilibrium notions, including subgame perfect and Bayesian equilibrium.22
22
The case of undominated Nash equilibria is easily settled: Propositions 3 and 7 apply also to SCCs that
are implementable in undominated Nash equilibria. The proofs go through verbatim.
20
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21